## Geometry and Physics Seminar

### An Intrinsic Batyrev Construction via Symplectic Topology

Wednesday, January 23, 2019, 5:30pm
Ungar Room 402

I will describe an intrinsic version of Batyrev's mirror construction associated to a general maximally degenerate log Calabi-Yau pair (M,D) using an invariant known as symplectic cohomology. The symplectic cohomology ring of log Calabi-Yau varieties comes equipped with a flat degeneration to the Stanley-Reisner ring of the dual intersection complex of a compactifying divisor. The deformation from the central fiber can be alternatively described using a symplectic version of log Gromov-Witten invariants, which modulo a certain technical conjecture enables us to relate our construction to recent mirror constructions of Gross-Hacking-Keel and Gross-Siebert.

## Combinatorics Seminar

### Stanley-Reisner Rings of Symmetric Simplicial Complexes

Tuesday, January 22, 2019, 5:00pm
Ungar Room 402

Abstract: A classical theme in algebraic combinatorics is the study of face rings of finite simplicial complexes (named after Stanley and Reisner, two of the pioneers of this field). In this talk I will examine the case where the simplicial complexes at hand carry a group action and are allowed to be infinite.

I will present the foundations of this generalized theory with a special focus on simplicial complexes associated to (semi)matroids, where the associated rings enjoy especially nice algebraic properties. A main motivation for our work comes from the theory of arrangements in Abelian Lie groups (e.g., toric and elliptic arrangements), and in particular from the quest of understanding numerical properties of the coefficients of characteristic polynomials and h-polynomials of arithmetic matroids. I will describe our current results in this direction and, time permitting, I will outline some open questions that arise in this new framework. (Joint work with Alessio D'Alì.)

## Geometry and Physics Seminar

### Fano Manifolds Old and New

Tuesday, January 15, 2019, 6:15pm
Ungar Room 402

## Geometric Analysis Seminar

### Ellipticity of the Bartnik Boundary Conditions

Monday, December 10, 2018, 3:30pm
Ungar Room 506

Abstract: The Bartnik quasi-local mass is defined to measure the mass of a bounded manifold with boundary, where a collection of geometric boundary data – the so-called Bartnik boundary data – plays a key role. Bartnik proposed the open problem whether, on a given manifold with boundary, there exists a stationary vacuum metric so that the Bartnik boundary conditions are realized. In the effort to answer this question, it is important to prove the ellipticity of Bartnik boundary conditions for stationary vacuum metrics.

In this talk, I will start with an introduction to the Bartnik quasi-local mass and the moduli space of stationary vacuum metrics. Then I will explain the ellipticity result for the Bartnik boundary conditions and, as an application, derive a local result to the existence question.

## Combinatorics Seminar

### Combinatorial Neural Codes

Monday, December 3, 2018, 4:00pm
Ungar Room 506

Abstract: Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus, where neurons exhibit convex receptive fields. I will discuss research with Curto et al and Itskov et al, characterizing combinatorial properties of convex neural codes as well as a restrictive variant – neural codes defined by dissections of a convex set.

## Applied Math Seminar

### How Do Diffusion and Heterogeneities Affect Competition?

Friday, November 30, 2018, 12:20pm
Ungar Room 411

Abstract: It has been nearly 90 years since Alfred J. Lotka and Vito Volterra proposed the models that established the basis of the study of the interaction between species. During this time, significant improvements have been achieved in this field, with the addition of stochastic techniques and the introduction of the space.

In this talk, we will discuss the last point, analyzing fascinating results that arise when the diffusion of the species and the heterogeneity of the environment are incorporated into the basic Lotka-Volterra competition model for two species. Particularly, the Singular Perturbation Theorem and the Principle of Induced Instability will be established as a link between non-diffusive and diffusive models and, among other consequences, either uniqueness or multiplicity will be obtained depending on the configuration of the habitat.

## Geometry and Physics Seminar

### Asymptotics of Primes in Short Intervals on Curves over Finite Fields

Wednesday, November 28, 2018, 5:00pm
Ungar Room 402

Abstract: I will talk about recent joint work with E. Bank, as well as work of A. Entin, which establishes several function field analogues of conjectures concerning the distribution of primes inside "short intervals" – intervals whose widths grow at a certain rate as they move off to infinity.

## Combinatorics Seminar

### Positivity Conjectures for Jack Polynomials

Monday, November 12, 2018, 5:00pm
Ungar Room 402

Abstract: In the course of investigating a statistical problem involving estimators for a parameter matrix, Donald Richards and Siddhartha Sahi have recently formulated certain positivity conjectures involving Jack polynomials. In this talk, I will present a strengthened version of the Richards-Sahi conjectures, which depends on a pair of partitions, and sketch a proof in a number of cases. This strengthened conjectures suggests new combinatorial identities involving Jack analogues of Kostka numbers and hook-length formulas.

## Geometry and Physics Seminar

### SL(2,C)-representations of Homology 3-spheres

Wednesday, November 7, 2018, 5:00pm
Ungar Room 402

Abstract: We prove that the splicing of any two non-trivial knots in the 3-sphere admits an irreducible SU(2)-representation of its fundamental group. This uses instanton gauge theory, and in particular a non-vanishing result of Kronheimer-Mrowka and some new results that we establish for holonomy perturbations of the ASD equation. Using a result of Boileau, Rubinstein and Wang (which builds on the geometrization theorem of 3-manifolds), it follows that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL(2,C).

## Geometric Analysis Seminar

### On Spectral Properties for Blow-up Solutions and Soliton Stability in Dispersive Equations

Monday, November 5, 2018, 4:00pm
Ungar Room 506

Abstract: When studying the blow-up dynamics and soliton stabilty in the NLS-type and KdV-type equations, we encounter spectral property questions, arising from linearization, or from the virial-type arguments. We will discuss examples of the L^2-critical blow-up dynamics, spectral properties and its consequences in the NLS and the Hartree equations, and then will describe the situation in Zakharov-Kuznetsov equation, which is a higher dimensional generalization of the KdV equation.

## Geometry and Physics Seminar

### Stable Reduction of Foliated Surfaces

Friday, November 2, 2018, 4:00pm
Ungar Room 402

Abstract: In 1977 Bogomolov proved that on surfaces of general type with c_1^2>c_2, curves of a given geometric genus form a bounded family. The role played by foliations in his proof was further investigated by McQuillan, who in 1998 proved the Green-Griffiths conjecture for the same class of surfaces. In this talk I will review some basic properties of foliations on algebraic surfaces, with a focus on birational geometry as initiated by Brunella, McQuillan and others. I will then discuss the problem of their variation in families, and present the main ideas behind the proof of the stable reduction theorem in this context.

## Combinatorics Seminar

### Reconstructing Spheres and Polytopes

Monday, October 29, 2018, 5:00pm
Ungar Room 402

Abstract: We review the historical progress of the problem of determining all faces of a sphere from partial information, starting in 1916 through the modern day. We culminate in a counterexample which disproves the strongest possible version of a conjecture made by Perles in 1960. This strengthened conjecture would imply that simplicial 3-spheres are reconstructible from their facet-ridge graph. While this conjecture fails, in its failure it leaves behind a new technique which may yet solve the problem of reconstructiblity of simplicial spheres.

## Geometric Analysis Seminar

### Extremal Metrics for the Min-max Width

Monday, October 29, 2018, 4:00pm
Ungar Room 506

Abstract: We present our study on the min-max width of Riemannian three-dimensional spheres. This is a natural geometric invariant which is closely related with critical values of the area functional acting on closed surfaces, and can be interpreted as the first eigenvalue of a non-linear spectrum of a Riemannian metric, as suggested by Gromov. We will focus first on optimal bounds for the above invariant involving their volumes in a fixed conformal classes. If time permits we will discuss some general properties of extremal metrics for the min-max width. This is all part of a joint work with Lucas Ambrozio.

## Applied Math Seminar

### Modeling the Trade-off between Transmissibility and Contact in Infectious Disease Dynamics

Friday, October 26, 2018, 12:20pm
Ungar Room 411

Abstract: Symptom severity affects disease transmission both by impacting contact rates, as well as by influencing the probability of transmission given contact. This involves a trade-off between these two factors, as increased symptom severity will tend to decrease contact rates, but increase the probability of transmission given contact (as pathogen shedding rates increase with symptom severity). This talk explores this trade-off between contact and transmission given contact, using a simple compartmental susceptible- infected-recovered type model. Under mild assumptions on how contact and transmission probability vary with symptom severity, we give sufficient, biologically intuitive criteria for when the basic reproduction number varies non-monotonically with symptom severity. Multiple critical points are possible. We give a complete characterization of the region in parameter space where multiple critical points are located in the special case where contact rate decreases exponentially with symptom severity. We consider a multi-strain version of the model with complete cross-immunity and no super-infection. In this model, we prove that the strain with highest basic reproduction number drives the other strains to extinction. This has both evolutionary and epidemiological implications, including the possibility of an intervention paradoxically resulting in increased infection prevalence. This is joint work with Kristen A. Deger and Joseph H. Tien.

## Geometric Analysis Seminar

### Mean Curvature Flow in a Ricci Flow Background

Monday, October 22, 2018, 4:00pm
Ungar Room 506

Abstract: Geometric flows have attracted much attention in the past years and have proven to be a powerful analytic tool leading to many groundbreaking results such as Hamilton's and Perelman's proof of the Poincare conjecture, Huisken's and Ilmanen's proof of the Penrose conjecture and Brendle's and Schoen's proof of the differentiable sphere theorem. We begin with giving a general introduction to geometric flows via the one dimensional heat equation before proceeding with the discussion of mean curvature flow in a Ricci flow backgroud. Using a variation of geodesics we derive a distance comparison theorem which was first observed by Huisken and can be considered a parabolic version of Frankel's theorem. In fact, we will obtain Frankel's theorem as a corollary of this result. Next, we discuss a long time existence result originating from our thesis, stating that suitable convex surfaces converge to a sphere after rescaling. This work is based on Huisken's approach to mean curvature flow in a fixed Riemannian manifold and employs Stampacchia iteration in order to obtain an a priori estimate pinching the eigenvalues of the second fundamental form. Finally, we present a novel rescaling technique which requires a slightly stronger gradient estimate but greatly simplifies the convergence arguments.

## Geometry and Physics Seminar

### Genus Minimization of Homology Classes of Knots in Lens Spaces

Wednesday, October 17, 2018, 5:00pm
Ungar Room 402

Abstract: The genus of torsion homology class in an orientable 3-manifold is the minimal (rational) Seifert genus among knots representing that class. Knots in lens spaces with integral surgeries to S^3 are known to be minimizers of this genus. On the other hand, a classification of knots in S^3 with Dehn surgeries to lens spaces would be complete (thereby resolving the Berge Conjecture) if we knew that such minimizers were unique. We'll discuss the current state of genus minimizers and a strategy for addressing this Dehn surgery classification problem.

## Geometry and Physics Seminar

### Singular Chains and the Fundamental Group

Tuesday, October 16, 2018, 5:00pm
Ungar Room 402

Abstract: I will explain how the singular chains on a connected space, considered as a coalgebra with extra algebraic structure, encodes the data of the fundamental group of the space. Then I will then introduce an algebraic notion of weak equivalence between differential graded coalgebras which is stronger than quasi-isomorphism to show the following version of a classical Whitehead theorem: a map between connected spaces is a weak homotopy equivalence if and only if the induced map at the level of singular chains is a weak equivalence (in the strong sense) of dg coalgebras.

## Combinatorics Seminar

### Convex Union Representability of Simplicial Complexes

Monday, October 15, 2018, 5:00pm
Ungar Room 402

Abstract: Given a collection of convex open sets, one can form an associated simplicial complex that records their intersection patterns. This complex retains important topological information about the sets; for example, Borsuk's Nerve Lemma states that it is homotopy equivalent to the union of the sets. I will discuss what happens when the union of the sets is convex. In this case, the associated simplicial complex has a number of rich combinatorial properties. I will also describe an application to the theory of convex codes.

## Geometry and Physics Seminar

### Categorical Curve Complex

Wednesday, October 10, 2018, 5:00pm
Ungar Room 402

Abstract: This is an attempt to present a category theory with a human face. In the talk we will recall classical topological and geometric constructions used in the calculations of mapping class groups and lamination conjectures. We will look at them from new prospective. Applications will be discussed.

## Combinatorics Seminar

### On the Intersection Lattice of the Homogenized Linial Arrangement

Monday, October 8, 2018, 5:00pm
Ungar Room 402

Abstract: In 2017, Hetyei introduced the homogenized Linial arrangement and showed that the number of regions is equal to a median Genocchi number. In this talk, I will discuss joint work with Wachs, in which we refine Hetyie's result by computing the Möbius function of the lattice of intersections of the arrangement. We show that the Möbius invariant of the intersection lattice is a Genocchi number. Our techniques also yield a type B analog of Hetyei's result and more generally a Dowling arrangement analog involving a new q-analog of the median Genocchi numbers.

## Geometric Analysis Seminar

### Formal Power Series Solutions of the Bach Equation

Monday, October 8, 2018, 4:00pm
Ungar Room 506

Abstract: Conformal gravity is an alternative to Einstein gravity in 4 dimensions, obtained by replacing the Einstein equation by the Bach equation, which has many more solutions. Maldacena has proposed that the theories are equivalent, provided one imposes certain boundary and physical conditions to remove the additional solutions of the Bach equation. We test this idea. Following the method laid out by Fefferman and Graham for the Einstein equation, we expand asymptotically hyperbolic solutions of the Bach equation in power series about conformal infinity, so as to identify the free data and find those data that yield Einstein metrics. There are infinitely many free data, reflecting the conformal invariance of the 4-dimensional Bach equation, but even if we choose to break conformal invariance by imposing a constant-scalar-curvature condition, the so-called mass aspect tensor remains freely specifiable.

In dimensions greater than 4, there are many different generalizations of the Bach tensor, most of which are not well-suited to the Fefferman-Graham method. We choose a well-suited definition and find that the free data separate into two pairs of data, reflecting the separation of data for the Einstein equation into "Dirichlet" and "Neumann" data.

This talk is based on joint work with Aghil Alaee.

## Applied Math Seminar

### Number and Stability of Relaxation Oscillations for Predator-Prey Systems with Small Death Rates

Friday, October 5, 2018, 12:20pm
Ungar Room 411

Abstract: Predator-prey models that possess limit cycles can be used to explain oscillatory phenomena in real-world data, and have been studied extensively in the literature. In this talk, we will consider predator-prey systems with small predator death rate as fast-slow systems, and derive new characteristic functions that determine the location and the stability of relaxation oscillations. This criterion determines the number and the global stability of limit cycles for some planar predator-prey systems. This criterion can also been extended (joint work with Gail S. K. Wolkowicz) to be applied on some three-dimensional systems, including chemostat predator-prey systems and a class of epidemic models.

## Geometry and Physics Seminar

### Decomposing Manifolds into Cartesian Products

Wednesday, October 3, 2018, 5:00pm
Ungar Room 402

Abstract: We study the decomposability of a Cartesian product of two non-decomposable manifolds into products of lower dimensional manifolds. For 3-manifolds we obtain an analog of a result due to Borsuk for surfaces, and in higher dimensions we show that similar analogs do not exist. This is a joint work with Reinhard Schultz.

## Combinatorics Seminar

### Random Preprocessing in Computational Topology

Monday, October 1, 2018, 5:00pm
Ungar Room 402

Abstract: Computational topology aims at understanding the 'shape' (=homotopy type, or sometimes just homology) of big data. In 2014 with Frank Lutz we introduced Random Discrete Morse theory as an experimental measure for the complicatedness of a triangulation. This measure depends both on the homotopy type of the space, and on how nicely the space is triangulated. Our approach was elementary, but sometimes successful even for huge inputs. I'll discuss some variants, drawbacks, and possible new ideas that were figured out in the meantime. At the same time, these approaches reveal that the existing libraries of examples in computational topology are all 'too easy' for testing algorithms. So let's build a new one!

## Geometric Analysis Seminar

### Some Recent Results Pertaining to Bartnik's Quasi-local Mass

Monday, October 1, 2018, 4:00pm
Ungar Room 506

Abstract: Bartnik's quasi-local mass is often said to be one of the most likely quantities to give a physical measure of the gravitational field mass/energy in general relativity, however it is effectively impossible to compute it general. The Bartnik mass of a domain (or its boundary data) is given by the infimum of the ADM mass over an appropriate space of asymptotically flat manifolds with nonnegative scalar curvature.

In this talk, we give a brief introduction to the Bartnik mass and give some related results that follow from a gluing technique. In particular, we give conditions that ensure two subtly different definitions of the Bartnik mass yield the same value and prove that the mass is continuous with respect to the boundary data.

## Geometry/Topology Seminar

### The Strong Slope Conjecture for Montesinos Knots

Friday, September 28, 2018, 3:00pm
Ungar Room 402

Abstract: The Strong Slope Conjecture by Garoufalidis and Kalfagianni-Tran relates the topology of essential surfaces to the colored Jones polynomial, which connects fundamental objects of 3-manifold topology to quantum topology, much like the Volume Conjecture. Much of the progress on the conjecture has been made by computing the topology of essential surfaces separately from the degree of the polynomial, and comparing the end results. This provides few insights to the reasons behind the conjecture.

To address this problem, we (joint with S. Garoufalidis and R. van der Veen) use Montesinos knots as a template and verify the conjecture for most knots in the family by establishing a close analogy between terms in the state sum defining the colored Jones polynomial of the knot, and properly embedded surfaces in the knot complement. In this talk, I will discuss these results and how they present a model for understanding the Strong Slope Conjecture for all knots, as well as the remaining difficulty.

## Combinatorics Seminar

### Round Polytopes

Monday, September 24, 2018, 5:00pm
Ungar Room 402

Abstract: It is known that the space of polytopes is dense in the space of closed bounded convex sets endowed with various different metrics. Given that polytopes come equipped with combinatorial structure it is reasonable to ask about the combinatorial structure of a polytope that is a good approximation to a given convex body K. We will discuss theorems about simplicial polytopes approximating convex bodies whose boundary is smooth (e.g an Euclidean ball of radius one).

## Geometric Analysis Seminar

### Isometric Embeddings into Spheres:A Conformal Pascal Triangle (Cone?)

Monday, September 24, 2018, 4:00pm
Ungar Room 506

Abstract: Special properties of a Riemannian manifold (M,g) isometrically embedded into abackground manifold (N,h are reflected in special properties of extrinsic quantities, second fundamental form \alpha, mean curvature vector H. Same if (M,J,g) is almost Hermitian, but we now add the extrinsic quantity \alpha( . , J. ) into the consideration.

Any (M,J,g) admits a solution to both the usual and almost Hermitian Yamabe problems. Nash's theorem produces isometric embeddings of M with these metrics into standard spheres. The twisting by J affects the conformal level of the embedding.

We sketch a conformal Pascal triangle of (almost Hermitian) manifolds isometrically embedded into standard spheres. The description is fairly detailed if the manifold has nonnegative Ricci curvature and the level of the critical embedding is small (critical relative to the functionals associated to the extrinsic quantities).

## Applied Math Seminar

### Resident-invader Dynamics in Infinite Dimensional Systems

Friday, September 21, 2018, 12:20pm
Ungar Room 411

Abstract: Motivated by evolutionary biology, we study general infinite-dimensional dynamical systems involving two species - the resident and the invader. Sufficient conditions for competitive exclusion phenomena are given when the two species play similar, but distinct, strategies. Those conditions are based on invasibility criteria, and allow, for example, the identification of evolutionarily stable strategies in the framework of adaptive dynamics. The results extend ideas developed by S. Geritz et al. for a class of ordinary differential equations.

## Applied Math Seminar

### Recent Work on the Ecology and Evolution of Dispersal

Friday, September 14, 2018, 12:20pm
Ungar Room 411

Abstract: This will be an informal talk describing some topics related to dispersal that I have been studying recently. Specifically, I will present some background, modeling, and and analysis of models related to optimal dispersal of organisms in time periodic environments, the use of nonlocal information in dispersal, and switching between multiple movement modes. The models are all based on reaction-diffusion-advection equations or systems.

## Geometry and Physics Seminar

### Integral Affine Structures in Non-Archimedian Geometry

Wednesday, September 12, 2018, 5:00pm
Ungar Room 402

Abstract: An integral affine structure on a manifold X consists of a flat connection and the choice of a full rank lattice inside each tangent space compatible with that connection. I will explain how integral affine structures arise in non-archimedian geometry, following the work of Kontsevich and Soibelman, with explicit examples.

## Geometry and Physics Seminar

### Jet Differentials on Complete Intersections and Applications

Thursday, June 7, 2018, 2:30pm
Ungar Room 402

Abstract: The goal of this talk is to describe a strategy allowing us to construct symmetric differential forms, and more generally jet differentials, on complete intersection varieties. We will also explain how one can deduce from this that general hypersurfaces of large degrees are hyperbolic, a result conjectured by Kobayashi in the 70's.

## Combinatorics Seminar

### The Homogenized Linial Arrangement

Monday, April 23, 2018, 5:00pm
Ungar Room 402

Abstract: The homogenized Linial arrangement was introduced by Hetyei in 2017 to prove enumerative results in graph theory. In this talk we present preliminary results of a deeper study of the combinatorics of this hyperplane arrangement.

## Geometric Analysis Seminar

### On the Schoen-Yau Positive Mass Theorem

Monday, April 23, 2018, 4:00pm
Ungar Room 506

Abstract: The Riemannian Positive Mass Theorem states that the mass of an asymptotically flat manifold of nonnegative scalar curvature is nonnegative and zero only when the manifold is actually Euclidean. The theorem was proved by Schoen and Yau initially for dimension less than 8 in 1979 and recently for all dimensions. In this talk, we propose a new approach and, in particular, express the mass in terms of other invariants including the scalar curvature.

## Hodge Theory Seminar

### The Dual Complex of a Semi-log Canonical Surface

Friday, April 20, 2018, 4:00pm
Ungar Room 506

Abstract: Semi-log canonical surfaces arise as the limits of canonically polarized surfaces. In this sense they are the natural generalization to surfaces of nodal curves. The goal of this talk is to explore how we can associate a 2-dimensional cell complex to a semi-log canonical surface, analogous to the dual graph of a nodal curve.

## Geometry and Physics Seminar

### Satake-Baily-Borel Completions of Moduli Spaces

Wednesday, April 18, 2018, 5:00pm
Ungar Room 402

Abstract: In the classical cases of curves, abelian varieties, K3 surfaces, etc. the SBB compactification of quotients of Hermitian symmetric domains by arithmetic groups gives a minimal way of completing moduli moduli spaces. In the non-classical case when one doesn't have an Hermitian symmetric and when the global monodromy groups may not be arithmetic, almost nothing is known about the global structure of the boundary of KSBA moduli spaces. We will give an informal account of the general SBB completion of moduli spaces explaining what is "behind the scenes" in the construction and in its applications to moduli.

## Combinatorics Seminar

### A Connection Between Tilings and Matroids on the Lattice Points of a Regular Simplex

Monday, April 16, 2018, 5:00pm
Ungar Room 402

Abstract: The set of lattice points T(n,d) inside the regular simplex obtained by intersecting the nonnegative cone of R^d with the affine hyperplane x_1 + ... + x_d = n-1 is the ground set of a matroid M(n,d) whose independent sets are precisely those subsets S of T(n,d) satisfying that the intersection of S and T has at most k elements for each parallel translate T of the regular simplex T(k,d). We will present some matroidal properties of M(n,3) in connection to certain tilings of holey triangular regions associated to the subsets of T(n,3). In particular, we will provide characterizations for the independent sets and circuits of M(n,3) related to certain tilings of their holey triangular regions, extending a characterization of the bases of M(n,3) already given by Ardila and Billey. If time permits, we will also exhibit connections between tilings and the flats and connectivity of the matroids M(n,3).

## Geometric Analysis Seminar

### Hooke, Euler, Lagrange

Monday, April 16, 2018, 4:00pm
Ungar Room 506

Abstract: Mathematical theories describing the motion of an ideal fluid and the deformation of an elastic solid have many challenging problems. I will discuss some of them and explain recent progress Lars Andersson and I have made.

## Combinatorics Seminar

### Parity Arguments in Combinatorics and Beyond

Monday, April 9, 2018, 5:00pm
Ungar Room 402

Abstract: We survey four cute ways to apply the combinatorial concept of parity to other fields. Namely:

• in algebra, the parity distinction for permutations (dating back at least to Cauchy, 1815);
• in topology, the combinatorial proof of Brouwer's fixed point theorem (Sperner, 1928);
• in geometry, the neighborlyness of cyclic polytopes (Gale, 1963);
• and in number theory, the 'one-sentence proof' of the sum-of-squares theorem (Zagier, 1990).

This talk is intended as didactical, rather than research-oriented; it does not assume expertise in any of the four fields above.

## Geometric Analysis Seminar

### Gauge, Peeling and Linearized Gravity on the Kerr Exterior Spacetime

Monday, April 9, 2018, 4:00pm
Ungar Room 506

Abstract: The Teukolsky Master Equation governs the dynamics of linearized gravity on the Kerr rotating black hole spacetime. In this talk I will discuss some aspects of scattering and peeling for the Teukolsky equation and the issue of gauge choice in the problem of linearized stability on the Kerr background. This is based on joint work with Thomas Bäckdahl, Pieter Blue, and Siyuan Ma.

## Geometry and Physics Seminar

### Mirror Symmetry and Filtrations

Wednesday, April 4, 2018, 5:00pm
Ungar Room 402

## Geometric Analysis Seminar

### Topological Defects in Anti de Sitter Space

Monday, April 2, 2018, 4:00pm
Ungar Room 506

Abstract: The solutions to the equations of motion for a topological defect in Minkowski space have been known for many years. In particular, the work of Bogomolny, Prasad, and Sommerfield allows a solution for these equations in the limit that mass of the scalar field vanishes. When the embedding space is AdS, another limit exists, which provides additional analytic solutions to appropriately-modified equations of motion.

## Geometry and Physics Seminar

### P=W and Algebraic Cycles

Wednesday, March 21, 2018, 5:00pm
Ungar Room 402

Abstract: This will be a provocative report of my scintilla of progress of learning from Carlos and Phillip.

## Combinatorics Seminar

### Threshold Hypergraphs Revisited

Monday, March 19, 2018, 5:00pm
Ungar Room 402

Abstract: Threshold graphs were introduced by Chvatal and Hammer(1974) as tools in optimization. They coincide with the class of shifted graphs and can be described and studied in three different ways: purely combinatorial, slicing the second hypersimplex or slicing a cube. A question of Golumbic(1978) answered in the negative by Reiterman, Rodl, Sinajova and Tuma(1985), asks for higher dimensional analogues. We will give a geometric explanation for the negative answer to such question and propose a corrected version of Golumbic's question. We will then highlight the relevance of this question in the theory matroid polytopes.

## Geometric Analysis Seminar

### Rigorous Justification of Modulation Approximations to the Full Water Wave Problem

Monday, March 19, 2018, 4:00pm
Ungar Room 506

Abstract: We consider solutions to the infinite depth water wave problem in 2D and 3D which are to leading order wave packets with small $O(\epsilon)$ amplitude and slow spatial decay that are balanced. In the case of zero surface tension, multiscale calculations formally suggest that such solutions have modulations that evolve on $O(\epsilon^{-2})$ time scales according to a version of a cubic nonlinear Schrodinger (NLS) equation. Justifying this rigorously is a real problem, since standard existence results do not yield solutions to the water wave problem that exist for long enough to see the NLS dynamics. Nonetheless, given initial data suitably close to such a wave packet in $L^2$ Sobolev space, we show that there exists a unique solution to the water wave problem which remains within $o(\epsilon)$ to the formal approximation on the natural NLS time scales. The key ingredient in the proof is a formulation of the evolution equations for the water wave problem developed by Sijue Wu with either no quadratic nonlinearities (in 2D) or mild quadratic nonlinearities that can be eliminated using the method of normal forms (in 3D). If time permits, we will discuss recent work towards a justification result in the presence of the effects of both gravity and surface tension in 2D.

## Combinatorics Seminar

### Sign Matrix Polytopes

Monday, March 5, 2018, 5:00pm
Ungar Room 402

Abstract: Motivated by the study of polytopes formed as the convex hull of permutation matrices and alternating sign matrices, we define several new families of polytopes as convex hulls of sign matrices, which are certain {0,1,-1}-matrices in bijection with semistandard Young tableaux. We investigate various properties of these polytopes, including their inequality descriptions, vertices, facets, and face lattices, as well as connections to alternating sign matrix polytopes and transportation polytopes.

## Geometry and Physics Seminar

### From Homotopical Mathematics to Emergent Geometry

Wednesday, February 28, 2018, 5:00pm
Ungar Room 402

Abstract: At the root of the fundamental mathematical notion of symmetry is the idea that it is useful to keep track of the multitude of ways in which two objects can be identified, rather than to simply ask if they are the same. Taking this idea to its logical conclusion leads to a mathematical universe where shapes (homotopy types) are the fundamental building blocks of mathematical structures instead of sets. Derived geometry is geometry in this homotopy-theoretic context. It provides an intuitive language for quantum field theory, and a powerful framework in which "classical geometry" can be seen to emerge from the structure of quantum field theory.

After introducing this paradigm, I will touch upon joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich, in which we attempt to formalize and understand the mathematical structures underlying the physical notion of stability for D-branes in string theory using the language of derived noncommutative geometry. Our work builds upon Bridgeland's notion of stability conditions on triangulated categories, and is inspired by ideas from symplectic geometry, non-Archimedean geometry, dynamical systems, geometric invariant theory, and the Donaldson-Uhlenbeck-Yau correspondence.

## Geometry and Physics Seminar

### BPS Cohomology and Character Varieties

Tuesday, February 27, 2018, 5:00pm
Ungar Room 506

Abstract: In this talk I will review a general definition of BPS invariants counting stable sheaves on a Calabi-Yau 3-fold. It turns out that the definition is really a theorem, which in turn can be categorified to give a notion of the cohomology of the space of BPS states. Although this cohomology can be explicitly defined, the explicit definition is not really the cohomology of a space at all, but the vanishing cycle cohomology of the intersection complex of a coarse moduli space.

Despite being more closely analogous to the category of sheaves on a K3 surface than a CY3 variety, the category of representations of the fundamental group of a Riemann surface fits naturally into this theory, and in contrast with the general case, the BPS cohomology has a (conjecturally) much more down-to-earth description: it is the cohomology of the twisted character variety, a central and mysterious object in the study of nonabelian Hodge theory.

## Geometric Analysis Seminar

### End-periodic Index Theory and Metrics of Positive Scalar Curvature

Monday, February 26, 2018, 4:00pm
Ungar Room 506

Abstract: We study metrics of positive scalar curvature on certain closed manifolds of even dimension. We provide a new obstruction to the existence of such metrics, and give examples of manifolds with infinitely many path components in the moduli space of metrics of positive scalar curvature. The methods include the index theorem for end-periodic Dirac operators (due to Mrowka, Ruberman, and the speaker) and some Seiberg-Witten theory. This is a joint project with Jianfeng Lin, Tom Mrowka, and Danny Ruberman.

## Geometry and Physics Seminar

### Simplicial Sets and Partially Ordered Sets

Wednesday, February 21, 2018, 5:00pm
Ungar Room 402

Abstract: I am going to discuss how to describe families of groupoids over simplicial sets in terms of families over partially ordered sets, and why this seems a good thing to do.

## Combinatorics Seminar

### On Enumerators of Smirnov Words by Descents and by Cyclic Descents

Monday, February 19, 2018, 5:00pm
Ungar Room 402

Abstract: Smirnov words are words over the alphabet of positive integers with no adjacent equal letters. The enumerator of these words by descent number is a symmetric function which arose in work with Shareshian on q-Eulerian polynomials, on Rees products of posets, and on chromatic quasisymmetric functions. In this talk I will discuss this work withShareshian and recent work with Ellzey on the enumerators of Smirnov words by cyclic descents.

## Geometric Analysis Seminar

### Some Recent Developments on Manifolds with Nonnegative Scalar Curvature

Monday, February 19, 2018, 4:00pm
Ungar Room 506

Abstract: Scalar curvature is a basic scalar quantity of curvature. It is tied to local energy density in relativity. For noncompact manifolds, the Riemannian positive mass theorem and Riemannian Penrose inequality are fundamental results formulated on asymptotically flat manifolds that model isolated systems. For compact manifolds with boundary, understanding the impact of scalar curvature on the manifold boundary is tied to the quasi-local mass problem in relativity. In this talk, I will discuss recent developments on compact Riemannian manifolds with nonnegative scalar curvature, with boundary. The talk will be based on my join work with Christos Mantoulidis, and with Siyuan Lu, respectively.

## Hodge Theory Seminar

### Brown Representability for Unpointed Spaces

Friday, February 16, 2018, 4:00pm
Ungar Room 411

## Geometry and Physics Seminar

### Toric Schobers and D-modules

Wednesday, February 14, 2018, 5:00pm
Ungar Room 402

Abstract: Many classical mirror symmetry results can be recast using the more recent language of perverse sheaves of categories and schobers. In this context, I will explain a Riemann-Hilbert type conjectural connection with the D-modules naturally appearing in mirror symmetry.

## Geometric Analysis Seminar

### Milne-like Spacetimes

Monday, February 12, 2018, 4:00pm
Ungar Room 506

Abstract: The study of the (in)-extendibility of spacetimes is motivated by the strong cosmic censorship conjecture in general relativity. Recently there has been an interest in extendibility results with regularity less than C^2. This began with Sbierski's work where he demonstrated that the maximally analytic extension of the Schwarzschild spacetime is C^0-inextendible. In his paper Sbierski posed the question of whether or not the FLRW cosmological models admit C^0 extensions. In this talk we present a class of cosmological models, dubbed Milne-like spacetimes, which admit C^0 extensions through the big bang. We discuss their properties and how they fit in the modern view of cosmology.

## Geometry and Physics Seminar

### Yang-Mills Theory and Definite Intersection Forms Bounding Homology 3-spheres

Friday, February 9, 2018, 4:00pm
Ungar Room 402

Abstract: Using Yang-Mills instanton Floer theory, we find new constraints on the possible definite intersection forms of smooth 4-manifolds that bound integer homology 3-spheres. We will give examples of 3-manifolds such that the set of all bounding negative definite lattices consists of essentially two distinct non-standard lattices. The methods used follow the work of Froyshov.

## Geometry and Physics Seminar

### Factorization Homology

Wednesday, February 7, 2018, 5:00pm
Ungar Room 402

Abstract: The Ran space Ran(X) is the space of finite subsets of X, topologized so that points can collide. Ran spaces have been studied in diverse works from Borsuk-Ulam and Bott, to Beilinson-Drinfeld, Gaitsgory-Lurie and others. The alpha form of factorization homology takes as input a manifold or variety X together with a suitable algebraic coefficient system A, and it outputs the sheaf homology of Ran(X) with coefficients defined by A. Factorization homology simultaneously generalizes singular homology, Hochschild homology, and conformal blocks or observables in conformal field theory. I'll discuss applications of this alpha form of factorization homology in the study of mapping spaces in algebraic topology, bundles on algebraic curves, and perturbative quantum field theory. I'll also describe a beta form of factorization homology, where one replaces Ran(X) with a moduli space of stratifications of X, designed to overcome certain strict limitations of the alpha form. One such application is to proving the Cobordism Hypothesis, after Baez-Dolan, Costello, Hopkins-Lurie, and Lurie. This is joint work with David Ayala.

## Geometry and Physics Seminar

### On the Notion of an Enhanced Category

Monday, February 5, 2018, 6:00pm
Ungar Room 411

Abstract: By now, it is a well-established general principle that when you localize a category with respect to a class of morphisms, you get a category "enriched in homotopy types". Several precise definition of this notion exist in the literature (complete Segal spaces of Rezk, infinity-categories of Lurie), and they are all equivalent in some sense, but they all depend on some auxiliary choices, and the precise sense in which they are all equivalent also depends on choices. In these lectures, I am going to sketch a somewhat more invariant approach to the subject that seems to be much more model-independent. This is based on Grothendieck's idea of a derivator, also well-established in the literature; however, and this seems to be new, there is also a theorem that states that when you pass to the derivator, you lose no information, so that the approach is equivalent to the earlier ones. In the first lecture, I am going to give a general overview, and then discuss a version of Brown representability theorem for unpointed topological spaces.

## Combinatorics Seminar

### Some Contractible 2-complexes Do Not Embed in R^4

Monday, February 5, 2018, 5:00pm
Ungar Room 402

Abstract: We discuss the problem of whether all contractible d-complexes can be drawn in R^{2d}. This is clear only for d=1 (in which case the answer is: "yes, all trees are planar graphs".) We also look at combinatorial strengthenings of contractibility, like collapsibility and non-evasiveness. This is work in progress with Karim Adiprasito.

## Geometric Analysis Seminar

### Causality and Geodesics in Low Regularity

Monday, February 5, 2018, 4:00pm
Ungar Room 506

Abstract: As a concrete example by Chruściel and Grant demonstrates, many classical results from causality theory fail for metrics that are not at least Lipschitz continuous; for example lightcones need no longer be hypersurfaces and one may have maximizing causal curves that are neither timelike nor null. In this talk I will try to give a bit of an overview over what is currently known for which regularity classes of metrics and some of the remaining open problems. We are also going to take a look at the geodesic equation for Lipschitz metrics: To deal with the right-hand-side of the geodesic equation being only locally bounded one can make use of a solution concept of Filippov allowing for a general existence result.

## Combinatorics Seminar

### Intersection Patterns of Sets

Monday, January 29, 2018, 5:00pm
Ungar Room 402

Abstract: We present Kneser's conjecture and its reformulation into a graph coloring problem. We then introduce the generalized Erdos-Kneser conjecture partially proven by Sarkaria in 1990 and its associated hypergraph coloring problem. We prove this conjecture for r-uniform hypergraphs with the size of intersection s, not too close to r. We discuss what's still open related to this conjecture and possible methods for further proofs.

## Hodge Theory Seminar

### Integral Virtual Fundamental Chains

Friday, January 26, 2018, 4:00pm
Ungar Room 402

Abstract: To define invariants using moduli spaces of holomorphic curves in general symplectic manifolds, a virtual technique is typically required, such as Kuranishi theory or polyfolds. All the methods in full generality use perturbation or duality, involve locally breaking the symmetry then taking the weighted averages, and thus yield virtual fundamental chains over rationals. We carry out a program of Fukaya-Ono outlined in their 2001 paper. The key notions are a group-normal structure that one can always construct for a good coordinate system, and a group-normal complex structure that is always present on the moduli space of holomorphic curves, and their combined group-normal complex good coordinate system. Using this, one can perform a single-valued group-normally polynomial perturbation to yield integral virtual fundamental chains/pseudocycles for Floer/GW moduli spaces on general symplectic manifolds. This method is expected to be applicable to all moduli spaces based on holomorphic curves. This is a joint work with Guangbo Xu.

## Geometry and Physics Seminar

### Smooth Fano Weighted Complete Intersections and Landau-Ginzburg Models

Wednesday, January 24, 2018, 5:00pm
Ungar Room 402

Abstract: Smooth Fano varieties are classified in dimensions up to three, while in higher dimensions only some examples are known. The typical examples in the most interesting Picard rank one case are smooth complete intersections in weighted projective spaces and Grassmannians. It turns out that smoothness is a strong restriction for weighted complete intersections that lets to get bounds on their numerical invariants and, thus, lets classify all of them for any given dimension. We observe what is known in this direction, as well as we discuss their numerical invariants like Hodge numbers. We also outline, by an analogy with three-dimensional case, the way to construct their Landau-Ginzburg models and to prove Hodge numbers mirror symmetry in the spirit of Katzarkov-Kontsevich-Pantev conjecture.

## Geometry and Physics Seminar

### On the (Non-)L-equivalence of Algebraic Varieties

Tuesday, January 23, 2018, 5:00pm
Ungar Room 506

Abstract: Recall that two complex algebraic varieties are called L-equivalent if they have the same classes in the localization of the Grothendieck ring of varieties with respect to L (the class of affine line).

In this talk I will disprove (the original versions of) two conjectures on L-equivalence, due to Huybrechts and to Kuznetsov-Shinder. The first one states that isogenous K3 surfaces are L-equivalent, and the second one states that derived equivalence of smooth projective varieties implies L-equivalence (the second conjecture fails already for abelian varieties). Moreover, it will be shown that both for K3 surfaces and for abelian varieties each L-equivalence class contains only finitely many isomorphism classes.

Our results on non-L-equivalence are deduced (via integral Hodge realization) from the very general (and quite surprisingly, new) results on the Grothendieck group of an additive category whose morphisms are finitely generated abelian groups.

## Combinatorics Seminar

### The Sperner Property

Monday, January 22, 2018, 5:00pm
Ungar Room 402

Abstract: A finite graded partially ordered set $P$ has the \emph{Sperner property} if the largest level of $P$ is an antichain of maximum size. Most of the talk will be a survey of the Sperner property, beginning with Sperner's result that the boolean algebra of all subsets of a finite set has the Sperner property. (Of course Sperner did not use this terminology.) We will focus our attention on the use of linear algebra. We conclude with a discussion of the weak Bruhat order of the symmetric group. It is an open problem whether this poset has the Sperner property. We will discuss a determinantal conjecture which would imply the Sperner property.

## Geometry and Physics Seminar

### Structures on the Boundary of the Character Variety of a Compact Riemann Surface

Wednesday, January 17, 2018, 5:00pm
Ungar Room 402

Abstract: Consider a compact Riemann surface or an orbicurve, and let M be the moduli space of representations of the fundamental group into a fixed complex algebraic group such as SL_r(C). In this talk, we'll describe a conjecture on the structure of the boundary of M. This conjecture, a relative of the "P=W conjecture", relates natural spherical structures that are visible, one on the "Dolbeault" side of Hitchin's description of M and the other on the "Betti" side of the most naive expression of M as an affine variety. Here we'll give an overview of what is known in this direction, and some of the techniques that can be used. Understanding in a detailed way the main examples, and theorems, will be the subject of my course.

## Geometry and Physics Seminar

### Categorical Brill Noether Invariants

Tuesday, January 16, 2018, 5:00pm
Ungar Room 402

## Combinatorics Seminar

### Hopf Algebras in Combinatorics(continuation)

Monday, December 18, 2017, 5:00pm
Ungar Room 402

## Combinatorics Seminar

### Hopf Algebras in Combinatorics(continuation)

Monday, December 11, 2017, 5:00pm
Ungar Room 402

## Combinatorics Seminar

### Hopf Algebras in Combinatorics(continuation)

Monday, December 4, 2017, 5:00pm
Ungar Room 402

## Geometry and Physics Seminar

### Classification of First Order Sesquilinear Forms

Wednesday, November 29, 2017, 5:00pm
Ungar Room 402

Abstract: We work with n complex-valued scalar fields over an m-dimensional real manifold M without boundary. Our object of study is a first order Hermitian sesquilinear form, i.e. an integral over the manifold whose integrand is a linear combination of terms "product of gradient of scalar field and scalar field" and "product of two scalar fields".

We call two sesquilinear forms equivalent if one is obtained from the other by some x-dependent GL(n,C) transformation, i.e. by a change of basis in the vector space of n-tuples of complex-valued scalar fields. Our aim is to provide a description of equivalence classes of sesquilinear forms.

The main result of the talk is that in the special case m=4, n=2 we provide explicit necessary and sufficient conditions for two sesquilinear forms to be equivalent. In the process of formulating these necessary and sufficient conditions we show that a first order Hermitian sesquilinear form implicitly contains geometric constructs such as Lorentzian metric, spin structure, connection coefficients and electromagnetic covector potential.

The talk is based on the paper Z. Avetisyan, Y.-L. Fang, N. Saveliev and D. Vassiliev, "Analytic definition of spin structure", Journal of Mathematical Physics 58 (2017), 082301.

## Combinatorics Seminar

### Tilings of Space and the Dedekind-MacNeille Completion of Bruhat Order

Monday, November 13, 2017, 5:00pm
Ungar Room 402

Abstract: It is quite ordinary to consider how a group acts on an object. What if instead, one fixes a representation and lets the set of linear transformations (id-g) act on the object? In 2005, Waldspurger showed that, for the regular representation of a finite reflection group, the action of (id-g) on the cone over the fundamental weights gives a tiling of the cone over the positive roots. Shortly thereafter, Meinrenken considered the case of affine Weyl groups, and showed that the action of (id-g) on a fundamental alcove gives a tiling of the whole vector space. Bibikov and Zhgoon then proved analogous results for all cocompact hyperbolic reflection groups. We will look at some combinatorial consequences of these theorems for finite and affine types A and B. In particular, we will investigate the Dedekind-MacNeille completion or Bruhat order—the smallest lattice containing Bruhat order as a subposet.

## Geometry and Physics Seminar

### Froyshov Invariants of Branched Double Covers

Wednesday, November 8, 2017, 5:00pm
Ungar Room 402

Abstract: Let Y be a double cover of the 3-sphere branched over a knot K. The Froyshov invariant of Y (for the unique spin structure) is a useful concordance invariant of the knot. When the knot is quasi-alternating, this invariant equals the signature of the knot divided by 8 (as proved by Manolescu-Owens and Lisca-Owens). While this relation does not hold in general, I will give a generalization that holds for all knots (even links) in the 3-sphere and sketch the proof. Various applications will be discussed, including an interesting relation between Seiberg-Witten type invariants and Donaldson type invariants of homology S1 cross S3. This is a joint work with Daniel Ruberman and Nikolai Saveliev.

## Applied Math Seminar

### Modelling Antimicrobial De-escalation:Implications for Stewardship Programs

Tuesday, November 7, 2017, 2:15pm
Ungar Room 402

Abstract: Antimicrobial de-escalation aims to minimize resistance to high-value broad-spectrum empiric antimicrobials by switching to alternative drugs when testing confirms susceptibility. Though widely practiced in ICUs, the effects of de-escalation are not well understood. We develop a high-dimensional ODE model to assess the effect of de-escalation on a broad range of outcomes, and clarify expectations. In this talk, I will present the medical background and conclusions of this study, and show how we numerically analyze the model output with a broad range of undetermined parameters and limited data. Ongoing work on the model simplification and relevant mathematical analysis will be discussed at the end.

## Combinatorics Seminar

### Chromatic Quasisymmetric Functions and Hessenberg Varieties

Monday, November 6, 2017, 5:00pm
Ungar Room 402

Abstract: I will discuss an algebro-geometric approach to proving the longstanding Stanley-Stembridge e-positivity conjecture for chromatic symmetric functions that was proposed by Shareshian and myself several years ago. Our approach to this conjecture involves a refinement of Stanley's chromatic symmetric functions. We conjectured a certain relationship between our refinement and Hessenberg varieties. Our conjecture was recently proved by Brosnan and Chow using techniques from algebraic geometry, and more recently by Guay-Paquet using Hopf algebras. I will describe this result, some of its consequences, and what still needs to be done to prove the Stanley-Stembridge conjecture.

## Geometry and Physics Seminar

### Berkovich Spaces and Dual Complexes of Degenerations

Wednesday, November 1, 2017, 5:00pm
Ungar Room 402

Abstract: In the late nineteen-nineties Berkovich developed a new approach to non-archimedean analytic geometry. This theory has quickly found many applications in algebraic and arithmetic geometry. In particular it turned out that there are strong connections between Berkovich spaces of degenerations of varieties and the birational geometry of dual complexes. In this talk, I will explain how the dual complex of a degeneration can be interpreted as a simplicial subset of a Berkovich space. I will introduce the central objects of this theory: the weight function and the essential skeleton of the degeneration. Finally, I will use them to study the dual complex of products and of some examples of hyperkahler varieties.

This is a joint work with Morgan Brown and a complementary talk to the seminar he gave on the dual complex of a product of degenerations.

## Applied Math Seminar

### Dynamics of Populations with Individual Variation in Dispersal on Bounded Domains(Joint work with Steve Cantrell and Xiao Yu)

Tuesday, October 31, 2017, 2:15pm
Ungar Room 406

Abstract: Most classical models for the movement of organisms assume that all individuals have the same patterns and rates of movement, but there is empirical evidence that movement rates and patterns may vary among individuals. One way to capture variation in dispersal is to allow individuals to switch between two distinct dispersal modes. We consider models for populations with logistic-type local population dynamics whose members can switch between two different nonzero rates of diffusion. The resulting reaction-diffusion systems can be cooperative at some population densities and competitive at others. We analyze the dynamics of such systems on bounded regions. (Traveling waves and spread rates have been studied by others for similar models in the context of biological invasions.) The analytic methods include ideas and results from reaction-diffusion theory, semi-dynamical systems, and bifurcation/continuation theory.

## Combinatorics Seminar

### Chromatic Quasisymmetric Functions of Directed Graphs

Monday, October 30, 2017, 5:00pm
Ungar Room 402

Abstract: I will be presenting my work on expansions (in various bases for the ring of symmetric and quasisymmetric functions) of chromatic quasisymmetric functions for digraphs. This is a version of the talk I will be giving at the Combinatorics Seminar at Brandeis.

## Combinatorics Seminar

### A Combinatorial Model for the Based Loop Space

Monday, October 23, 2017, 5:00pm
Ungar Room 402

Abstract: To any topological space we may associate a topological monoid called the based loop space: as a set it consists of all loops in the space based at a fixed point and the multiplication is given by concatenation of loops. The homology of the based loop space has the structure of a Hopf algebra: the product is induced by concatenation of loops, the coproduct by the Alexander-Whitney diagonal, and the antipode by the map sending a loop to its inverse. From a classical result of homotopy theory we know that sufficiently nice topological spaces may be modeled by combinatorial objects called simplicial sets. I will explain how to model the above construction in purely combinatorial terms, namely, to any connected simplicial set S I will construct a natural differential graded Hopf algebra, based on the combinatorics of S, having the property that its homology is isomorphic to the homology Hopf algebra of the the based loop space of the geometric realization of S. This is joint work with Samson Saneblidze and generalizes classical results of Adams and Baues.

## Geometry and Physics Seminar

### Perverse Sheaves of Categories

Wednesday, October 18, 2017, 5:00pm
Ungar Room 402

Abstract: A perverse sheaf of categories is a graph on a punctured Riemann surface with categorical data associated to each edge and vertex. In this talk, I will explain how these thing can be used to encode the derived category of coherent sheaves on certain algebraic varieties and what this means for homological mirror symmetry.

## Combinatorics Seminar

### Hopf Algebras in Combinatorics(continuation)

Monday, October 16, 2017, 5:00pm
Ungar Room 402

## Geometry and Physics Seminar

### Categorical and Algebraic Constructions Related to Path Spaces

Wednesday, October 11, 2017, 5:00pm
Ungar Room 402

Abstract: In this talk I will describe explicitly how the following three functors are related:

1) the path space functor and its relatives (based path space, based loop space, free loop space, etc...)
2) the cobar functor from the category of differential graded coalgebras to the category of differential graded algebras
3) the rigidification functor from simplicial sets to simplicial categories

1) is a classical and important construction which appears all over through geometry, topology, and mathematical physics. 2) is a purely algebraic construction introduced by Frank Adams in the 1950's to obtain an algebraic model for the based loop space of a simply connected space, which is suitable for computations. 3) was introduced by Jacob Lurie in order to compare different models for infinity categories.

The key to relate these three functors is to introduce a cubical version of 3). Understanding these relationships reveals a few interesting consequences, for example: it tells us how to remove the simply connectedness hypothesis in Adams' theorem, we obtain a strict adjoint functor for the differential graded nerve functor and a transparent algebraic definition for infinity local systems, and we also obtain a direct proof of the fact that the chains on the based loop space of a space with Poincaré duality has a Calabi-Yau algebra structure.

## Combinatorics Seminar

### Discrete Line Field

Monday, September 25, 2017, 5:00pm
Ungar Room 402

Abstract: The discrete line field is our proposal for a possible discretization of the theory of line fields. The discrete object will be a Morse matching just between the vertices and edges of a cellular complex. The objective is to define the critical objects and their indices, and then show that the complex is homotopy equivalent to a cellular complex with just the critical objects.

## Geometry and Physics Seminar

### The Quantum GIT Conjecture

Wednesday, September 6, 2017, 5:00pm
Ungar Room 402

Abstract: For a compact symplectic manifold X with a Hamiltonian action of a compact group, one can define the gauges Gromov-Witten theory which involves integration over the moduli space of G-bundles and twisted maps to X. The space of states of this theory has a description as the equivariant quantum cohomology of X for an action of the free loop group of G. When X is Fano, with orbifold GIT quotient, this agrees conjecturally with QH*(X//G). (For G a torus and X a vector space, this is a vast generalization of Batyrev's description of QH^* of toric Fanos.) I describe a tentative outline of the proof, and the relation to the open string version of this conjecture.

## Geometry and Physics Seminar

### The Dual Complex of a Product of Degenerations

Wednesday, August 30, 2017, 5:00pm
Ungar Room 402

Abstract: To a simple normal crossing degeneration of algebraic varieties, we can associate an invariant called the dual complex, which is the intersection complex of the special fiber. In this talk I will investigate how this construction behaves under products. Unfortunately, the product of two simple normal crossings degenerations over a curve in general fails to remain snc, but belongs to the broader class of toroidal singularities. I will introduce toroidal geometry and use it to show that the dual complex of a product of semistable degenerations is PL homeomorphic to the product of the individual dual complexes.

## Combinatorics Seminar

### Hopf Algebras in Combinatorics(continuation)

Monday, August 28, 2017, 5:00pm
Ungar Room 402

## Combinatorics Seminar

### Hopf Algebras in Combinatorics(continuation of the summer seminar)

Monday, August 21, 2017, 5:00pm
Ungar Room 402

## Combinatorics Seminar

### Deformation Cones for Polytopes

Friday, June 23, 2017, 11:00am
Ungar Room 411

Abstract: Given a lattice polytope, the set of all polytopes having the same (or a coarsening) normal fan is a polyhedral cone. This cone has appeared in different contexts, for instance, it is closely related to the nef cone of the associated toric variety. In the case of the regular permutohedron we get the cone of submodular functions. The purpose of this talk is to survey known results and show how to compute this deformation cones in further combinatorial examples. This is joint work with Fu Liu.

## Combinatorics Seminar

### The Gamma-coefficients of the Tree Eulerian Polynomials

Wednesday, June 21, 2017, 2:30pm
Ungar Room 411

Abstract: We consider the generating polynomial T_n(t) of the number of rooted trees on the set {1,2,...,n} counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. In particular it has palindromic coefficients and hence it can be expressed in the the basis $\left \{ t^i(1+t)^{n-1-2i}\,\mid\, 0\le i \le \lfloor \frac{n-1}{2}\rfloor\right \}$, known as the $\gamma$-basis. We show that $\T_n(t)$ has nonnegative $\gamma$-coefficients and we present various combinatorial interpretations for them.

## Geometry and Physics Seminar

### Hidden Symmetries and Conservation Laws

Wednesday, May 3, 2017, 5:00pm
Ungar Room 402

Abstract: Maxwell's theory of electromagnetism is a relativistic field theory on Minkowski space which is symmetric under the 15-dimensional conformal group of Minkowski space. However, it admits further less obvious symmetries including the Heaviside-Larmor-Rainich symmetry, as well as hidden symmetries associated with the 20-dimensional space of conformal Killing-Yano 2-tensors on Minkowski space. I will give some background to these facts and discuss their relation to conservation laws for the Maxwell and gravitational field on Minkowski space as well as on non-trivial geometries including the Kerr black hole spacetime.

## AMS Graduate Student Chapter Seminar

### NSA: The Secret Life of Mathematicians

Friday, April 28, 2017, 4:00pm
Ungar Room 402

Abstract: The National Security Agency is one of the largest employers of mathematicians in the United States. In this talk, Dr. Horta will discuss his career at the NSA, job opportunities for both mathematicians and computer scientists, and what "A Day in the Life of an NSA Mathematician" is like. Also, sample mathematical applications to NSA problems will be presented, including the applications of Number Theory to Cryptography and applications of Graph Theory to Data Science.

## Geometry and Physics Seminar

### Categorifying the Alexander Polynomial as a Reshetikhin-Turaev Invariant

Wednesday, April 26, 2017, 5:00pm
Ungar Room 402

Abstract: The Reshetikhin-Turaev construction for the standard representation of the quantum group gl(1|1) sends tangles to C(q)-linear maps in such a way that a knot is sent to its Alexander polynomial. After a brief review of this construction, I will give an introduction to tangle Floer homology -- a combinatorial generalization of knot Floer homology which sends tangles to (homotopy equivalence classes of) bigraded dg bimodules. Finally, I will discuss how to see tangle Floer homology as a categorification of the Reshetikhin-Turaev invariant. This is joint work with Alexander Ellis and Vera Vertesi.

## Applied Math Seminar

### Modeling Evolution in the Galapagos

Tuesday, April 25, 2017, 11:00am
Ungar Room 411

## Combinatorics Seminar

### Labeled Binary Trees, Schur-positivity and Generalized Tamari Lattices

Monday, April 24, 2017, 5:00pm
Ungar Room 402

Abstract: Gessel introduced a multivariate formal power series tracking the distribution of ascents and descents in labeled binary trees. In addition to showing that it was a symmetric function, he conjectured that it was Schur-positive.

In this talk, I will present a proof of this conjecture which utilizes an extension of a beautiful bijection of Preville-Ratelle and Viennot concerning generalized Tamari lattices. I will subsequently discuss connections between specializations of Gessel's symmetric function and Frobenius characteristics of symmetric group actions on certain Coxeter deformations, focusing in particular on semiorder and Linial arrangements. Finally, I will discuss some potential avenues to pursue.

This is joint work with Ira Gessel and Sean Griffin.

## Geometry and Physics Seminar

### Equivariant Invarians of GIT Quotients and Vanishing Cycles

Wednesday, April 19, 2017, 5:00pm
Ungar Room 402

Abstract: I will describe a relation between Hodge structure on vanishing cohomolgy of weighted homogeneous singularities and equivariant cohomology motivated by Landau-Ginzburg/Calabi Yau correspondence. Extension of this relation to correspondence between hybrid models also will be discussed.

## Combinatorics Seminar

### On r-inversions and Symmetric Functions

Monday, April 17, 2017, 5:00pm
Ungar Room 402

Abstract: The r-inversion number is a statistic on words of length n (over the positive integers), which interpolates between the descent number (r=2) and the inversion number (r=n). We consider a symmetric function U_{n,r} that enumerates words of length n by this statistic. The symmetric function U_{n,r} is an example of an LLT polynomial. The LLT polynomials were shown to be Schur-positive by Grojnowski and Haiman by means of Kazhdan-Lusztig theory.  It is an open question to give a combinatorial description of the coefficients in the Schur basis expansion. For r = 2 and r=n, such descriptions are well known. For r = 3, a description (in a more general setting) was conjectured by Haglund and was proved by Blasiak using noncommutative Schur functions and Lam's algebra of ribbon Schur operators. In this talk I will describe a more elementary proof for the r = 3 case, which uses classical RSK theory. I will also discuss results for some other cases, and a consequence involving an r-analog of the q-binomial coefficients. This is joint work with Yuval Roichman.

## Geometry and Physics Seminar

### Toward Ulrich Bundles on ACM Varieties

Wednesday, April 5, 2017, 5:00pm
Ungar Room 402

Abstract: Let X be a variety embedded in projective space. Ulrich bundles on X are those vector bundles which admit a linear resolution when viewed as sheaves on the ambient projective space. Existence of an Ulrich bundle implies that the homogeneous coordinate ring of X satisfies Lech's conjecture and that cone of cohomology tables of X agrees with that of a projective space of the same dimension. Motivated by the problem of constructing Ulrich bundles, I will describe a result relating Ulrich bundles to higher rank Brill-Noether theory on certain curves on X. Furthermore, I will explain a result that an arithmetically Cohen-Macaulay (ACM) variety always admits a reflexive sheaf whose restriction to a general one-dimensional linear section is Ulrich.

## Combinatorics Seminar

### Dyck Paths and Positroids from Unit Interval Orders

Monday, April 3, 2017, 5:00pm
Ungar Room 402

Abstract: It is well known that the number of non-isomorphic unit interval orders on [n] equals the n-th Catalan number. Combining work of Skandera and Reed and work of Postnikov, we will assign a rank n positroid on [2n] to each unit interval order on [n]. We call such positroids "unit interval positroids." Then we will give a characterization of the unit interval positroids by describing their associated decorated permutations, showing that each one must be a 2n-cycle encoding a Dyck path of length 2n.

## Combinatorics Seminar

### Determinants in "Wonderland"

Monday, March 27, 2017, 5:00pm
Ungar Room 402

Abstract: Determinants are found everywhere in mathematics and other scientific endeavors. Their particular role in Combinatorics does not need any cynical introduction or special advertisement. In this talk, we will illustrate certain techniques which proved to be useful in the evaluation of several class of determinantal evaluations. We conclude this seminar with an open problem. The content of our discussion is accessible to anyone with "an intellectual appetite".

## Applied Math Seminar

### Diffusion of a Population on a Landscape:Does a Heterogeneous Landscape Lead to a Higher Population Size than a Homogeneous Landscape?

Tuesday, March 21, 2017, 11:00am
Ungar Room 411

Abstract: Single population reaction-diffusion models have shown that a population diffusing in an environment with a spatially heterogeneous carrying capacity can reach a higher total population size than when the same total carrying capacity, K(x), is distributed homogeneously as a function of distance, x (Lou 2006). A similar result had been pointed out earlier by Holt (1985) for a two-patch model with different growth rates and carrying capacities on the two patches, which he termed a "paradox". These results suggest that a higher population size can be attained by configuring the same total carrying capacity in a heterogeneous, as opposed to a homogeneous, manner.

However, it is biologically impossible to create carrying capacities and growth rates the way they are formulated in these models. Biological populations require input of energy and nutrients to survive and grow, and the populations affect those resources through exploitation. We consider a two-variable chemostat-type model of a consumer population and an exploitable resource (e.g., a limiting nutrient), in which the same total amount of resource input can be spatially distributed in any possible way across the landscape. Spatially varying carrying capacities and growth rates emerge through exploitation by the diffusing population. We proved for this model the following. (1) A diffusing population in a heterogeneous environment can exceed in size a non-diffusing population under certain conditions on the parameters. (2) However, the population size of a diffusing population, when the resource inputs are heterogeneously distributed, will always be less than (or at most equal to) the size of the population that is either diffusing or not diffusing when the same total amount of resource input is homogeneously distributed. This resolves the paradox noted by Holt (1985). Experiments using yeast (Zhang et al. submitted) corroborate these results.

Based on work by Don DeAngelis, Wei-Ming Ni, and Bo Zhang.

## Combinatorics Seminar

### Partition into Distinct Parts and Unimodality

Monday, March 20, 2017, 5:00pm
Ungar Room 402

Abstract: We discuss the (non)unimodality of the rank-generating function, $F_{\lambda}$, of the poset of partitions with distinct parts contained inside a given partition $\lambda$. This work, in collaboration with Richard Stanley (European J. Combin., 2015), is in part motivated by an attempt to place into a broader context the unimodality of $F_{\lambda}(q)=\prod_{i=1}^n(1+q^i)$, the rank-generating function of the staircase'' partition $\lambda=(n,n-1,\dots,1)$, for which determining a combinatorial proof remains an outstanding open problem.

We will present a number of results on the polynomials $F_{\lambda}$. Surprisingly, these results carry a remarkable similarity to those proven in 1990 by Dennis Stanton. His work extended, to any partition $\lambda$, the study of the unimodality of $q$-binomial coefficients --- that is, the rank-generating functions of the \emph{arbitrary} partitions contained inside given rectangular partitions.

We will also discuss some open problems and recent developments. These include a (prize-winning) paper by Levent Alpoge, who solved our conjecture on the unimodality of $F_{\lambda}$ when $\lambda$ is the "truncated staircase" $(n,n-1, \dots,n-c)$, for $n\gg c$.

## Combinatorics Seminar

### Rhombic Tilings and Bott-Samelson Varieties

Monday, March 6, 2017, 5:00pm
Ungar Room 402

Abstract: Elnitsky gave an elegant bijection between rhombic tilings of 2n-gons and commutation classes of reduced words in the symmetric group on n letters. We explain a natural connection between Elnitsky's and Magyar's construction of the Bott-Samelson resolution of Schubert varieties. This suggests using tilings to encapsulate Bott-Samelson data and indicates a geometric perspective on Elnitsky's combinatorics. We also extend this construction by assigning desingularizations to the zonotopal tilings considered by Tenner. This is based on joint work with Pechenik, Tenner and Yong.

## Combinatorics Seminar

### Cyclic Descents of Standard Young Tableaux

Monday, February 27, 2017, 5:00pm
Ungar Room 402

Abstract: Permutations in the symmetric groups, as well as standard Young tableaux, are equipped with a well-established notion of descent set. The cyclic descent set of permutations was introduced by Cellini and further studied by Dilks, Petersen and Stembridge, while cyclic descents on standard Young tableaux (SYT) of rectangular shapes were introduced by Rhoades.

The existence of cyclic descent maps for SYT of all non-ribbon skew shapes was recently proved, using nonnegativity properties of Postnikov's toric Schur polynomials. The proof and its implications will be explained by Ron Adin in tomorrow's colloquium talk.

In this talk we will focus on explicit combinatorial interpretations of the concept, applications to Schur-positivity and open problems.

Based on joint works with Ron Adin, Sergi Elizalde and Vic Reiner.

## Applied Math Seminar

### Modeling the Transmission Dynamics of Avian Influenza H7N9 Virus

Tuesday, February 21, 2017, 11:00am
Ungar Room 411

Abstract: In March 2013, a novel avian-origin influenza A (H7N9) virus was identified among human patients in China and a total of 124 human cases with 24 related deaths were confirmed by May 2013. There were no reported cases in the summer and fall 2013. However, the virus has been coming back in November every year. In fact, the second outbreak from November 2013 to May 2014 caused 130 human cases with 35 deaths, the third outbreak from November 2014 to June 2015 caused 216 confirmed human cases with 99 deaths, the fourth outbreak from November 2015 to July 2016 caused 114 confirmed human cases and 45 deaths, respectively. From November 2016 to January 2017, there have already been 304 cases with 99 deaths. In this talk, I will introduce some recent studies on modeling the transmission dynamics of the avian influenza A (H7N9) virus from birds to humans and apply our models to simulate the open data for numbers of the infected human cases and related deaths reported by the Chinese Center for Disease Control and Prevention. The basic reproduction number is estimated and sensitivity analysis of in terms of model parameters is performed. Our studies demonstrate that H7N9 virus has been well established in birds and will cause regular outbreaks in humans again in the future.

## Combinatorics Seminar

### On Local-Global Phenomena in the Betti Tables of Stanley-Reisner Ideals

Monday, February 20, 2017, 5:00pm
Ungar Room 402

Abstract: Let I be an homogeneous ideal in a polynomial ring S over a field. The Betti table of I describes the graded minimal free resolution of I over S. When I is a Stanley-Reisner ideal of a simplicial complex C, the Betti table can be used to compute the h- and f-vectors of C. In this talk I will describe several recent results about what I call local-global phenomena in the Betti tables. Namely, information on a small part of the table forces strong result on the whole resolution, and give structural information about C such as its depth, regularity or chordality. If time permits, I will also explain the connection of these results to classical commutative algebra, and some new connections to group cohomological dimensions. The talk will be based on various joint works with Schweig-Huneke, Schweig, and Vu.

## Combinatorics Seminar

### Simplicial Rook Graphs:Algebraic and Combinatorial Properties

Friday, February 17, 2017, 4:00pm
Ungar Room 402

Abstract: A few years ago, Jeremy L. Martin and Jennifer D. Wagner introduced the simplicial rook graphs SR(d,n) as the graph whose vertices are the lattice points in the n-th dilate of the standard simplex in Rd, with two vertices adjacent if they differ in exactly two coordinates. Martin and Wagner proved that SR(3,n) has integral eigenvalues and determined other interesting properties of these graphs. In this talk, I will describe our work proving some conjectures made by Martin and Wagner as well as determining other algebraic and combinatorial facts about these graphs. This is joint work with Andries Brouwer (TU Eindhoven, The Netherlands), Willem Haemers (Tilburg University, The Netherlands) and Jason Vermette (Missouri Baptist Univ., USA).

## Geometry and Physics Seminar

### Emergent Gravity and H. Weyl's Volume Formula

Wednesday, February 15, 2017, 5:00pm
Ungar Room 402

Abstract: In physical theories where the energy is localized near a submanifold of Euclidean space, there is a universal expression for the energy. We derive a multipole-like expansion for the energy that has a finite number of terms, and depends on intrinsic geometric invariants of the submanifold and extrinsic invariants of the embedding of the submanifold. This universal expression is a generalization of an exact formula of Hermann Weyl for the volume of a tube. In special situations, dictated by spherical symmetry of the energy density, the expression is a generalized Lovelock lagrangian for gravity constructed using only the Lipschitz-Killing curvatures. This class of theories is interesting because there are no negative metric states. We discuss in what sense this is an emergent theory of gravity and how it is related to the local isometric embedding problem.

## Combinatorics Seminar

### A Directed Graph Generalization of Chromatic Quasi-symmetric Functions

Monday, February 13, 2017, 5:00pm
Ungar Room 402

Abstract: Chromatic quasisymmetric functions of labeled graphs were defined by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric functions. In this talk, we present an extension of their definition from labeled graphs to directed graphs, suggested by Richard Stanley. We show that the chromatic quasisymmetric functions of proper circular arc digraphs are symmetric functions, which generalizes a result of Shareshian and Wachs on natural unit interval graphs. The directed cycle on n vertices is contained in the class of proper circular arc digraphs, and we give a generating function for the e-basis expansion of the chromatic quasisymmetric function of the directed cycle, refining a result of Stanley for the undirected cycle. We discuss a generalization of the Shareshian-Wachs refinement of the Stanley-Stembridge e-positivity conjecture. We present our F-basis expansion of the chromatic quasisymmetric functions of all digraphs and our p-basis expansion for all symmetric chromatic quasisymmetric functions of digraphs, which extends work of Shareshian-Wachs and Athanasiadis.

## Geometry and Physics Seminar

### Minimal Hypersurfaces with Free Boundary and PSC-bordism

Wednesday, February 8, 2017, 5:00pm
Ungar Room 402

Abstract: There is a well-known technique due to Schoen-Yau from the late 70s which uses (stable) minimal hypersurfaces to study the topological consequences of a (closed) manifold's ability to support a Riemannian metric with positive scalar curvature. In this talk, we describe a version of this technique for manifolds with boundary and discuss how it can be used to study bordisms of positive scalar curvature metrics.

## Applied Math Seminar

### Some Mathematical Models of Criminal Behavior

Tuesday, February 7, 2017, 11:00am
Ungar Room 411

Abstract: In recent years there has been considerable interest in developing mathematical models of socioeconomic phenomena in general and criminal activity in particular. One topic that has been treated in some detail is the spatial distribution of criminal behavior, especially burglary. It has been observed that burglaries tend to be clustered in "hotspots" with sizes and locations that do not seem to be determined in an obvious way by the geographic distribution of socio-economic factors. Two influential modeling approaches to understanding the problem of spatial distribution of crime were introduced in the papers (M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Mathematical Models and Methods in Applied Science, 18 (2008), 1249–1267) and (H. Berestycki and J.-P. Nadal, Self-organised critical hot spots of criminal activity, European Journal of Applied Mathematics 21 (2010), 371–399). Models of the first type were initially formulated as agent based models, but continuum limits were derived from those. The resulting continuum models have features somewhat similar to chemotaxis models. Models of the second type of include spatially nonlocal terms and in some cases diffusion. Both types support spatial patterns. Various versions of those models have been studied from the viewpoints of existence theory, bifurcation theory, and traveling waves. This talk will describe some results and problems related to such models.

## Combinatorics Seminar

### On Conjectural Relatives of Matroid Polytopes

Monday, February 6, 2017, 5:00pm
Ungar Room 402

Abstract: We will present various curious similarities between shifted simplicial complexes and matroid independence complexes and provide evidence that all these similarities should be proved geometrically by extending the theory of matroid polytopes. Along the way we will pose questions, conjectures and explain some of the final goals. Based on joint work with Jeremy Martin.

## Geometry and Physics Seminar

### Quantum Toric Varieties

Wednesday, February 1, 2017, 5:00pm
Ungar Room 402

Abstract: In this talk I will talk about the theory of quantum toric virieties as developed by Katzarkov, Meersseman, Verjovsky and myself. Classical toric varieties have proved important in geometry. They are built out of tori and the combinatorics of raitonal polytopes. Our quantum toric varieties are built up out of quantum tori and the combinatorics of irrational polytopes. I will speak about their construction and also relations with the work of Kaledin, Shkolnikov and myself regarding sandpiles.

## Combinatorics Seminar

### Mogami Constructions of Manifolds from Trees of Tetrahedra

Monday, January 30, 2017, 5:00pm
Ungar Room 402

Abstract: A 3-ball is a simplicial complex homeomorphic to the unit ball in R^3. A "tree of tetrahedra" is a 3-ball whose dual graph is a tree. It is easy to see that every (connected) 3-manifold can be obtained from some tree of tetrahedra by recursively gluing together two boundary triangles.

The quantum physicist Tsugui Mogami has studied "Mogami manifolds", that is, those manifolds that can be obtained from a tree of tetrahedra by recursively gluing together two *incident* boundary triangles. In 1995 he conjectured that all 3-balls are Mogami. Mogami's conjecture would imply a much desired exponential bound (crucial for the convergence of certain models in quantum gravity) for the number of 3-balls with N tetrahedra. Unfortunately, we show that Mogami's conjecture does not hold.

## Combinatorics Seminar

### Counting with Congruence Conditions

Monday, January 23, 2017, 5:00pm
Ungar Room 402

Abstract: The archetypal result is the theorem of Lucas that the number of coefficients of the polynomial $(1+x)^n$ not divisible by a prime $p$ is $\prod(1+a_i)$, where $n=\sum a_ip^i$ is the base $p$ expansion of $n$. We will discuss numerous generalizations and analogues of this result. For example, the number of partitions of $n$ for which the number of standard Young tableaux of shape $\lambda$ is odd is equal to $2^{\sum b_i}$, where $n=\sum 2^{b_i}$ is the binary expansion of $n$ (due to I. G. Macdonald).

## Combinatorics Seminar

### LLT Polynomials and the Chromatic Symmetric Function of Unit Interval Orders

Monday, November 21, 2016, 5:00pm
Ungar Room 402

Abstract: Shareshian and Wachs have conjectured that a certain symmetric function, which depends on a Dyck path and a parameter t, has positive coefficients when expressed as a polynomial in the elementary symmetric functions. Their conjecture implies an earlier conjecture of Stanley and Stembridge. We show how some elements of the preprint of Carlsson and Mellit "A proof of the shuffle conjecture" imply that the Shareshian-Wachs symmetric function can be expressed, via a plethystic substitution, in terms of LLT polynomials, specifically LLT products of single cells. As corollaries we obtain combinatorial formulas for the expansion of Jack polynomials into the Schur basis, and also the power-sum basis. These formulas are signed, not always positive, but perhaps could be simplified. Other applications include a quick method for computing the chromatic symmetric function using plethystic operators. Based on joint work with Per Alexandersson, Greta Panova, and Andy Wilson.

## Geometry and Physics Seminar

### Canonical Rings of Stacky Curves

Wednesday, November 16, 2016, 5:00pm
Ungar Room 402

Abstract: We give a generalization to stacks of the classical (1920's) theorem of Petri -- we give a presentation for the canonical ring of a stacky curve. This is motivated by the following application: we give an explicit presentation for the ring of modular forms for a Fuchsian group with cofinite area, which depends on the signature of the group. This is joint work with John Voight.

## Combinatorics Seminar

### Subrack Lattices of Group Racks

Monday, November 14, 2016, 5:00pm
Ungar Room 402

Abstract: Let G be a finite group. A subset S of G is called a subrack if S is closed under conjugation. The set R(G) of all subracks of G is partially ordered by inclusion. With Istvan Heckenberger and Volkmar Welker of Philipps-University at Marburg, we have studied relations between the combinatorial structure of R(G) and the algebraic structure of G. I will discuss our results.

## Geometry and Physics Seminar

### The Prism Manifold Realization Problem

Wednesday, November 9, 2016, 5:00pm
Ungar Room 402

Abstract: The spherical manifold realization problem asks which spherical three-manifolds (equivalently, three-manifolds with finite fundamental groups) arise by surgery on knots in S^3. In recent years, the realization problem for C,T,O,I-type spherical manifolds has been solved, leaving the D-type spherical manifolds (aka prism manifolds) as the only remaining case. Every prism manifold can be parametrized as P(p,q), for a pair of relatively prime integers p>1 and q. We determine a complete list of prism manifolds P(p, q) that can be realized by positive integral surgery on knots in S^3 when q<0. The general methodology undertaken to obtain the classification is similar to that of Greene for lens spaces. The arguments rely on tools from Floer homology and lattice theory, and are primarily combinatorial in nature. This is joint work with Ballinger, Hsu, Mackey, Ni, and Ochse.

## Combinatorics Seminar

### Filtered Geometric Lattices

Monday, November 7, 2016, 5:00pm
Ungar Room 402

Abstract: In order to address some questions in tropical geometry, Mikhalkin and Ziegler introduced the notion of a filtered geometric lattice. These posets can be seen as generalizations of geometric semilattices (introduced by Wachs and Walker), which are themselves generalizations of geometric lattices.

In this talk, we will discuss some topological results of Adiprasito and Bjorner about filtered geometric lattices, as well as some open questions about these posets.

## Applied Math Seminar

### Stabilization of Compressible Fluid Models

Tuesday, November 1, 2016, 12:30pm
Ungar Room 411

Abstract: Compressible fluids are modeled through Navier-Stokes equations for density and velocity. In this talk, I consider a model in a bounded interval and the stabilization (steer the system to a steady state as time goes to infinity) of the solution around constant steady states. The control acts only on the velocity. After describing the problem for simple ODE systems, I will discuss the PDE systems.

## Combinatorics Seminar

### Discrete vs. Continuous Valuations: Similarities and Differences

Monday, October 31, 2016, 5:00pm
Ungar Room 402

Abstract: The prototypical valuation is presumably the volume. It has various favorable properties such as homogeneity, monotonicity and translation-invariance. In the continuous setting, valuations are well-studied and the volume plays a prominent role in many classical and structural results. In the less examined discrete setting, the number of lattice points in a polytope - its discrete volume - takes a central role. Although homogeneity and continuity are lost, some striking parallels can be drawn. In this talk, I will discuss some similarities, analogies and differences between the continuous and discrete world of translation-invariant valuations.

## Geometry and Physics Seminar

### Shifted Symplectic Structures and Applications

Wednesday, October 26, 2016, 5:00pm
Ungar Room 402

## Combinatorics Seminar

### Positive Semidefinite Matrix Completion and Free Resolutions

Monday, October 24, 2016, 5:00pm
Ungar Room 402

Abstract: I will discuss the positive semidefinite matrix completion problem arising e.g. in combinatorial statistics and explain how we can use results in algebraic geometry to understand it better. The object linking the two different areas is the cone of sums of squares and its properties as a convex cone.

## Geometry and Physics Seminar

### Twisted Symmetric Differentials and the Quadric Algebra

Wednesday, October 19, 2016, 5:00pm
Ungar Room 402

Abstract: It was shown by M. Schneider in the nineties that there are no symmetric differentials of degree m on projective subvarieties of low codimension even if twisted by O(k) for k<m. In this talk I will discuss the geometric nature of the extremal case k=m and its relation to the quadric hypersurfaces containing the subvariety. In particular, it is expected that these differentials are given by polynomials in the quadrics that vanish on the variety. I will give the proof for complete intersections and discuss the general case for varieties of very low codimension.

## Combinatorics Seminar

### The Dehn-Sommerville Relations and the Catalan Matroid

Monday, October 17, 2016, 5:00pm
Ungar Room 402

Abstract: The f-vector of a d-dimensional polytope P stores the number of faces of each dimension. When P is a simplicial polytope the Dehn-Sommerville relations condense the f-vector into the g-vector, which has length $\lceil{\frac{d+1}{2}}\rceil$. Thus, to determine the f-vector of P, we only need to know approximately half of its entries. This raises the question: Which $(\lceil{\frac{d+1}{2}}\rceil)$-subsets of the f-vector of a general simplicial polytope are sufficient to determine the whole f-vector? We prove that the answer is given by the bases of the Catalan matroid.

## Geometry and Physics Seminar

### Equivariant Rho-Invariants and Instanton Homology of Torus Knots

Wednesday, October 12, 2016, 5:00pm
Ungar Room 402

Abstract: The equivariant rho-invariants are a version of the classical rho-invariants of Atiyah, Patodi, and Singer in the presence of an isometric involution. In this talk we discuss these rho-invariants for allere involutions on 3-dimensional lens spaces with 1-dimensional fixed point sets, as well as for some involutions on Brieskorn homology spheres. As an application, we compute the Floer gradings in the singular instanton chain complex of (p, q)-torus knots with odd p and q.

CANCELLED

## Geometry and Physics Seminar

### Twisted Symmetric Differentials and the Quadric Algebra

Wednesday, October 5, 2016, 5:00pm
Ungar Room 402

Abstract: It was shown by M. Schneider in the nineties that there are no symmetric differentials of degree m on projective subvarieties of low codimension even if twisted by O(k) for k<m. In this talk I will discuss the geometric nature of the extremal case k=m and its relation to the quadric hypersurfaces containing the subvariety. In particular, it is expected that these differentials are given by polynomials in the quadrics that vanish on the variety. I will give the proof for complete intersections and discuss the general case for varieties of very low codimension.

## Applied Math Seminar

### The Importation, Establishment and Transmission Dynamics of the Mosquito-borne Disease

Tuesday, October 4, 2016, 12:30pm
Ungar Room 411

Abstract: Because of the development of the global transportation system, it is frequently found that the tourists are exposed to infections in the countries where certain infectious diseases are prevalent and the local residents are exposed to infections brought by the tourists from epidemic areas. These frequent activities lead to a possibility of international spread of infectious disease and bring threats to individuals who are geographically far from the original outbreak. Thus currently mosquito-borne diseases, including Dengue fever, Chikungunya and Zika virus, bring an emerging public health threat to some developed countries such as United States while previous attention was mainly put to the tropical countries considering the climate features and the socioeconomic conditions.

Our hypothesis is the imported cases first spread the mosquito-borne disease to local mosquitoes, which later cause local infections on humans. Based on this, we propose a mathematical model to study how these movements of humans affect the establishment and transmission dynamics of the mosquito-borne disease.

## Combinatorics Seminar

### Bruhat Order on Twisted Identities and KLV Polynomials

Monday, October 3, 2016, 5:00pm
Ungar Room 402

Abstract: We study the Bruhat order on the sets of twisted involution and twisted identities in a Coxeter group W equipped by an involutive automorphism. When W is the symmetric group of odd rank, we define the Kazhdan-Lusztig-Vogan polynomials indexed by elements in the set of twisted identities and we prove that they are combinatorially invariant for intervals that start with the identity. This generalizes the combinatorial invariance of the classical Kazhdan-Lusztig polynomials for lower bound intervals in a symmetric group.

This is joint work with Axel Hultman.

## Geometry and Physics Seminar

### Spectral Networks and Buildings

Wednesday, September 28, 2016, 5:00pm
Ungar Room 402

Abstract: The theory of spectral networks was introduced by Gaiotto, Moore and Neitzke for WKB problems as well as for physics motivations. We relate these combinatorial geometric objects that come from spectral curves, to actions of fundamental groups on euclidean buildings. The WKB asymptotics can be encoded in an equivariant harmonic map to a building, and the spectral network is a main part of the pullback of the singular locus of the building. This is joint work with Katzarkov, Noll and Pandit.

## Applied Math Seminar

### Fitness Based Prey Dispersal and Prey Persistence in Intraguild Predation Systems(co-authors Robert Stephen Cantrell, King-Yeung Lam, Tian Xiang, Xinru Cao)

Tuesday, September 27, 2016, 12:30pm
Ungar Room 411

Abstract: We establish prey persistence in intraguild predation systems in bounded habitats under mild conditions when the prey disperses using its fitness as a surrogate for the balance between resource acquisition and predator avoidance. The model is realized as a quasilinear parabolic system where the dimension of the underlying spatial habitat is arbitrary.

## Geometry and Physics Seminar

### A Characterization of Toric Varieties

Wednesday, September 21, 2016, 5:00pm
Ungar Room 402

Abstract: Toric varieties are ubiquitous in algebraic geometry. They have a rich combinatorial structure, and give the simplest examples of log Calabi-Yau varieties.

We give a simple criterion for characterizing when a log Calabi-Yau pair is toric, which answers a case of a conjecture of Shokurov. This is joint work with James McKernan, Roberto Svaldi, and Runpu Zong.

## Combinatorics Seminar

### Blowup Algebras of Rational Normal Scrolls

Monday, September 19, 2016, 5:00pm
Ungar Room 402

Abstract: The Rees ring and the special fiber ring of a polynomial ideal I, also known as the blowup algebras of I, play an important role in commutative algebra and algebraic geometry. A central problem is to describe the defining equations of these algebras. I will discuss the solution to this problem when I is the homogeneous ideal of a rational normal scroll.

## Geometry and Physics Seminar

### Asymmetric L-space Knots

Wednesday, September 14, 2016, 5:00pm
Ungar Room 402

Abstract: An L-space is a rational homology 3-sphere for which the cardinality of its first homology is the rank of its Heegaard Floer homology. It's been conjectured that the symmetry group of any knot in S^3 with a non-trivial surgery to an L-space contains an involution. We demonstrate that this is false through a general construction of the first examples of such knots for which the symmetry group is trivial.

## Applied Math Seminar

### Reaction-diffusion Models with Individual Movement Response to Habitat Edges

Tuesday, September 13, 2016, 12:30pm
Ungar Room 411

Abstract: How the interplay between population growth and individual movement behavior determines large scale spread and persistence in heterogeneous landscapes has been one of the central points in ecological research. Reaction-diffusion models that take into account habitat preference of individuals as a movement bias at the interface between two habitat types have been recently derived. Ovaskainen and Cornell (2003) showed that in such cases population density across the interface between patches should be discontinuous, both the level of bias and the ratio between diffusivities determining the jump in density. In this talk, I will show results for the persistence and rates of spatial spread in reaction-diffusion models that include movement response to habitat edges in patchy environments.

## Combinatorics Seminar

### Random Toric Surfaces and a Threshold for Smoothness

Monday, September 12, 2016, 5:00pm
Ungar Room 402

Abstract: I will present a construction of a random toric surface inspired by the construction of a random graph. With this construction we show a threshold result for smoothness of the surface. The hope is that this inspires further application of randomness to Algebraic Geometry. This talk does not require any background in Algebraic Geometry or Toric Geometry.

## Geometry and Physics Seminar

### Categories and Filtrations

Wednesday, September 7, 2016, 5:00pm
Ungar Room 402

## Geometry and Physics Seminar

### String Topology: Chain Level Transversality and Algebraic Models

Wednesday, August 31, 2016, 5:00pm
Ungar Room 402

Abstract: I will start by describing the construction of several string topology operations; these are transversal intersection type operations on the homology of the free loop space of a manifold. I will proceed with an outline of a framework (joint work with Dingyu Yang (IAS)) which allows us to work with these operations at the chain level. I will focus on a "secondary" coproduct operation for which the transversality is much more subtle than the original intersection product introduced by Chas and Sullivan. I will also explain how these operations are related with the algebraic theory of operations on Hochschild homology/cohomology of Frobenius algebras.

## Applied Math Seminar

### Spatial Population Models with Fitness Based Dispersal

Tuesday, August 30, 2016, 12:30pm
Ungar Room 411

Abstract: Traditional continuous time models in spatial ecology typically describe movement in terms of linear diffusion and advection, which combine with nonlinear population dynamics to produce semilinear equations and systems. However, if organisms are assumed to move up gradients of their reproductive fitness, and fitness is density dependent (for example logistic), the resulting models are quasilinear and may have other novel features. This talk will describe some models involving fitness dependent dispersal and some results and challenges in the analysis of such models.

## Combinatorics Seminar

### A Characterization of Simplicial Manifolds with g2 ≤ 2

Monday, August 29, 2016, 5:00pm
Ungar Room 402

Abstract: The celebrated low bound theorem states that any simplicial manifold of dimension ≥ 3 satisfies g2 ≥ 0, and equality holds if and only if it is a stacked sphere. Furthermore, more recently, the class of all simplicial spheres with g2 = 1 was characterized by Nevo and Novinsky, by an argument based on rigidity theory for graphs. In this talk, I will first define three different retriangulations of simplicial complexes that preserve the homeomorphism type. Then I will show that all simplicial manifolds with g2 ≤ 2 can be obtained by retriangulating a polytopal sphere with a smaller g2. This implies Nevo and Novinsky's result for simplicial spheres of dimension ≥ 4. More surprisingly, it also implies that all simplicial manifolds with g2 = 2 are polytopal spheres.

## Geometry and Physics Seminar

### Tautness of Foliations

Wednesday, August 24, 2016, 5:00pm
Ungar Room 402

Abstract: I will give a brief overview of the role tautness plays in the study of foliations of 3-manifolds. Elementary examples will be constructed to show that notions of tautness that are equivalent for fairly smooth foliations are not equivalent in the world of less smooth foliations. These examples of "phantom tori" have implications in the study of approximations of taut foliations by contact structures. This is joint work with Rachel Roberts.

## Partial Differential Equations Seminar

### Regularity Properties of Solutions of PDE and Systems

Tuesday, June 28, 2016, 3:00pm
Ungar Room 411

View Abstract

and

### Regularity Results for Some Classes of Higher Order Nonlinear Elliptic Systems

Tuesday, June 28, 2016, 3:45pm
Ungar Room 411

View Abstract

## Probability Seminar

### Quantum Feedback Control and Filtering Problem

Thursday, May 5, 2016, 3:30pm
Ungar Room 406

Abstract: The ability to control quantum systems is becoming an essential step towards emerging technologies such as quantum computation, quantum cryptography and high precision metrology. In this talk, we consider a controlled quantum system whose finite dimensional state is governed by a discrete-time nonlinear Markov process. By assuming the quantum non-demolition (QND) measurements in open-loop, we construct a strict control Lyapunov function which is based on the open-loop stationary states. We propose a measurement-based feedback scheme which ensures the almost sure convergence towards a target state. Moreover, I discuss the estimation and filtering problem for continuous-time quantum systems which are described by continuous-time stochastic master equations.

## PDE and Geometric Analysis Seminar

### Mathematics of Gravitational Waves

Tuesday, April 26, 2016, 2:30pm
Ungar Room 506

Abstract: I will review the mathematical issues arising when attempting to describe gravitational radiation.

## Combinatorics Seminar

### Weak Separation, Pure Domains and Cluster Distance

Monday, April 25, 2016, 5:00pm
Ungar Room 402

Abstract: Following the proof of the purity conjecture for weakly separated sets, recent years have revealed a variety of wider classes of pure domains in different settings. In this paper we prove the purity for domains consisting of sets that are weakly separated from a pair of "generic" sets I and J. Our proof also gives a simple formula for the rank of these domains in terms of I and J. This is a new instance of the purity phenomenon which essentially differs from all previously known pure domains. We apply our result to calculate the cluster distance and to give lower bounds on the mutation distance between cluster variables in the cluster algebra structure on the coordinate ring of the Grassmannian. Using a linear projection that relates weak separation to the octahedron recurrence, we also find the exact mutation distances and cluster distances for a family of cluster variables.

This is a joint work with Pavel Galashin.

## Probability Seminar

### Statistics of Nonlinear Biochemical Reaction Networks in Living Cells

Thursday, April 21, 2016, 3:00pm
Ungar Room 406

Abstract: Chemical reactions within cells involve sequences of random events among small numbers of interacting molecules. As a consequence, biochemical reaction networks are extremely noisy. These reactions are also non-linear, making analytical treatment of these systems difficult. I will present a method for approximating the statistics of molecular species in arbitrarily connected networks of non-linear biochemical reactions in small volumes, which I validate with stochastic simulations. I demonstrate that noise slow flux through biochemical networks with nonlinear reaction kinetics, with implications for the evolution of robustness in living cells.

## Applied Math Seminar

### Structural and Practical Identifiability Issues in an Immuno-epidemiological Model

Wednesday, April 20, 2016, 5:00pm
Ungar Room 506

Abstract: In this talk, I will present a mathematical model that links immunological model and epidemiological model. This model allows us to understand dynamical interplay of infectious diseases at two different scales; immunological response of the host at individual scale and the disease dynamics at population scale. Once the host is infected, it triggers the immune response which produces antigen-specific antibodies to clear the pathogen. The pathogen and antibody levels are often monitored in laboratory experiments. Can we use the data generated in the laboratory experiments to identify the parameters of the immunological model? Clearly, the parameters of the within-host immunological model have an effect on the epidemiological characteristics of disease such as reproduction number and prevalence. Epidemiological data is also available for the epidemiological model. I will present the both structural and practical identifiability issues in parameter estimation of the immuno-epidemiological model.

## Geometry and Physics Seminar

### Symmetric Differentials on Projective Varieties

Wednesday, April 20, 2016, 5:00pm
Ungar Room 402

Abstract: A symmetric differential on a complex variety is a section of a symmetric power of the cotangent bundle. The existence of non-trivial symmetric differentials is related to the topological properties of the variety, implying that the fundamental group is large in a suitable sense. I will review some recent results on symmetric differentials, and describe a necessary and sufficient condition for a symmetric differential of rank three on a complex surface to be expressible as a product of closed holomorphic 1-forms.

Joint work with Federico Buonerba.

## Combinatorics Seminar

### The Chromatic Quasisymmetric Function of the Cycle

Monday, April 18, 2016, 5:00pm
Ungar Room 402

Abstract: Chromatic quasisymmetric functions were introduced by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric functions. The results of Shareshian and Wachs focus primarily on incomparability graphs of natural unit interval orders. In this talk I will present my recent work on the chromatic quasisymmetric functions of other graphs, specifically the n-cycle, as well as a generalization of the n-cycle. I will give expansions of the chromatic quasisymmetric functions for these graphs in terms of Gessel's fundamental quasisymmetric basis and in terms of the power sum basis and see how these expansions compare to those obtained by Shareshian-Wachs and Athanasiadis.

## Geometry and Physics Seminar

### K3 Fibered Calabi-Yau Threefolds

Wednesday, April 13, 2016, 5:00pm
Ungar Room 402

Abstract: I will describe some classification results on specific K3 fibered Calabi-Yau threefolds and how these results possibly fit in the context of mirror symmetry.

## Applied Math Seminar

### Lyme Disease, Rash and Mathematics

Wednesday, April 13, 2016, 5:00pm
Ungar Room 506

Abstract: In this talk, I will introduce the basic principles behind mathematical immunology. Then we will discuss a few temporal and spatio-temporal models that describe how our immune system responds to various pathogens. We also establish a PDE chemotaxis model for the innate response to Borrelia burgdorferi, the causative agent of Lyme disease. We illustrate the key factors that lead to the characteristic skin rash that is often associated with Lyme disease. We finish the talk with a few comments regarding modeling in immunology.

## Applied Math Seminar

### Noise-induced Slowdown in Biochemical Reaction Networks

Wednesday, April 6, 2016, 5:00pm
Ungar Room 506

Abstract: Chemical reactions within cells are inherently noisy, but little is known about whether noise affects cell function or, if so, how and by how much. In my talk I will demonstrate that noise causes a quantifiable loss in cell fitness by slowing the average rate of biosynthesis. I present an analytical framework for solving the steady-state statistics of molecular species in arbitrarily connected networks of non-linear chemical reactions in mesoscopic volumes, which I validate with extensive stochastic simulations. In general, I find that reactions obeying hyperbolic kinetics, including Michaelis-Menten reactions, experience "hyperbolic filtering" which attenuates high amplitude substrate fluctuations leading to 1) reduced average reaction flux and 2) reduced product noise. Gene expression is particularly sensitive to noise-induced slowdown because, unlike metabolic reactions, translation is not buffered by network-level feedback. Additionally, I find that translation propagates less noise than a kinetically equivalent metabolic reaction, that mRNA noise directly reduces fitness by reducing the average translation rate, and that increasing mRNA/ribosome binding affinity actually reduces protein noise even though it increases translational bursting. These phenomena are missing from previous stochastic analyses of gene expression because translation is typically modeled as linear, rather than hyperbolic.

## Combinatorics Seminar

### New and Old Graph Dimension Parameters

Monday, April 4, 2016, 5:00pm
Ungar Room 402

Abstract: In this talk, we will explore various intersection and containment based representations of graphs and posets along with their associated parameters. Among these are the boxicity and cubicity of graphs, the dimension and interval dimension of posets and their comparability graphs, the bending number of intersecting paths on a grid, and the grid dimension of a graph.

We will also present recent work on the new notions of the separation dimension of a graph and the induced separation dimension of a graph. One of our main aims has been to find significant interconnections between such dimensional parameters. For example, we establish bounds relating the bending number to the partial order dimension for co-comparability graphs, and relating the induced separation dimension with the separation dimension and boxicity.

## Applied Math Seminar

### Modeling the Dynamics of Woody Plant-Snowshoe Hare Interactions with Twig Age-dependent Toxicity

Wednesday, March 30, 2016, 5:00pm
Ungar Room 506

Abstract: Modeling is used to study the effects that woody plant chemical defenses may have on population dynamics of boreal hares that feed almost entirely on twigs during the winter. The model takes into account that toxin concentration often varies with the age of twig segments. In particular, it incorporates the fact that the woody internodes of the youngest segments of the twigs of the deciduous angiosperm species that these hares prefer to eat are more defended by toxins than the woody internodes of the older segments that subtend and support the younger segments. Thus, the per capita daily intake of the biomass of the older segments of twigs by hares is much higher than their intake of the biomass of the younger segments of twigs. This age-dependent toxicity of twig segments is modeled using age-structured model equations, which are reduced to a system of delay differential equations involving multiple delays in the woody plant–hare dynamics. A novel aspect of the modeling was that it had to account for mortality of non-consumed younger twig segment biomass when older twig biomass was bitten off and consumed. Basic mathematical properties of the model are established together with upper and lower bounds on the solutions. Necessary and sufficient conditions are found for the linear stability of the equilibrium in which the hare is extinct, and sufficient conditions are found for the global stability of this equilibrium. Numerical simulations confirmed the analytical results and demonstrated the existence of limit cycles over ranges of parameters reasonable for hares browsing on woody vegetation in boreal ecosystems. This showed that age dependence in plant chemical defenses has the capacity to cause hare-plant population cycles, a new result.

## Geometry and Physics Seminar

### On Foliations Related to the Center of Mass in General Relativity

Wednesday, March 30, 2016, 5:00pm
Ungar Room 402

Abstract: In many situations in Newtonian gravity, understanding the motion of the center of mass of a system is key to understanding the general trend of the motion of the system. It is thus desirable to also devise a notion of center of mass with similar properties in general relativity.

However, while the definition of the center of mass via the mass density is straightforward in Newtonian gravity, there is a priori no definitive corresponding notion in general relativity. We will pursue a geometric approach to defining the center of mass, using foliations by hypersurfaces with specific geometric and physical properties. I will first illustrate this approach in the (easier) Newtonian setting and then review previous work in the relativistic situation, most prominently a fundamental result by Huisken and Yau from 1996. After introducing the foliation approach to defining the center of mass, I will discuss explicit counter-examples (partially joint work with Nerz) and discuss the analytic, geometric, and physical issues they illustrate. I will then present a new approach (joint work with Cortier and Sakovich) that remedies these issues.

## Combinatorics Seminar

### The Combinatorics of the Waldspurger Decomposition

Monday, March 28, 2016, 5:00pm
Ungar Room 402

Abstract: In 2005 J.L. Waldspurger proved a remarkable theorem. Given a finite reflection group G, the closed cone over the positive roots is equal to the disjoint union of images of the open weight cone under the action of 1-g. When G is taken to be the symmetric group the decomposition is related to the familiar combinatorics of permutations but also has some surprising features. To see this, we give a nice combinatorial description of the decomposition.

The decomposition is not a simplicial, or even CW complex and attempts to complete it to one are problematic. It does, however, define a dual graph on n-cycles. We prove some basic facts about this graph and state a few conjectures and open problems relating to it.

## Probability Seminar

### On the Volatile Correlation of Two Independent Wiener Processes

Wednesday, March 23, 2016, 3:30pm
Ungar Room 406

Abstract: In this paper, we resolve a longstanding open statistical problem. The problem is to analytically determine the second moment of the empirical correlation coefficient \beqn \theta := \frac{\int_0^1W_1(t)W_2(t) dt - \int_0^1W_1(t) dt \int_0^1 W_2(t) dt}{\sqrt{\int_0^1 W^2_1(t) dt - \parens{\int_0^1W_1(t) dt}^2} \sqrt{\int_0^1 W^2_2(t) dt - \parens{\int_0^1W_2(t) dt}^2}} \eeqn of two {\em independent} Wiener processes, $W_1,W_2$. Using tools from Fredholm integral equation theory, we successfully calculate the second moment of $\theta$ to be .240522. This gives a value for the standard deviation of $\theta$ of nearly .5. As such, we are the first to offer formal proof that two Brownian motions may be independent and yet can also be highly correlated with significant probability. This spurious correlation, unrelated to a third variable, is induced because each Wiener process is self-correlated'' in time. This is because a Wiener process is an integral of pure noise and thus its values at different time points are correlated. In addition to providing an explicit formula for the second moment of $\theta$, we offer implicit formulas for higher moments of $\theta$.

## Combinatorics Seminar

### Higher Dimensional Colorings and Flows, Arithmetic Tutte Polynomials, and Convolution Formulae

Monday, March 21, 2016, 5:00pm
Ungar Room 402

Abstract: In a recent series of papers by various authors, the theory of colorings and flows on graphs has been extended to the higher-dimensional case of CW complexes. We will survey this theory and show how the arithmetic Tutte polynomial naturally comes into play. (Joint work with E. Delucchi).

After recalling the basic properties of this polynomial, we will show some convolution formulae and their applications to the case of CW complexes (ongoing joint work with S. Backman, A. Fink and M. Lenz).

## PDE and Geometric Analysis Seminar

### Weird Symmetry, Nice Space

Thursday, March 17, 2016, 5:00pm
Ungar Room 411

Abstract: We will talk about joint work with Ruberman, Melvin and Kim establishing that there are nice geometric structures with weird symmetry groups. In particular we will prove that every $3$-manifold is invertible, equivariantly, $Z[\pi]$-homology cobordant to a hyperbolic manifold.

## Combinatorics Seminar

### Toric Arrangements and Group Actions on Semimatroids

Thursday, March 17, 2016, 5:00pm
Ungar Room 402

Abstract: Recent work of De Concini, Procesi and Vergne on vector partition functions gave a fresh impulse to the study of toric arrangements from an algebraic, topological and combinatorial point of view. In this context, many new combinatorial structures have recently appeared in the literature, each tailored to one of the different facets of the subject. Yet, a comprehensive combinatorial framework is lacking. As a unifying structure, in this talk I will propose the study of group actions on semimatroids and of related polynomial invariants, recently introduced in joint work with Sonja Riedel. In particular, I will outline some new open problems brought to the fore by this new point of view.

## Probability Seminar

### Asymptotic Behaviour of Some Locally-interacting Processes

Thursday, March 17, 2016, 3:30pm
Ungar Room 406

Abstract: We study locally-interacting birth-and-death processes on nodes of a finite connected graph; the model which is motivated by modelling interactions between populations, adsorption-desorption processes, and is related to interacting particle systems, Gibbs models, and interactive urn models.

Alongside with general results, we obtain a more detailed description of the asymptotic behaviour in the case of certain special graphs.

Based on a joint work with Vadim Scherbakov (Royal Holloway, University of London).

## Applied Math Seminar

### Chaos in the Plankton

Wednesday, March 16, 2016, 5:00pm
Ungar Room 506

Abstract: The free floating communities in the sun lit surface of the ocean, the euphotic zone, are supported by photosynthesis in the phytoplankton (P). The lower reaches of this zone are bound by the compensation depth where the photosynthetic fixation of carbon in the phytoplankton just meet their own respiratory requirements. In simple models that include P with a nutrient (N) and either a grazer (Z) or a dissolved organic pool (D) without vertical movement or diffusion, this level is characterized by a double infinity. The solutions, however, reproduce the near surface and deep nutrient and phytoplankton curves. It is the underlying nature of the upper solution going to +infinity and the lower solution to -infinity that underlie the dynamics of the lower portion of the euphotic zone. The infinities can be eliminated by adding either sinking in P or by diffusion in N. The fixed points of the system, however, still exchange stabilities at the compensation depth leading to chaotic states. Addition of more complete terms with more phytoplankton (Pi) or zooplankton grazers only lead to more complete dynamics. Observations of the deep chlorophyll maximum are in agreement with the idea that the bottom of the euphotic zone is fundamentally chaotic.

## Geometry and Physics Seminar

### Neighbors of Knots in the Gordian Graph

Wednesday, March 16, 2016, 5:00pm
Ungar Room 402

Abstract: The Gordian graph is the graph with vertex set the set of knot types and edge set consisting of pairs of knots which have a diagram wherein they differ at a single crossing. Bridge number is a classical knot invariant which is a measure of the complexity of a knot. It can be refined by another, recently discovered, knot invariant known as "bridge distance". We show, using arguments that are almost entirely elementary, that each vertex of the Gordian graph is adjacent to a vertex having arbitrarily high bridge number and bridge distance. This is joint work with Ryan Blair, Marion Campisi, Jesse Johnson, and Maggy Tomova.

## Combinatorics Seminar

### Evaluations of the Power Sum Traces at Kazhdan-Lusztig Basis Elements of the Hecke Algebra

Monday, March 14, 2016, 4:00pm
Ungar Room 402

Abstract: In 1993, Haiman studied certain functions from the Hecke algebra to Z[q] called monomial traces. He conjectured that the evaluation of these at Kazhdan-Lusztig basis elements resulted in polynomials in N[q]. A weakening of this conjecture is that the evaluations of other traces, called power sum traces, results in polynomials in N[q]. We will discuss several combinatorial interpretations of these polynomials for Kazhdan-Lusztig basis elements indexed by permutations which avoid the patterns 3412 and 4231. These interpretations come from joint work with Matthew Hyatt, and results of Athanasiadis, Shareshian, and Wachs.

## Geometry and Physics Seminar

### Degenerations of Hodge Structure

Wednesday, March 2, 2016, 5:00pm
Ungar Room 402

Abstract: I will explain how a polarized limiting mixed Hodge structure (PLMHS) may be viewed as a degeneration of a (pure, polarized) Hodge structure. There is a notion of "polarized relation" between PLMHS that encodes information on how varieties may degenerate within a family. I will give a classification of PLMHS and their polarized relations in terms of Hodge diamonds (discrete data associated with a PLMHS).

## Applied Math Seminar

### Resident-Invader Dynamics in Infinite Dimensional Dynamical Systems

Wednesday, March 2, 2016, 5:00pm
Ungar Room 506

Abstract: We discuss an extension of the Tube Theorem from adaptive dynamics to infinite dimensional contexts, including that of reaction-diffusion equations. This is joint work with Chris Cosner and King-Leung Lam.

## Combinatorics Seminar

### Real Stable Polynomials, Determinants, and Matroids

Monday, February 29, 2016, 4:00pm
Ungar Room 402

Abstract: Real stable polynomials define real hypersurfaces that are maximally nested ovaloids. These polynomials appear in many areas of mathematics, including optimization, combinatorics and differential equations. In 2004, Choe, Oxley, Sokal and Wagner established a tight connection between matroids and multiaffine real stable polynomials. Branden recently used this theory and the Vamos matroid to disprove the generalized Lax conjecture, which concerns representing polynomials as determinants. I will discuss the fascinating connections between these fields and some extensions to some varieties associated to hyperplane arrangements.

## Geometry and Physics Seminar

### Causal R-actions and the Rigidity of Brinkmann Spaces

Wednesday, February 24, 2016, 5:00pm
Ungar Room 402

Abstract: We study the interplay between the global causal and geometric structures of a spacetime (M,g) and the features of a given smooth R-action on M whose orbits are all causal curves, building on classic results about Lie group actions on manifolds by Palais. Although the dynamics of such an action can be very hard to describe in general, simple restrictions on the causal structure of (M,g) can simplify this dynamics dramatically. We show how this can in turn be used in some cases to constrain the global geometry of (M,g), illustrating this fact by obtaining a rigidity result for the so-called Brinkmann spacetimes.

## Combinatorics Seminar

### A Broad Class of Shellable Lattices

Monday, February 22, 2016, 5:00pm
Ungar Room 402

Abstract: Jay Schweig and I recently discovered a large class of shellable lattices. Our original motivation were the order congruence lattices of finite posets. Afterwards, we noticed that the subgroup lattices of solvable groups are also contained in the class, and indeed, the definition may be seen as a lattice-theoretic abstraction of solvable groups.

In this talk, I'll review some of the theory of supersolvable lattices, and show how to extend similar ideas to our class of lattices.

## Applied Math Seminar

### From Malaria to Zika:Modeling the Transmission Dynamics and Control of Mosquito-Borne Diseases

Wednesday, February 17, 2016, 5:00pm
Ungar Room 506

Abstract: Mathematical modeling is a powerful and important tool in studying the transmission dynamics and control of many infectious diseases including vector-borne diseases (such as malaria, West Nile virus, dengue, chikungunya). In this talk I will use malaria as an example to introduce some basic concepts and models in mathematical epidemiology. Then I will review some basic features about Zika virus and the status of the ongoing Zika epidemic. Finally I will briefly talk about our preliminary results on modeling the spread and control of Zika virus infection.

## Geometry and Physics Seminar

### Asymptotically Poincaré-Einstein Metrics

Wednesday, February 17, 2016, 5:00pm
Ungar Room 402

Abstract: I will discuss asymptotically hyperbolic manifolds and sub-types, especially Asymptotically Poincaré-Einstein (APE) manifolds. These are manifolds whose metrics admit a Fefferman-Graham type expansion near conformal infinity, such that the first few leading terms are determined by the Einstein equations. APEs appear both as time slices of asymptotically Anti-de Sitter (AdS) spacetimes and as Wick rotations of static AdS spacetimes. Among the invariants that can (sometimes) be associated to APEs are mass and renormalized volume. For APEs that represent static AdS black holes in 4 dimensions, the renormalized volume and the mass of a static slice can be related. Remarkably, this relationship, which can be established purely on geometric grounds, is the free energy relation of black hole thermodynamics. A related result is that APEs which are time-symmetric slices of globally static AdS spacetimes cannot have positive mass. Examples are provided by Horowitz-Myers geons, which are complete APEs with scalar curvature $S=-n(n-1)$ and negative mass. They serve as time-symmetric slices of so-called AdS solitons, which are Einstein spacetimes with toroidal conformal infinity. The Horowitz-Myers "positive" energy conjecture proposes that Horowitz-Myers geons minimize the mass over all APEs with scalar curvature $S\ge -n(n-1)$ and the same toroidal conformal infinity. The conjecture is an open problem, but it is possible to show that if this conjecture is true then there is a rigidity theorem. Mass minimizing metrics must be static Einstein metrics and, for $n=3$ spatial dimensions, the least-mass geon is the unique complete APE having that mass and obeying $S\ge -n(n-1)$.

## Combinatorics Seminar

### Dual Graphs and the Castelnuovo-Mumford Regularity of Subspace Arrangements

Monday, February 15, 2016, 5:00pm
Ungar Room 402

Abstract: In the talk I will discuss an ongoing joint work with Bruno Benedetti and Michela Di Marca on the Castelnuovo-Mumford regularity of subspace arrangements. The Castelnuovo-Mumford regularity of an embedded projective variety is an important invariant measuring its complexity. For a Gorenstein subspace arrangement, it turns out that this invariant has an amazingly simple description in terms of the dual graph of the arrangement. The goal of the talk is to discuss the concepts of dual graph (keeping in mind the motivating case of simplicial complexes) and Castelnuovo-Mumford regularity, to explain the connection we found out for Gorenstein subspace arrangements, and to provide several examples.

## Geometry and Physics Seminar

### Geometry of Nilpotent Cones in Hodge Theory

Wednesday, February 10, 2016, 5:00pm
Ungar Room 402

Abstract: The local monodromy of a degeneration of smooth complex projective varieties gives rise to a monodromy cone which plays a central role in constructing analogs of Mumford's toroidal compactifications for Hodge structures of arbitrary weight. In this talk, I will describe several methods for describing the possible monodromy cones which can arise in a given period domain using topological boundary components and signed Young diagrams.

## Combinatorics Seminar

### Geometric Grid Classes and Symmetric Sets of Permutations

Tuesday, February 9, 2016, 5:00pm
Ungar Room 402

Abstract: Characterizing sets of permutations whose associated quasisymmetric function is symmetric is a long-standing problem in algebraic combinatorics. We present a general method to construct symmetric and Schur-positive sets and multisets, based on algebraic and geometric operations on grid classes. This approach produces new instances of Schur-positive sets and explains the existence of known such sets that until now were sporadic cases.

Joint with Sergi Elizalde.

## Applied Math Seminar

### Evolution of Dispersal in Spatial Population Models with Multiple Timescales

Thursday, February 4, 2016, 4:00pm
Ungar Room 506

Abstract: In many cases the timescale of dispersal for the organisms is sufficiently fast compared to timescale of their population dynamics that it is reasonable to assume the spatial distribution of the organisms is always effectively at equilibrium when viewed on the timescale of population dynamics. Starting from that assumption it is possible to construct models for population dynamics and species interaction that are based on ordinary differential equations but still contain information about the how the dispersal strategies being used by the focal populations interact with environmental heterogeneity. This talk will describe the modeling approach and discuss some applications that are related to the evolution of dispersal.

## Probability Seminar

### Contribution Restrictions in a Global Game

Thursday, February 4, 2016, 3:30pm
Ungar Room 406

Abstract: We investigate a manager's decision to restrict the contribution of an agent with private information. We explore the link between this problem and contribution restrictions in global games with a continuum of agents and continuous actions.

## Geometry and Physics Seminar

### From the Point of View of ModuliWhat Are the Most Extreme Examples of Curves and Surfaces?

Wednesday, February 3, 2016, 5:00pm
Ungar Room 402

Abstract: What does Hodge theory looks like for these examples? We will illustrate a general set of problems about the relationship between the boundary of moduli spaces and the boundary of the period domains. This is a joint work with Griffiths, Laza, Robles.

## Combinatorics Seminar

### Parking Functions and their Generating Function

Monday, February 1, 2016, 4:00pm
Ungar Room 402

Abstract: A parking function is a sequence (a_1,...,a_n) of positive integers whose increasing rearrangement (b_1,...,b_n) satisfies b_i \leq i.

We first explain the connection with parking cars and give the elegant proof of Pollak that the number of parking functions of length n is (n+1)^{n-1}. The symmetric group S_n acts on parking functions of length n by permuting coordinates. This action corresponds to a symmetric function F_n with many interesting properties. We discuss properties of the generating function \sum F_n z^n and show its connection with some variants of parking functions. Finally we consider a q-analogue of the previous theory.

## Probability Seminar

### How to Control a Process to a Goal

Thursday, January 28, 2016, 3:30pm
Ungar Room 406

## Combinatorics Seminar

### Boolean Statistics

Monday, January 25, 2016, 5:00pm
Ungar Room 402

Abstract: We introduce a new function b(x) associated to simplicial complexes that measures how versatile you can decompose it into boolean lattices. The function interpolates between three important cases: b(2) is the number of faces; b(1) is a good upper bound on the number of maximal faces; and b(0) is the minimal number of critical cells in a discrete Morse matching, which is a natural upper bound on the total dimension of cohomology. In contemporary graph theory b(1) is central, and in topological combinatorics b(2) is. We try to unify some approaches in these branches of discrete mathematics by this setup and manage for example to prove some new theorems regarding the structure of maximal independent sets in triangle-free graphs, improving on recent work by Balogh et al. In the talk we will also explain why all of the bounds mentioned above are optimal for shellable simplicial complexes.

## Geometry and Physics Seminar

### Categorical Donaldson Uhlenbeck Yau Correspondence

Wednesday, January 20, 2016, 5:00pm
Ungar Room 402

Abstract: In the last two years a parallel between theory of Higgs bundles and theory of shaves of categories was created. We take this parallel to the next level giving classical correspondence a new categorical meaning. Some applications will be discussed.

## Geometry and Physics Seminar

### Introduction to Theory of Algebraic Surfaces

Wednesday, January 13, 2016, 5:00pm
Ungar Room 402

## Geometry and Physics Seminar

### Introduction to Theory of Algebraic Curves

Wednesday, December 9, 2015, 5:00pm
Ungar Room 402

## Probability Seminar

### Stochastic Models of Cancer Cell Growth

Tuesday, December 8, 2015, 4:00pm
Ungar Room 406

Abstract: Recent research suggests that cancer is a genetic disease caused by DNA alterations which induce cells to divide in uncontrolled fashion and become invasive to the organism. In the evolution of the disease, some of these pernicious cells can mutate either because they divide faster or because they become resistant to some particular treatment. Several mathematical models are presented. Is natural to investigate the distribution of the number $Z_0$ of cancer cells present in the human body at a given time and the distribution of the first time $T_m$ for this amount to reach a certain threshold, when the disease can be detected. Additionally, we shall look at $Z_{1}$, the time of the first occurrence of mutation in cancer cells. The basic model proposed is a two-type branching process in which $Z_0$ cells give birth to $Z_1$ type cells. This talk is based on the paper, "Evolutionary dynamics of tumor progression with random fitness values" by Rick Durrett, Jasmine Foo, Kevin Leder, John Mayberry and Franziska, and the survey "Branching Process Models of Cancer" written by Dr. Richard Durrett. Other discussion in reference of the paper "Modeling the Manipulation of Natural Populations by Mutagenic Chain Reaction" by Robert L. Unckless, Philipp W. Messer, Tim Connallon, and Andrew G. Clark may be address as well.

## Combinatorics Seminar

### Chromatic Symmetric Functions on Simplicial Complexes

Monday, December 7, 2015, 5:00pm
Ungar Room 402

Abstract: Inspired by the theory pioneered by Stanley's chromatic symmetric function and its connection to Hopf algebras, we will see how abstract simplicial complexes can be endowed naturally with a combinatorial Hopf structure that gives rise to chromatic generating functions. Using principal specializations on these generating functions we derive certain combinatorial identities involving acyclic orientations related to the complexes. We will also discuss work in progress aiming to derive the analogous of Shareshian-Wachs' "chromatic quasisymmetric functions" for simplicial complexes.

This is joint work with J. Hallam and J. Machacek. No familiarity with Hopf algebras is required, we will give the necessary background.

## Probability Seminar

### Progress Report on the Fluctuation Limit for an AIMD Model

Tuesday, December 1, 2015, 4:00pm
Ungar Room 406

Abstract: We discuss the central limit theorem for the fluctuation random field from the fluid limit (LLN) of an Additive Increase Multiplicative Decrease (AIMD) protocol used in internet traffic modeling. The random field is unusual but can be characterized inductively using polynomial test functions. These are preliminary results and a brief survey of methodology.

## Combinatorics Seminar

### Dual Graphs of Projective Curves

Monday, November 30, 2015, 5:00pm
Ungar Room 402

Abstract: In 1962, Hartshorne proved that the dual graphs of an arithmetically Cohen-Macaulay scheme is connected. After establishing a correspondence between the languages of algebraic geometry, commutative algebra and combinatorics, we are going to refine Hartshorne's result and measure the connectedness of the dual graphs of certain projective schemes in terms of an algebro-geometric invariant of the projective schemes themselves, namely their Castelnuovo-Mumford regularity. Time permitting, we are also going to address briefly the inverse problem of Hartshorne's result, by showing that any connected graph is the dual graph of a projective curve with nice geometric properties.

## Geometry and Physics Seminar

### Non-Archimedean Geometry in Rank >1

Wednesday, November 18, 2015, 5:00pm
Ungar Room 402

Abstract: Recent work of Nisse-Sottile, Hrushovski-Loeser, Ducros, and Giansiracusa-Giansiracusa has demonstrated that valuation rings of rank >1 play an important role in the geometry of analytic and tropical varieties over non-Archimedean valued fields of rank 1. In this talk, I will present recent work with Dhruv Ranganathan in which we prove several foundational results on the geometry of analytic and tropical varieties over higher rank valued fields, and recent work with Max Hully in which we use rank 2 valuations to give a new, non-analytic proof of Rabinoff's theorem on the correspondence between tropical and algebraic intersection multiplicities.

## Probability Seminar

### Reducing the Error-propagation Effect Associated with Stacking Classifiers

Wednesday, November 18, 2015, 3:30pm
Ungar Room 406

Abstract: Multi-label classification targets domains characterized by examples that may belong to more than one category at the same time. A common way to address such problems is to induce a separate classifier for each class. Thus, each classifier determines whether its respective category is relevant for a new example or not. By targeting each class independently, this technique, known as Binary Relevance (BR), assumes that classes are independent of each others, which may not necessarily be the case in all domains. For example, an image of a 'Beach' scene will likely be tagged with the concept 'Ocean' as well. Conversely, the same image is unlikely to represent the topic 'Industry'. To incorporate such class correlations in the BR framework, researchers suggest using information about the classes an example is already known to belong to as inputs in addition to the example's original attributes. Since this information is typically unknown for a previously unseen example, several approaches fill in these values using the outputs of independent classifiers such as those in the BR framework. In real-world scenarios, these outputs are prone to errors. Consequently, using them as inputs may reduce the accuracy of the classifiers. The presented work suggests two ways to address this so-called error-propagation problem. The first method reduces the dependency on the error-prone inputs by eliminating weak class dependencies from the final model. The second proposed solution is to compare the probabilistic classification confidences of the independent models with their dependent counterparts, and then choose the more confident classification. Experiments on a broad set of benchmark datasets indicate that a combination of the two approaches yields a boost in classification accuracy when compared to using the dependent models alone.

## Combinatorics Seminar

### Optimal Discrete Morse Vectors Are Not Unique

Monday, November 16, 2015, 5:00pm
Ungar Room 402

Abstract: In classical Morse theory, for any given manifold there is always a unique optimal Morse vector (=the vector counting the number of critical points of index 0,1,..., up to the dimension). It turns out that in Forman's discrete version of Morse theory, this is no longer the case. I will sketch how to construct a contractible 3-complex on which the 'best' discrete Morse vectors are (1,0,1,1) and (1,1,1,0), because (1,0,0,0) is out of reach.

## Applied Math Seminar

### An Introduction to Pattern Formation in Developmental Biology and Ecology

Friday, November 13, 2015, 4:00pm
Ungar Room 402

Abstract: In 1952, Alan Turing (one of the founders of computer science) proposed a reaction-diffusion model wherein two homogeneously distributed substances, a fast-diffusing inhibitor and a slow-diffusing activator, would interact to produce stable patterns during morphogenesis. These patterns would represent regional differences in the concentrations of the two substances. Their interactions would produce an ordered structure out of random chaos. Turing's model has become one of the most important mathematical models in developmental biology.

In this talk, I will first introduce the spatial dynamics in some reaction-diffusion systems induced by Turing instability and their applications in the formation of skin patterns of some animals and fish. Then I will talk about the spatial, temporal, and spatiotemporal patterns in ecological models, in particular predator-prey models with mutual interference.

## Geometry and Physics Seminar

### On Fillings of the Canonical Contact Structure on the Unit Cotangent Bundle of a Surface

Wednesday, November 11, 2015, 5:00pm
Ungar Room 402

Abstract: The classic example of a contact manifold is the unit sphere bundle of the cotangent bundle of a smooth manifold: $ST^*M$; and the classic example of a symplectic filling is the unit disk bundle of the cotangent bundle $DT^*M.$ It turns out that, using hard applications of holomorphic curves by McDuff, Hind and Wendl, for $Y = S^2$ or $T^2$, these are in fact the only symplectic fillings. Using tools from Li and Mak, we now can extend this list to include all surfaces, at least if we consider classifying symplectic fillings up to homotopy equivalence. This is joint work with Steven Sivek.

## Combinatorics Seminar

### Relaxations of the Matroid Axioms

Tuesday, November 10, 2015, 5:00pm
Ungar Room 402

Abstract: Motivated by a question of Duval and Reiner about eigenvalues of combinatorial Laplacians, we develop various generalisations of (ordered) matroid theory to wider classes of simplicial complexes. In addition to all independence complexes of matroids, each such class contains all pure shifted simplicial complexes, and it retains a little piece of matroidal structure. To achieve this, we relax many cryptomorphic definitions of a matroid. In contrast to the matroid setting, these relaxations are independent of each other, i.e., they produce different extensions. Imposing various combinations of these new axioms allows us to provide analogues of many classical matroid structures and properties. Examples of such properties include the Tutte polynomial, lexicographic shellability of the complex, the existence of a meaningful nbc-complex and its shellability, the Billera-Jia-Reiner quasisymmetric function, and many others. We then discuss the h-vectors of complexes that satisfy our relaxed version of the exchange axiom, extend Stanley's pure O-sequence conjecture about the h-vector of a matroid, solve this conjecture for the special case of shifted complexes, and speculate a bit about the general case. Based on joint works with Jeremy Martin, Ernest Chong and Steven Klee.

## Applied Math Seminar

### Hydrodynamic Limit for a Supercritical Branching Process

Friday, November 6, 2015, 4:00pm
Ungar Room 402

Abstract: In 1993, Bak and Sneppen proposed a model aiming to describe an ecosystem of interacting species that evolve by mutation and natural selection. Thereafter various mathematical attempts have been made to study the model in its equilibrium. In this talk we'll investigate a variant of the Bak-Sneppen model and its hydrodynamic limit. The solution solves a heat equation with mass creation at a source inside the domain, normalized to have mass one. We discuss its representation as the average of the empirical measure of an auxiliary branching system with mass growing exponentially fast and the relationship between the stationary measure and quasi-stationarity for the auxiliary semigroup.

## Probability Seminar

### Shortest-weight Paths in Random Graphs

Tuesday, November 3, 2015, 4:00pm
Ungar Room 406

Abstract: We study the impact of random exponential edge weights on the distances in a random graph and, in particular, on its diameter. Our main result consists of a precise asymptotic expression for the maximal weight of the shortest weight paths between all vertices (the weighted diameter) of sparse random graphs, when the edge weights are iid exponential random variables. This is based on a joint work with Marc Lelarge.

## Combinatorics Seminar

### Power Sum Expansion of the Chromatic Quasisymmetric Functions

Monday, November 2, 2015, 5:00pm
Ungar Room 402

Abstract: Shareshian and Wachs introduced the chromatic quasisymmetric function of a graph as a refinement of Stanley’s chromatic symmetric function. In their paper, Shareshian and Wachs conjecture a formula for the expansion of the chromatic quasisymmetric function of incomparability graphs of natural unit interval orders in the power sum basis. Recently, Athanasiadis proved the conjecture by using a formula of Roichman for the irreducible characters of the symmetric group. In this talk, we will present Athanasiadis' work.

## Geometry and Physics Seminar

### A Cup Product Lemma, Bounded Geometry, and the Bochner-Hartogs Dichotomy

Wednesday, October 28, 2015, 5:00pm
Ungar Room 402

Abstract: We will consider a version of Gromov's cup product lemma in which one factor is the (1,0)-part of the differential of a continuous pluri-subharmonic function, as well as consequences for the structure of complete Kaehler manifolds with bounded geometry along levels of suitable plurisubharmonic functions. For example, we will see that the Bochner-Hartogs dichotomy holds for any connected one-ended covering of a weakly 1-complete non-compact complete Kaehler manifold; that is, either the first compactly supported cohomology with values in the structure sheaf vanishes, or there exists a proper holomorphic mapping onto a Riemann surface.

## Probability Seminar

### Quasi-stationarity and the Fleming-Viot Particle System

Tuesday, October 27, 2015, 4:00pm
Ungar Room 406

Abstract: We discuss a general class of stochastic processes obtained from a given Markov process whose behavior is modified upon contact with a catalyst, from the perspective of a particle system that undergoes branching with conservation of mass (Fleming-Viot mechanism). We explain the relation of the process and its scaling limit to the existence of quasi-stationary distributions and their simulation. Non-explosion and large deviations for the soft catalyst case will be discussed if time permits. Joint work with Min Kang.

## Combinatorics Seminar

### A Non-partitionable Cohen-Macaulay Complex

Monday, October 26, 2015, 5:00pm
Ungar Room 402

Abstract: In joint work with Art Duval, Caroline Klivans, and Jeremy Martin, we construct a non-partitionable Cohen-Macaulay simplicial complex. This construction disproves a longstanding conjecture by Stanley that would have provided an interpretation of h-vectors of Cohen-Macaulay complexes. Due to an earlier result of Herzog, Jahan, and Yassemi, this construction also disproves the conjecture that Stanley depth is always greater than or equal to depth. Time permitting, we will also discuss Garsia's open conjecture that every Cohen-Macaulay poset has a partitionable order complex.

## Applied Math Seminar

### Effects of Diffusion on Total Biomass in Heterogeneous Continuous and Discrete-Patch Systems

Friday, October 23, 2015, 4:00pm
Ungar Room 402

Abstract: Theoretical models of populations on a system of two connected patches have shown that, when the two patches differ in maximum growth rate and carrying capacity, and in the limit of high diffusion, conditions exist for which the total population size at equilibrium exceeds that of the Ideal Free Distribution, which predicts that the total population would equal the total carrying capacity of the two patches. However, this result has only been shown for the Pearl-Verhulst growth function on two patches and for a single-parameter growth function in continuous space. Here we provide a general criterion for total population size, exceeding total carrying capacity for three commonly used population growth rates for both heterogeneous continuous and multi-patch heterogeneous landscapes with high population diffusion. We show that a necessary condition for this situation is that there is a convex positive relationship between the parameter for the maximum growth rate and the carrying capacity, as both vary across a spatial region. Because this relationship occurs in biological populations, the result has ecological implications.

## Applied Math Seminar

### Evolution of Dispersal in Spatially Heterogeneous Temporally Constant Habitats

Friday, October 16, 2015, 4:00pm
Ungar Room 402

Abstract: In this talk we survey some recent results on the evolution of dispersal in spatially heterogeneous but temporally constant environments. We focus on the evolutionary advantage that arises from moving so as to match underlying heterogeneous resource patterns. Collaborators on this program of study include Lee Altenberg, Chris Cosner, Yuan Lou, Dan Ryan, Mark Lewis, Sebastian Schreiber and King-Yeung Lam.

## Probability Seminar

### A Game Theoretic Approach to Modeling Debt Capacity

Tuesday, October 13, 2015, 4:00pm
Ungar Room 406

Abstract: We propose a dynamic model that explains the build-up of short term debt when the creditors are strategic and have different beliefs about the prospects of the borrowers' fundamentals. We define a dynamic game among creditors, whose outcome is the short term debt. As common in the literature, this game features multiple Nash equilibria. We give a refinement of the Nash equilibrium concept that leads to a unique equilibrium.

For the resulting debt-to-asset process of the borrower we define a notion of stability and find the debt ceiling which marks the point when the borrower becomes illiquid. We show existence of early warning signals of bank runs: a bank run begins when the debt-to-asset process leaves the stability region and becomes a mean-fleeing sub-martingale with tendency to reach the debt ceiling. Our results are robust across a wide variety of specifications for the distribution of the capital across creditors' beliefs. (joint with J Wissel)

## Combinatorics Seminar

### Arrangements of Subspaces for Finite Groups and Their Geometrical Applications

Monday, October 12, 2015, 5:00pm
Ungar Room 402

Abstract: Given a faithful representation V of a group G one can consider the partially ordered set of conjugacy classes of stabilizer subgroups. Using this combinatorial object we proved that the motivic class of the classifying stack of every finite linear (or projective) reflection group is trivial.

This poset is a key combinatorial tool also in the study of the motivic class of the quotient variety U/G, where U is the open set of V where the group acts trivially. We discuss the study of such classes by starting from a theorem of Aluffi in the reflection groups case and we conclude by showing that a similar result holds for finite subgroups of GL_3(k) for an algebraically closed field of characteristic zero.

These results relate naturally to Noether's Problem and to its obstruction, the Bogomolov multiplier.

(Part of this is joint work with Emanuele Delucchi.)

## Combinatorics Seminar

### Tverberg-type Theorems and Zero Sum Problems

Monday, October 5, 2015, 5:00pm
Ungar Room 402

Abstract: Tverberg-type theorems are concerned with the intersection pattern of faces in a simplicial complex when mapped to Euclidean space. One has to distinguish between results for affine maps (with straight faces) and continuous maps: In the topological case, number-theoretic conditions on the multiplicity of intersections play a role. We will show that most Tverberg-type results, which were believed to require proofs using involved techniques from algebraic topology, follow from a simple combinatorial reduction via the pigeonhole principle. We will construct counterexamples to the topological Tverberg conjecture by Bárány from 1976 building on recent work of Mabillard and Wagner, and we will apply similar ideas to investigate zero-sum problems in Euclidean space. Joint work with Pavle Blagojevic and Günter M. Ziegler.

## Geometry and Physics Seminar

### Mirror Symmetry for C^2 - Point

Thursday, October 1, 2015, 5:00pm
Ungar Room 402

## Geometry and Physics Seminar

### Berkovich Spaces and Birational Geometry

Wednesday, September 30, 2015, 5:00pm
Ungar Room 402

Abstract: A Berkovich space is a type of analytic space associated to an algebraic variety over a field $K$ with valuation $v$, such as $\mathbb{Q}_p$ or $\mathbb{C}((t))$. These spaces are intimately connected with other areas of mathematics including tropical geometry and number theory. I will give an introduction to the theory of Berkovich spaces, and explain how connections with birational geometry can help us understand the geometry of Berkovich spaces.

## Combinatorics Seminar

### Weighted Bond Posets and Graph Associahedra, Part 2

Monday, September 28, 2015, 5:00pm
Ungar Room 402

Abstract: This talk is a continuation of the September 21 talk.

## Applied Math Seminar

### The Reduction Principle, the Ideal Free Distribution, and the Evolution of Dispersal Strategies

Friday, September 25, 2015, 4:00pm
Ungar Room 402

Abstract: The problem of understanding the evolution of dispersal has attracted much attention from mathematicians and biologists in recent years. For reaction-diffusion models and their nonlocal and discrete analogues, in environments that vary in space but not in time, the strategy of not dispersing at all is often convergence stable within in many classes of strategies. This is related to a "reduction principle" which states that that in general dispersal reduces population growth rates. However, when the class of feasible strategies includes strategies that generate an ideal free population distribution at equilibrium (all individuals have equal fitness, with no net movement), such strategies are known to be evolutionarily stable in various cases. Much of the work in this area involves using ideas from dynamical systems theory and partial differential equations to analyze pairwise invasibility problems, which are motivated by ideas from adaptive dynamics and ultimately game theory. The talk will describe some past results and current work on these topics.

## Combinatorics Seminar

### Weighted Bond Posets and Graph Associahedra

Monday, September 21, 2015, 5:00pm
Ungar Room 402

Abstract: We consider a weighted version of the bond lattice of a graph. This generalizes the poset of weighted partitions introduced by Dotsenko and Khoroshkin and studied in a previous paper of the authors. We show that for cordal graphs, each interval of the weighted bond poset has the homotopy type of a wedge of spheres, and we present an intriguing connection with h-vectors of graph associahedra studied by Postnikov, Reiner and Williams, and others. This is joint work with Rafael Gonzalez D'Leon.

## Geometry and Physics Seminar

### New Surgeries between the Poincare Homology Sphere and Lens Spaces

Wednesday, September 2, 2015, 5:00pm
Ungar Room 402

Abstract: We exhibit an infinite family of hyperbolic knots in the Poincare Homology Sphere with tunnel number 2 and a lens space surgery and discuss the implications. Notably, this is in contrast to the previously known examples due to Hedden and Tange which are all doubly primitive.

## Geometry and Physics Seminar

### Categorical Kaehler Metrics

Friday, August 28, 2015, 5:00pm
Ungar Room 411

## Combinatorics Seminar

### Determinant, Permanent, Tensors and Words

Monday, April 20, 2015, 5:00pm
Ungar Room 402

Abstract: Using words of operators in tensor product we present an inequality for positive operators on Hilbert space. The proof of the main result is combinatorial. As applications of the operator inequality and by a multilinear approach, we show some matrix inequalities concerning induced operators and generalized matrix functions (including determinants and permanents as special cases).

## Geometry and Physics Seminar

### On the Center of Mass in General Relativity

Wednesday, April 15, 2015, 5:00pm
Ungar Room 402

Abstract: In many situations in Newtonian gravity, understanding the motion of the center of mass of a system is key to understanding the general "trend" of the motion of the system. It is thus desirable to also devise a notion of center of mass with similar properties in general relativity. However, while the definition of the center of mass via the mass density is straightforward in Newtonian gravity, there is a priori no definitive corresponding notion in general relativity. Instead, there are several alternative approaches to defining the center of mass of a system. We will discuss some of these different approaches for both asymptotically Euclidean and asymptotically hyperbolic systems and present some explicit (counter-)examples.

## Combinatorics Seminar

### Combinatorics of Acyclic Orientations of Graphs:Algebra, Geometry and Probability

Monday, March 30, 2015, 5:00pm
Ungar Room 402

Benjamin Iriarte Giraldo will defend his MIT Ph.D. thesis (under the direction of Richard Stanley) at the University of Miami.

## Combinatorics Seminar

### Generalizations of Bjorner's NBC Basis for the Homology of the Partition Lattice

Monday, March 23, 2015, 5:00pm
Ungar Room 402

Abstract: Bjorner's NBC basis for the homology of the partition lattice has a very nice description in terms of fundamental cycles obtained by splitting increasing rooted trees. We present two generalizations of this basis. One of these, obtained in joint work with Rafael Gonzalez D'Leon, is for a weighted partition poset and the other, obtained in joint work with John Shareshian, is for the 1 mod k partition poset.

## Geometry and Physics Seminar

### Towards a Lagrangian-Floer Theory for Representation Spaces of Tangles

Wednesday, March 18, 2015, 5:00pm
Ungar Room 402

Abstract: We describe how to use SU(2) character varieties of fundamental groups of 3-manifolds to construct a Lagrangian-Floer theory counterpart to Kronheimer-Mrowka's singular instanton knot Floer homology.

## Geometry and Physics Seminar

### Spectra Sets for Inoue Surfaces

Wednesday, March 4, 2015, 5:00pm
Ungar Room 402

## Combinatorics Seminar

### Period Collapse of Ehrhart Quasipolynomials

Monday, February 9, 2015, 5:00pm
Ungar Room 402

Abstract: Let P be a convex polytope with rational vertices. For a positive integer n, let i(P,n) be the number of integer points in nP. A basic theorem of Ehrhart theory says that if p is the gcd of the denominators of all coordinates of the vertices of P, then for 0\leq j<p the function i(P,n) is a polynomial f_j(n) when n is congruent to j mod p. In some cases, however, the "quasiperiod" p can be smaller. After a general discussion we focus on the very simple case of a triangle with vertices (0,0), (a/b,0), (0,a/b), for which some surprising results hold. Most of this is joint work with Daniel Gardiner, who was motivated by symplectic geometry. We assume no prior knowledge of polytopes, Ehrhart theory, or symplectic geometry.

## Geometry and Physics Seminar

### Kashiwara Conjugation for Twisted D-modules

Wednesday, February 4, 2015, 5:00pm
Ungar Room 506

Abstract: In 1987, Kashiwara introduced a functor taking D-modules on a complex manifold X to D-modules on the complex conjugate of X. Moreover, he showed that this functor, which is called Kashiwara conjugation, is an (anti)-equivalence from the category of regular holonomic D-modules on X to those on the complex conjugate of X. Motivated by applications to representation theory, Barlet and Kashiwara extended this functor to modules over rings of twisted differential on generalized flag varieties. I will explain a simple way to extend the Barlet-Kashiwara result to more general rings of twisted differential operators on arbitrary complex manifolds. As some of my motivation for thinking about this comes from conjectures of Schmid and Vilonen on representation theory, I will also give some examples coming out of those conjectures.

## Geometry and Physics Seminar

### Categorical Linear Systems and Oscillating Integrals

Saturday, November 15, 2014, 6:00pm
Ungar Room 411

## Applied Math Seminar

### Modeling Queueing Networks

Friday, November 14, 2014, 4:30pm
Ungar Room 402

Abstract: Our research group in Mathematics at UF has recently come up with a simulation model of the flow of patients through an Emergency Department. The model was developed in collaboration with Adrian Tyndall, Head of Emergency Services at Shands Teaching Hospital at UF.

The model is based on current practices in providing emergency care at a university hospital. The triage of patients, the prioritizing of treatment, the waiting times for various services, and the requirements for facilities and attendants are all based on data from the ED at Shands.

The mathematics involved in creating this model can be applied in many other situations. One project is to expand the ED model to the flow of patients through the whole hospital. Another project is to develop a sophisticated model the spread of hospital acquired infections. We have developed some new techniques in analyzing queueing networks that make the theory better applicable in a wide variety of situations such as the ones mentioned above.

## Geometry and Physics Seminar

### Linear Systems and Stability Mixed Structures

Sunday, November 9, 2014, 6:00pm
Ungar Room 411

## Probability Seminar

### Some Remarks on Finitely-additive Probability

Thursday, November 6, 2014, 2:15pm
Ungar Room 411

## Applied Math Seminar

### Inverse Boundary Value Problems with Incomplete Data

Friday, October 31, 2014, 4:30pm
Ungar Room 402

Abstract: Inverse boundary value problems arise when one tries to recover internal parameters of a medium from data obtained by boundary measurements. In many of these problems the physical situation is modeled by partial differential equations. The goal is to determine the coefficients of the equations from measurements of the solutions on the boundary. However, collecting data from the whole boundary is sometimes either not possible or extremely expensive in practice. In this talk we present the recent developments on inverse problems with incomplete data for Schroedinger types of equations in both bounded domains and unbounded domains.

## Probability Seminar

### On Random Geometric Subdivisions

Thursday, October 30, 2014, 2:15pm
Ungar Room 411

Abstract: I will present several models of random geometric subdivisions, similar to that of Diaconis and Miclo (Combinatorics, Probability and Computing, 2011), where a triangle is split into 6 smaller triangles by its medians, and one of these parts is randomly selected as a new triangle, and the process continues ad infinitum. I will show that in a similar model the limiting shape of an indefinite subdivision of a quadrilateral is a parallelogram. I will also show that the geometric subdivisions of a triangle by angle bisectors converge (but only weakly) to a non-atomic distribution, and that the geometric subdivisions of a triangle by choosing a uniform random points on its sides converges to a flat triangle, similarly to the result of the paper mentioned above.

## Combinatorics Seminar

### On q-gamma Positivity

Tuesday, October 28, 2014, 5:00pm
Ungar Room 402

Abstract: Gamma-positivity is a property of polynomials that implies palindromicity and unimodality. It has received considerable attention in recent times because of Gal's conjecture, which asserts gamma-positivity of the h-polynomial of flag homology spheres. Eulerian polynomials and the Narayana polynomials are examples of such h-polynomials that are known to be gamma-positive. In this talk I will present q-analogs of formulas establishing gamma-positivity of these and other polynomials. Geometric interpretations involving toric varieties and consequences such as q-unimodality will also be discussed. This is based on joint work with John Shareshian and with Christian Krattenthaler.

## Geometry and Physics Seminar

### Categorical Multiplier Ideal Sheaf

Sunday, October 26, 2014, 5:00pm
Ungar Room 411

## Geometry and Physics Seminar

### On Categorical Multiplier Ideal Sheaf

Tuesday, October 14, 2014, 5:00pm
Ungar Room 411

## Applied Math Seminar

### Turing Instability and Hopf Bifurcation: Spatio-temporal Dynamics

Friday, October 3, 2014, 4:30pm
Ungar Room 402

Abstract: For a physical or biological system described by reaction-diffusion equations, spatial patterns can occur via Turing instability mechanism (that is bifurcation induced by the diffusion); temporal patterns can occur via Hopf bifurcation induced by the change of parameters. At the points where the Turing instability curve and Hopf bifurcation curve intersect, the model can undergo Turing-Hopf bifurcation and exhibit spatiotemporal patterns. As examples, spatial, temporal, and spatiotemporal dynamics of biological and physical systems are presented and numerical simulations are carried out to verify andillustrate the bifurcation phenomena.

## Geometry and Physics Seminar

### Thin Position, Graph Clustering, and Applications

Thursday, October 2, 2014, 5:00pm
Ungar Room 506

Abstract: We describe a novel algorithm for clustering vertices of graphs. The method is inspired by the technique of thin position in low dimensional topology. We show that a version of our algorithm works well on an important real world data set from biology. This is joint work with Doug Heisterkamp, Jesse Johnson, and Danielle O'Donnol.

## Applied Math Seminar

### Avoidance Behavior in Intraguild Predation Communities: A Cross-Diffusion Model

Friday, September 26, 2014, 4:30pm
Ungar Room 402

Abstract: A cross-diffusion model of an intraguild predation community where the intraguild prey employs a fitness based avoidance strategy is examined. The avoidance strategy employed is to increase motility in response to negative local fitness. Global existence of trajectories and the existence of a compact global attractor are proved. It is shown that if the intraguild prey has positive fitness at some point in the habitat when trying to invade, then it will be uniformly persistent in the system if its avoidance tendency is sufficiently strong. This type of movement strategy can lead to coexistence states in which the intraguild prey is marginalized to areas with low resource productivity while the intraguild predator maintains high densities in regions with abundant resources, a pattern observed in many real world intraguild predation systems.

## Functors and Complexity Seminar

### Landau-Ginzburg Models and Secondary Polytopes II

Tuesday, September 16, 2014, 5:00pm
Ungar Room 411

Abstract: We will continue our review of recent efforts by Kapranov, Kontsevich, and Soibelman to understand categories associated to Lefschetz fibrations by means of a combinatorial deformation theory. Time permitting, will conjecture possible connections to other geometric approaches to decomposing these structures.

## Applied Math Seminar

### Spatial Population Dynamics in a Producer-Scrounger Model

Friday, September 12, 2014, 4:30pm
Ungar Room 402

Abstract: The spatial population dynamics of an ecological system involving producers and scroungers is studied using a reaction-diffusion model. The two populations move randomly and increase logistically, with birth rates determined by the amount of resource acquired. Producers can obtain the resource directly from the environment, but must surrender a proportion of their discoveries to nearby scroungers through a process known as scramble kleptoparasitism. The proportion of resources stolen by a scrounger from nearby producers decreases as the local scrounger density increases. Producer persistence depends in general on the distribution of resources and producer movement, whereas scrounger persistence depends on the ability to invade an environment when producers are at steady-state. It is found that (i) both species can persist when the habitat has high productivity, (ii) neither species can persist when the habitat has low productivity, and (iii) slower dispersal of both the producer and scrounger is favored when the habitat has intermediate productivity. This research is in collaboration with Andrew Nevai.

## Functors and Complexity Seminar

### Landau-Ginzburg Models and Secondary Polytopes

Tuesday, September 9, 2014, 5:00pm
Ungar Room 411

Abstract: We will review recent efforts by Kapranov, Kontsevich, and Soibelman to understand categories associated to Lefschetz fibrations by means of a combinatorial deformation theory. Time permitting, will conjecture possible connections to other geometric approaches to decomposing these structures.

## Geometry and Physics Seminar

### Linear Systems – Old and New

Wednesday, September 3, 2014, 5:00pm
Ungar Room 506

## Combinatorics Seminar

### Face Rings of Cycles, Associahedra, and Standard Tableaux

Monday, April 22, 2014, 5:00pm
Ungar Room 402

Abstract: Let Jn denote the quadratic monomial ideal generated by the diagonals of an n-gon (i.e. the Stanley-Reisner ideal of an n-cycle). One way to realize a free resolution of Jn is to utilize a natural monomial labeling of the faces of the (dual) associahedron - the complex computing cellular homology supports a resolution of Jn. This resolution is not minimal since the dual associahedron has too many high-dimensional faces.

On the other hand, some years ago Richard Stanley gave a simple bijection between the faces of the associahedron and standard Young tableaux of certain shapes. We show that the Betti numbers of Jn (the ranks of the free modules in a minimal resolution) are given by the number of standard Young tableaux of certain sub-shapes. While we do not have a good description of the differentials with this basis, this does suggest a connection with the face poset structure of the associahedron.

## Combinatorics Seminar

This seminar will be a mini-symposium consisting of 3 half hour talks. Tuesday, March 18, 2014, 5:00pm Ungar Room 402

### A Short Trip to the Riordan Group

Abstract: The Riordan group is a group of infinite lower triangular matrices defined by a pair of formal power series. It has been studied for the last twenty years over a wide range of different areas. In this talk we invite you on a short trip to the Riordan group. A brief survey of results on the algebraic structure of the Riordan group will be given. We will also go through applications of Riordan group theory and explore a few of them in more detail.

and

### A Jacobi-Trudi Formula for Macdonald Functions of Rectangular Shapes

Abstract: We express Macdonald functions of rectangular shapes using vertex operators, thereby giving a generalized Jacobi-Trudi formula for them. The proof relies on a splitting formula for the q-Dyson Laurent polynomial, from which two results follow immediately: Kadell's orthogonality conjecture proved by Károlyi, Lascoux, and Warnaar, and Andrews' q-Dyson constant term conjecture proved by Zeilberger and Bressoud. This is joint work with Naihuan Jing.

and

### Sperner Theorems for Convex Families

Abstract: Sperner's theorem states that the density of the largest Sperner family (or antichain) in the boolean algebra Bnis $\binom{n}{n/2}/2^n$. It is conjectured that this is an upper bound for any convex family. We provide further evidence for the conjecture by exhibiting a number of examples of convex families satisfying the bound.

## Geometry and Analysis Seminar

### Stable Equivalence of Surfaces in 4-manifolds

Monday, March 17, 2014, 2:30pm
Ungar Room 506

Abstract: It is well known that there are homeomorphic 4-manifolds that are not smoothly equivalent, that become smoothly equivalent after taking the connected sum with one copy of a special manifold. Similar behavior may be found in other geometric settings, including diffeomorphisms up to isotopy, positive scalar curvature and knotted surfaces. In this talk we will prove that there is an infinite family of spheres in a connected sum of complex projective spaces with assorted orientations, so that no two spheres in the family are smoothly equivalent, yet every pair is topologically isotopic. Furthermore we will show that the spheres become smoothly isotopic after one stabilization. We will do so with an explicit description of the spheres and the isotopies. This is joint work with Danny Ruberman, Paul Melvin, and Hee Jung Kim.

## Applied Math Seminar

### Spatially Heterogeneous Cholera Models

Friday, February 28, 2014, 5:00pm
Ungar Room 506

Abstract: Cholera was one of the most feared diseases in the 19th century, and remains a serious public health concern today. It can be transmitted to humans directly by person-to-person contact or indirectly through ingestion of contaminated water. Spatial heterogeneity of both humans and water may influence the spread of cholera. To incorporate these spatial effects, two cholera models are proposed that both include direct and indirect transmission. The first is a multi-group model and the second is a multi-patch model. New mathematical tools from graph theory are used to understand the dynamics of both heterogeneous cholera models, and to show that each model satisfies a sharp threshold property. Specifically, Kirchhoff's matrix tree theorem is used to investigate the dependence of the disease threshold on the patch connectivity and water movement (multi-patch model), and also to establish the global dynamics of both models.

## Seminar on Cluster Algebras

### Polyhedral Cones and Categories

Friday, February 28, 2014, 4:00pm
Ungar Room 506

## Combinatorics Seminar

### Enumeration of Vacua of String and M-theory: Formulas for Counting Conjugacy Classes of Elements of Finite Order in Lie Groups

Tuesday, February 25, 2014, 5:00pm
Ungar Room 402

## Seminar on Cluster Algebras

### Cluster Algebras, Character Varieties, and Cremona Groups

Friday, February 21, 2014, 4:00pm
Ungar Room 506

## Combinatorics Seminar

### A Geometric Interpretation of an Eulerian Number Identity

Monday, February 17, 2014, 5:00pm
Ungar Room 402

Abstract: We discuss an identity of Chung, Graham and Knuth involving Eulerian numbers and binomial coefficients. We give a geometric interpretation of this identity, of a q-analog due to Chung-Graham and Han-Lin-Zeng, and of a symmetric function analog due to Shareshian-Wachs. Our interpretation involves the h-vector of the stellohedron and the representation of the symmetric group on the cohomology of the associated toric variety. This is joint work with John Shareshian.

## Applied Math Seminar

### Study of Some Mathematical Models on Communicable Diseases

Friday, February 14, 2014, 5:00pm
Ungar Room 402

Abstract: In the first part of this talk, I will focus on mathematical modeling of the transmission of a vector-borne disease, Malaria, considering the following three factors: 1) disease latency; 2) spatial dispersal; 3) Multiple strains. In the second part, I will talk about some statistical analysis on epidemiological data of H1N1 Influenza in 2009.

## Seminar on Cluster Algebras

### Can We Obtain an Invariant of Contact Structures from Automorphisms of Cluster Algebras?

Friday, February 7, 2014, 3:30pm
Ungar Room 411

## Combinatorics Seminar

### The Combinatorial Structure behind Multibracketed Free Lie Algebras

Tuesday, February 4, 2014, 5:00pm
Ungar Room 402

Abstract: We explore a beautiful interaction between algebra and combinatorics in the heart of the free Lie algebra on n generators: The multilinear component of the free Lie algebra Lie(n) is isomorphic as a representation of the symmetric group to the top cohomology of the poset of partitions of an n-set tensored with the sign representation. Then we can understand the algebraic object Lie(n) by applying poset theoretic techniques to the poset of partitions whose description is purely combinatorial. We will show how this relation generalizes further in order to study free Lie algebras with multiple compatible brackets. In particular we obtain combinatorial bases and compute the dimensions of these modules. Part of the talk is based on joint work with M. Wachs.

## Combinatorics Seminar

### Two Enumerative Tidbits

Tuesday, January 28, 2014, 5:00pm
Ungar Room 402

Abstract: The two tidbits are:

1. The Smith normal form of some matrices connected with Young diagrams

2. A Distributive lattice associated with three-term arithmetic progressions

## Applied Math Seminar

### Why the Present Is More Complicated than the Future

Friday, January 17, 2014, 4:00pm
Ungar Room 506

Abstract: Models from e.g. fluid dynamics or systems biology often can be described by ordinary differential equations. While the long-term behavior of those models as time tends to infinity is quite well understood, the theory of transient solution behavior is still in its infancy. We present some results as well as open questions.

## Geometry and Physics Seminar

### Differentiably Finite Functions and Noncommutative Mordell Lang Conjecture

Wednesday, January 15, 2014, 5:00pm
Ungar Room 402

Abstract: In this talk we will connect some classical and new results from logic, number theory, combinatorics and dynamical systems to theory of categories.

## Combinatorics Seminar

### Discretized Morse Theory vs. Knots

Monday, December 2, 2013, 5:00pm
Ungar Room 402

Abstract: Morse theory studies smooth manifolds up to homotopy by looking at generic real-valued functions defined on them. Discrete Morse theory relies on Whitehead's "simple homotopy" theory, and it applies to triangulations of manifolds -- or, more generally, to arbitrary (regular CW) complexes. It yields a valid tool to `simplify' a complex without changing its homotopy. In this talk we plan to sketch:

(1) the relation between smooth and discrete Morse theory. (For example, how to reconstruct a smooth function from a discrete one, or how to characterize the Heegaard genus as "best discrete Morse vector" over all possible triangulations.)

(2) obstructions coming from knot theory. (For example, how to build "nasty" triangulations of the 3-sphere, over which any discrete Morse function has lots of critical faces.)

## Seminar on Cluster Algebras

### Nonrational Clusters

Monday, December 2, 2013, 4:00pm
Ungar Room 411

## Seminar on Cluster Algebras

### Surfaces and Cluster Algebras

Monday, November 18, 2013, 4:00pm
Ungar Room 411

## Geometry and Physics Seminar

### Kobayashi Pseudometric on Hyperkahler Manifolds

Tuesday, November 12, 2013, 5:00pm
Ungar Room 402

## Seminar on Cluster Algebras

### The Numerology of Finite File Type Clusters

Monday, November 11, 2013, 4:00pm
Ungar Room 411

## Geometry and Physics Seminar

### Pretzel Knots Admitting L-space Surgeries

Wednesday, November 6, 2013, 5:00pm
Ungar Room 402

Abstract: A rational homology sphere whose Heegaard Floer homology is the same as that of a lens space is called an L-space. We will classify pretzel knots with any number of tangles which admit L-space surgeries, and discuss some questions regarding essential Conway spheres and knot Floer homology that arise from this classification.

## Applied Math Seminar

### Interaction of Traveling Waves and Entire Solutions for Several Evolution Systems

Friday, November 1, 2013, 5:00pm
Ungar Room 402

Abstract: Wave propagation occurs in many applied fields such as materials science, biology and life science. In addition to the traveling waves, one can also observe interaction between different waves. Mathematically, this phenomenon can be described by the so-called wave-like entire solution that is defined for all space and time and behaves like a combination of traveling waves as $t\rightarrow-\infty$. In this talk we show the existence and various qualitative properties of wave-like entire solutions for several evolution systems. Some open problems and issues are also suggested for future research.

## Geometry and Physics Seminar

### Verified Computations for Hyperbolic 3-manifolds

Wednesday, October 16, 2013, 5:00pm
Ungar Room 402

Abstract: Given a triangulated 3-manifold M a natural question is: Does M admit a hyperbolic structure?

While this question can be answered in the negative if M is known to be reducible or toroidal, it is often difficult to establish a certificate of hyperbolicity, and so computer methods have developed for this purpose. In this talk, I will describe a new method to establish such a certificate via verified computation and compare this method to existing techniques.

This is joint work with Kazuhiro Ichihara, Masahide Kashiwagi, Hidetoshi Masai, Shin'ichi Oishi, and Akitoshi Takayasu.

## Seminar on Cluster Algebras

### The Birational Geometry of Mutations (d'apres Gross, Hacking, and Keel)

Monday, October 14, 2013, 4:30pm
Ungar Room 411

## Applied Math Seminar

### Metapopulation Models with Nonlocal Dispersal

Friday, October 11, 2013, 5:00pm
Ungar Room 402

Abstract: In this work we study evolutionary stability of nonlocal dispersal strategies in metapopulation prospective. (Holt and Timothy 2000) complement the classic Levins model (Hanski 1997) by considering the environmental gradients. Our model is a continuum generated from this metapopulation model, based on integrodifferential equations, using nonlocal operators to describe colonization at each site on a finite domain. We extend the Maximum principle, Comparison theorem and study an eigenvalue problem for our model. We also study coexistence and extinction in the competition system.

## Combinatorics Seminar

### About the Combinatorics of Multibracketed Free Lie Algebras

Friday, October 11, 2013, 4:00pm
Ungar Room 402

Abstract: It is a classical result that the multilinear component of the free Lie algebra with n generators Lie(n) has dimension (n-1)!. It is also well known that Lie(n) is isomorphic as a representation of the symmetric group to the top cohomology of the poset of set partitions tensored with the sign representation. I will discuss how these results generalize as to consider free Lie algebras with multiple compatible brackets.

## Seminar on Cluster Algebras

Monday, October 7, 2013, 4:00pm
Ungar Room 411

## Applied Math Seminar

### How the Latency Impacts the Disease Dynamics

Friday, October 4, 2013, 5:00pm
Ungar Room 402

Abstract: In this work, we modified the classic Ross-Macdonald model for malaria disease dynamics by incorporating latencies both for human beings and female mosquitoes. We introduced two general probability functions (P1(t) and P2(t)) to reflect the fact that the latencies differ from individuals to individuals, and investigated the impact of the latencies on disease outbreak.

## Combinatorics Seminar

### Cellular Resolutions via Mapping Cones

Wednesday, October 2, 2013, 5:00pm
Ungar Room 411

Abstract: Suppose I is a monomial ideal. One can iteratively obtain a free resolution of I by considering the "mapping cone" of a map of complexes associated to adding one generator at a time. Herzog and Takayama have shown that this procedure yields a minimal resolution if the ideal I "has linear quotients".

Here we consider cellular realizations of these resolutions. Extending a construction of Mermin we describe a regular CW complex that supports the HT resolution. By varying the choice of chain map we recover other known cellular resolutions and obtain combinatorially distinct complexes with interesting structure, suggesting a notion of a "space of resolutions". This is joint work with Fatemeh Mohammadi.

## Seminar on Cluster Algebras

Monday, September 30, 2013, 4:00pm
Ungar Room 411

## Applied Math Seminar

### Modeling the Transmission and Control of Schistosomiasis in China

Friday, September 27, 2013, 5:00pm
Ungar Room 402

Abstract: Schistosomiasis is one of the main tropic diseases. In some provinces of China, Schistosoma japonicum is endemic. There are three types of regions: (i) plain regions with waterway networks, (ii) mountainous and hilly regions, and (iii) marshland and lake regions. The transmission of the first type of regions has been eliminated. The second type has been controlled. The disease is still endemic in some regions of the third type, although it is becoming better. Based on the human-cattle-snail transmission of schistosomiasis, a model consisting of six ordinary differential equations that describe susceptible and infected human, cattle and snail subpopulations is proposed. We perform some numerical simulations and analysis for different types of regions separately.

## Geometry and Physics Seminar

### Stable Pairs and Coercive Estimates for the Mabuchi Functional II

Friday, September 27, 2013, 4:00pm
Ungar Room 402

Abstract: We show that a projective manifold is "stable" if and only if the Mabuchi Energy is proper on the corresponding space of Bergman metrics. We also show that properness implies finite automorphism group.

Background and Motivation will be provided. The talk may be of interest to people working in algebraic geometry and invariant theory as well as P.D.E.'s and differential geometry.

## Geometry and Physics Seminar

### Stable Pairs and Coercive Estimates for the Mabuchi Functional

Thursday, September 26, 2013, 3:15pm
Florida International University
DM 409

Abstract: We show that a projective manifold is "stable" if and only if the Mabuchi Energy is proper on the corresponding space of Bergman metrics. We also show that properness implies finite automorphism group.

Background and Motivation will be provided. The talk may be of interest to people working in algebraic geometry and invariant theory as well as P.D.E.'s and differential geometry.

## Seminar on Cluster Algebras

Monday, September 23, 2013, 4:00pm
Ungar Room 411

## PechaKucha Global Night

### 3D Printing & Mathematics

Friday, September 20, 2013, 7:00pm
The LAB Miami
400 NW 26th Street
Miami, FL 33127

Abstract: 3D printing experts explore developments in the education, medical, and corporate sectors. Live demonstrations will follow after the program.

Admission is free. Everyone is welcome to attend.

## Seminar on Cluster Algebras

### Clusters, Categories and Cremona

Friday, September 20, 2013, 4:00pm
Ungar Room 402

## Seminar on Cluster Algebras

Monday, September 16, 2013, 4:00pm
Ungar Room 411

## Applied Math Seminar

### Multi-patch Models for Vector-borne Diseases

Friday, September 13, 2013, 5:00pm
Ungar Room 402

Abstract: We develop spatial models of vector-borne disease dynamics on a network of patches to examine how the movement of humans in heterogeneous environments affects transmission. We show that the movement of humans between patches is sufficient to maintain disease persistence in patches with zero transmission. We construct two classes of models using different approaches: (i) Lagrangian that mimics human commuting behavior and (ii) Eulerian that mimics human migration. We determine the basic reproduction number R0 for both modeling approaches and study the transmission dynamics in terms of R0. We also study the dependence of R0 on some parameters such as the travel rate of the infectives.

## Seminar on Cluster Algebras

Monday, September 9, 2013, 4:00pm
Ungar Room 533

## Geometry and Physics Seminar

### New Surgeries between Lens Spaces and Non-prime Manifolds

Wednesday, September 4, 2013, 5:00pm
Ungar Room 402

Abstract: The Cabling Conjecture asserts that any knot in S^3 with a surgery to a non-prime 3-manifold must be a cabled knot. In contrast, two families of hyperbolic knots in lens spaces with non-prime surgeries have been known. Recently we have greatly generalized these families and suggested that these might form the foundation for a classification of surgeries between lens spaces and non-prime manifolds. In this talk we'll describe this generalization and discuss the construction of yet further new families of such knots.

## Geometry and Physics Seminar

### Categories Dynamical Systems and Cremona Groups

Wednesday, August 28, 2013, 5:00pm
Ungar Room 402

## Geometry and Physics Seminar

### On Landau-Ginzburg Models

Friday, July 26, 2013, 4:30pm
Ungar Room 506

Abstract: Some concrete examples of Landau-Ginzburg models can be obtained from Letschetz fibrations. I will discuss various features of Hodge diamonds obtained in such examples, and how they vary according to the choices of compactification.

## Applied Math Seminar

### Bifurcations in Predator-Prey Models with Seasonal Prey Harvesting

Friday, April 26, 2013, 5:00pm
Ungar Room 402

Abstract: In last three decades, some previous work is about the application of bifurcation theory to predator-prey models with a variety of functional responses. Meanwhile it is also important to understand the nonlinear dynamics of predator-prey systems with harvesting. The most common types of harvesting are constant-effort harvesting and constant-yield harvesting. Now people begin to think the seasonal harvesting which is usually described by a periodic function and investigate the change of dynamical behaviors caused by seasonal harvesting.

My model is based on the predator-prey model with Ivlev type functional response. First considering the constant-yield harvesting, we reveal far richer dynamics compared to the case without harvesting. Secondly, we will consider the seasonal prey harvesting. There is some work about seasonal harvesting in one species system. Now we are trying to give some results in this two species system.

## Geometry and Physics Seminar

### Index Theory of the de Rham Complex on End-periodic Manifolds

Wednesday, April 17, 2013, 4:00pm
Ungar Room 402

Abstract: The analytic index of the de Rham complex on a compact orientable manifold is known to equal its Euler characteristic; the same holds for manifolds with product ends, for a properly understood L2 index. We show that this is no longer true for more general manifolds with periodic ends, by providing an explicit formula for the difference between the L2 index of the de Rham complex and the Euler characteristic of the manifold in terms of topology of the end.

## Geometry and Physics Seminar

### Hertz Potentials and Asymptotic Behavior of Massless Fields

Wednesday, April 10, 2013, 5:00pm
Ungar Room 506

Abstract: Hertz and related potentials provide a method to construct massless spin-sfields from solutions of wave equations. In this talk I will explain recent work which gives decay estimates for massless fields using a representation in terms of Hertz potentials. The method extends to the Kerr spacetime and I will discuss some equations which arise in this context.

## Combinatorics Seminar

### Positivity Results and Conjectures for Chromatic Quasisymmetric Functions

Tuesday, April 9, 2013, 5:00pm
Ungar Room 402

Abstract: The chromatic quasisymmetric functions, introduced by Shareshian and Wachs, are a refinement of Stanley's chromatic symmetric functions, which in turn specialize to the classical chromatic polynomials. For incompatibility graphs of natural unit interval orders the chromatic quasisymmetric functions are actually symmetric functions, and are conjecturally related to Tymoczko's representation of the symmetric group on cohomology of Hessenberg varieties and to Iwahori-Hecke algebra characters. In this talk I will present our refinement of Gasharov's Schur-positivity result for incomparability graphs of natural unit interval orders and show how we use it to obtain coefficients in the expansion of chromatic quasisymmetric functions in the power-sum basis. This is joint work with John Shareshian.

## Geometry and Physics Seminar

### From WKB Method to Category Theory and Their Ergodic Nature

Wednesday, April 3, 2013, 4:00pm
Ungar Room 402

Abstract: We will look at the 200-year-old procedure from the point of view of categories.

## Geometry and Physics Seminar

### The Kaehler Moduli Space of a Toric Stack

Tuesday, March 26, 2013, 5:00pm
Ungar Room 402

Abstract: The Kaehler moduli space is an interesting string theoretical notion whose mathematical definition is still unsatisfactory. In this talk, I will describe an attempt to give a categorical interpretation of this moduli space in the case of toric varieties/stacks. The construction is inspired by the grade restriction rule for categories of graded modules and coherent sheaves as introduced by Herbst-Hori-Page and Ballard-Favero-Katzarkov.

## Jang Soo KimUniversity of Minnesota

### Dyck Tilings and Related Topics

Tuesday, March 19, 2013, 5:00pm
Ungar Room 402

Abstract: Recently, Kenyon and Wilson introduced a certain matrix M in order to compute pairing probabilities of a double-dimer model. They showed that the absolute value of each entry of the inverse matrix of M is equal to the number of certain Dyck tilings of a skew shape. They conjectured two elegant formulas on the sum of the absolute values of the entries in a row or a column of M^{-1}. In this talk we will see bijective proofs of the two conjectures due to Kim, Meszaros, Panova, and Wilson. In the two bijective proofs Dyck tilings correspond to increasing labelled trees and complete matchings. We will also see a connection between Dyck tilings and the (q,t)-Catalan numbers.

## Combinatorics Seminar

### Hecke Algebra Characters and Quantum Chromatic Symmetric Functions

Tuesday, March 12, 2013, 4:00pm
Ungar Room 402

Abstract: We discuss generating functions for Hecke algebra characters, formulas for the evaluation of these characters at Kazhdan-Lusztig basis elements of the Hecke algebra, and (conjecturally) related symmetric functions defined by Shareshian and Wachs. While certain posets called unit interval orders may provide the key to connecting the Hecke algebra elements and symmetric functions, we propose to describe the connection in terms of a class of directed graphs which arose implicitly in papers of Goulden-Jackson, Greene, Stanley-Stembridge, and Haiman. Using these directed graphs, we conjecture a combinatorial formula for coefficients in the power sum expansion of the quantum chromatic symmetric functions.

## Geometry and Physics Seminar

### Floer Theory and Frobenius Manifolds

Wednesday, March 6, 2013, 4:00pm
Ungar Room 402

Abstract: While it is known that the axioms of Gromov-Witten theory can be encoded into the geometrical notion of Frobenius manifolds, Floer theory can be viewed as a generalization of Gromov-Witten theory. In this talk I show how the Frobenius manifolds of Gromov-Witten theory generalize to Floer theory, employing the symplectic field theory of Eliashberg-Givental-Hofer. In particular, I show that the symplectic cohomology of an open symplectic manifold can be equipped with the structure of a cohomology F-manifold in the sense of Merkulov. Here the first obstruction against smoothness is given by the BV bracket on symplectic cohomology.

## Applied Math Seminar

### Modeling Transmission Dynamics of Rabies in China

Friday, March 1, 2013, 5:00pm
Ungar Room 402

Abstract: Human rabies is one of the major public-health problems in China. The number of human rabies cases has increased dramatically in the last 15 years, partially due to the poor understanding of the transmission dynamics of rabies and the lack of effective control measures of the disease. In order to explore effective control and prevention measures we propose a deterministic model to study the transmission dynamics of rabies in China. The model consists of susceptible, exposed, infectious, and recovered subpopulations of both dogs and humans and describes the spread of rabies among dogs and from infectious dogs to humans. The model simulations agree with the human rabies data reported by the Chinese Ministry of Health. We also modify the model to include stray dogs into account and use the model to simulate the human rabies cases reported in Guangdong Province. Furthermore, we consider the seasonal effect on the transmission of rabies. Sensitivity analysis of R0 in terms of the model parameters is performed and different control and prevention measures, such as culling and immunization of dogs, are compared. Our study demonstrates that (i) reducing dog birth rate and increasing dog immunization coverage rate are the most effective methods for controlling rabies in China; (ii) large scale culling of susceptible dogs can be replaced by immunization of them; (iii) enhancing public education and awareness about rabies; and (iv) strengthening supervision of pupils and children in the summer and autumn.

## Combinatorics Seminar

### On the Tutte-Krushkal-Renardy Polynomial for Cell Complexes

Tuesday, February 26, 2013, 5:00pm
Ungar Room 402

Abstract: Recently V. Krushkal and D. Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A. Duval, C. Klivans, and J. Martin. Moreover, after a slight modification, the Tutte-Krushkal-Renardy polynomial evaluated at the origin gives a weighted count of cellular spanning trees, and therefore its free term can be calculated by the cellular matrix-tree theorem of Duval et al. In the case of cell decomposition of a sphere, this modified polynomial satisfies the same duality identity as before. We find that evaluating the Tutte-Krushkal-Renardy along a certain line is the Bott polynomial.

## Geometry and Physics Seminar

### Cobordism of Lens Spaces and Instantons

Wednesday, February 20, 2013, 4:00pm
Ungar Room 402

Abstract: In this talk we consider a special type of cobordisms among lens spaces. We determined the inclusion homomorphisms on integral homology from the boundary components of cobordisms under some (strong) conditions.

Let X be a cobordim among n copies of L(p,q) and n copies of L(p,-q) such that the second Betti number is zero and the inclusion homomorphism on integral homology from the boundary components is surjective. A typical example of X is the connected sum of n copies of L(p,q) x [0,1]. We show that when q = 1 the kernel of the inclusion homomorphism is of the "same form" as in the case of the typical example. In the proof we use Donaldson theory to construct a non-trivial reducible SU(2)-flat connection on X. A key point is "bubbling" phenomena of instantons on orbifolds.

The argument can be seen as an illustration of one aspect of Donaldson theory whose counterpart in Heegaard Floer theory or Seiberg-Witten theory is still missing. This is a joint work with M. Furuta.

## Applied Math Seminar

### Population Models with Nonlocal Dispersal in Discrete and Continuous Space

Friday, February 15, 2013, 5:00pm
Ungar Room 402

## Applied Math Seminar

### Nonlinear Diffusion and Resource Matching in Population Dynamics

Friday, February 8, 2013, 5:00pm
Ungar Room 402

## Geometry and Physics Seminar

### Closed Symmetric Differentials, Foliations and Fibrations Part 2: The Proofs

Wednesday, February 6, 2013, 4:00pm
Ungar Room 402

Abstract: We will prove the two main theorems described in the previous lecture. We will start and the emphasis will be given to theorem 2.

Theorem 1: If a projective manifold has a closed symmetric 2-differential w of the 1st kind, then π1(X) is infinite and w comes from a map from X to a quotient of an Abelian variety by a cyclic or dihedral group.

The next theorem deals with a class of closed symmetric differentials that contains differentials not of the 1st kind.

Theorem 2: If a projective manifold has a closed symmetric 2-differential of rank 2 which is decomposable as a product of two closed meromorphic 1- differentials, then the 1st Betti of X is non trivial and the Albanese dimension of X is 2.

## Combinatorics Seminar

### Conjectures on Rational Catalan Numbers

Tuesday, February 5, 2013, 5:00pm
Ungar Room 402

Abstract: Given coprime positive integers a and b, there are some things we don't understand about the relationship between b^{a-1} and a^{b-1}. There are categorifications of these numbers into bigraded representations of the symmetric groups S_a and S_b, respectively; which we call "rational parking spaces". The multiplicity of the trivial character in each equals the "rational Catalan number"

Cat(a,b) = (a+b-1)! / (a!b!).

What explains the a,b-symmetry? I will describe an encoding of this theory in terms of Shi hyperplane arrangements, and state several interesting conjectures.

The discussion will be informal(!), as I will not have time to write a script.

## Combinatorics Seminar

### Bigraphical Arrangements

Tuesday, January 29, 2013, 5:00pm
Ungar Room 402

Abstract: Associated to any graph G, Hopkins and Perkinson recently defined a family of hyperplane arrangements that they call 'bigraphical'. Specifying various parameters of the bigraphical arrangements recover the classical Shi and interval order arrangements. Hopkins and Perkinson show that a certain 'Pak-Stanley' labeling of the regions of the bigraphical arrangement recover the G-parking functions, and furthermore that the set of labels is invariant under a notion of sliding hyperplanes. We will discuss these notions and speculate on connections to some underlying commutative algebra. Time permitting, we will discuss a generalization to higher dimensional complexes.

## Geometry and Physics Seminar

### Closed Symmetric Differentials, Foliations and Fibrations

Wednesday, January 23, 2013, 4:00pm
Ungar Room 402

Abstract: We will show that if a projective surface has a symmetric differential of degree 2 which decomposes as a product of two closed meromorphic differentials, then the fundamental group of X is infinite (in fact we will describe the geometric structures that produce such differentials).

If time permits we will explore the properties of the 2-webs (2 foliations) associated to closed symmetric differentials. Here we are thinking of results that describe when does the presence of a closed symmetric 2-differential implies the existence of a fibration (associated to a map to a curve). These results are somewhat related to the De Franchis-Castelnuovo result and the result that states that a foliation associated to a closed holomorphic 1-form is a fibration if it contains a compact leaf that it is not exceptional.

## Geometry and Physics Seminars

### Strong L-spaces

Friday, November 30, 2012, 4:00pm
Ungar Room 402

Abstract: Strong L-spaces are a family of 3-manifolds with a concrete, combinatorial description. However, they owe their definition to Heegaard Floer homology. I will explain the special role they play in Heegaard Floer homology and then turn to the question of their classification. Some surprising combinatorics enters into the picture (perfect matchings and Pfaffian orientations on graphs).

## Combinatorics Seminar

### On Bounded Regions of Hyperplane Arrangements (Continued)

Tuesday, November 27, 2012, 5:00pm
Ungar Room 402

## Combinatorics Seminar

### On Bounded Regions of Hyperplane Arrangements

Tuesday, November 20, 2012, 5:00pm
Ungar Room 402

## Geometry and Physics Seminar

### Evolving Hypersurfaces by Their Inverse Null Mean Curvature

Wednesday, November 14, 2012, 4:00pm
Ungar Room 402

Abstract: We introduce a new second order parabolic evolution equation where the speed is given by the reciprocal of the null mean curvature. This flow is a generalisation of inverse mean curvature flow and it is motivated by the study of black holes and mass/energy inequalities in general relativity. We present a theory of weak solutions using level-set methods and an appropriate variational principle, and outline a natural application of the flow as a variational approach to constructing marginally outer trapped surfaces (MOTS), which play the role of quasi-local black hole boundaries in general relativity.

## Geometry and Physics Seminar

### Geometrostatics: The Geometry of Static Spacetimes in General

Tuesday, November 13, 2012, 5:00pm
Ungar Room 402

Abstract: Geometrostatics is an important subdomain of Einstein's General Relativity. It describes the mathematical and physical properties of static isolated relativistic systems such as stars, galaxies, or black holes. For example, geometrostatic systems have a well-defined ADM- mass (Chrusciel, Bartnik) and (if this is nonzero) also a center of mass (Huisken-Yau, Metzger, Huang) induced by a CMC-foliation at infinity. We will present surface integral formulas for these physical properties in general geometrostatic systems. Together with an asymptotic analysis, these can be used to prove that ADM-mass and center of mass 'converge' to the Newtonian mass and center of mass in the Newtonian limit c→∞(using Ehler's frame theory). We will discuss geometric similarities of geometrostatic and classical static Newtonian systems along the way.

## Geometry and Physics Seminar

### Special Holonomy and the ADHM Construction

Monday, November 5, 2012, 5:00pm
Ungar Room 506

Abstract: I will explain the construction of instantons over P^3 via a complexified version of the ADHM construction. This gives a holomorphic connection with special holonomy on the moduli of mathematical instantons on P^3. The same geometry, called trisymplectic, appears whenever one attempts to build a complexification of a hyperkaehler manifold.

A trisymplectic structure on a complex 2n-manifold is a triple of holomorphic symplectic forms such that any linear combination of these forms has rank 2n, n or 0. We show that a trisymplectic manifold is equipped with a holomorphic 3-web and the Chern connection of this 3-web is holomorphic, torsion-free, and preserves the three symplectic forms. We construct a trisymplectic structure on the moduli of rational curves in the twistor space of a hyperkaehler manifold.

The space of moduli of mathematical instantons on CP^3 can be identified with a component on the moduli space of rational curves on a twistor space of moduli of framed instantons on CP^2. The trisymplectic structure on this space is applied to obtain the space of instantons using a new geometric reduction procedure, called trihyperkaehler reduction. This is used to prove that the space of mathematical instantons on CP^3 is smooth, settling a long-standing conjecture.

## Geometry and Physics Seminar

### Stochastic Processes on Lorentzian Manifolds

Wednesday, October 31, 2012, 4:00pm
Ungar Room 402

Abstract: Stochastic processes on Lorentzian manifolds were first considered by Dudley in 1965. The first physically realistic Lorentzian process was proposed in 1997 by Debbasch, Mallick and Rivet. Since then, the field has been developing quickly at the interface between the physics and mathematics communities. I will review the basic ideas underlying the construction of Lorentzian processes and compare all existing processes with each other. A very robust and elegant H-theorem will be presented on this occasion. I will also discuss some important recent results which highlight connections between diffusions on Lorentzian manifolds and geometrical flows, black hole horizons and quantum walks.

## Geometry and Physics Seminar

### On Bartnik's Construction of Prescribed Scalar Curvature

Wednesday, October 24, 2012, 4:00pm
Ungar Room 402

Abstract: Riemannian 3-manifolds with prescribed scalar curvature arise naturally in general relativity as spacelike hypersurfaces in the underlying spacetime. In 1993, Bartnik introduced a quasi-spherical construction of metrics of prescribed curvature on 3-manifolds. Under quasi-spherical ansatz, the problem is converted to the initial value problem for a semi-linear parabolic equation of the lapse function. In this talk, we consider background foliations given by conformal round metrics, and by the Ricci flow on 2-speres. We discuss conditions on the scalar curvature function and on the foliation that guarantee the solvability of the parabolic equation, and thus the existence of flat 3-matrics with a prescribed inner boundary.

## Geometry and Physics Seminar

### Annular Twists and Bridge Numbers of Knots

Wednesday, October 10, 2012, 4:00pm
Ungar Room 402

Abstract: Performing +1/n and -1/n Dehn surgery on the boundary components of an annulus A in a 3-manifold M provides a homeomorphism of M similar to a Dehn twist. If a knot intersects the interior of A in an essential manner, then this twisting produces an infinite family of knots. In joint work with Gordon and Luecke, we show (under certain hypotheses) that if the bridge numbers of this family with respect to a given Heegaard surface of M are bounded, then the annulus may be isotoped to embed in the Heegaard surface. With this we construct genus 2 manifolds that each contain a family of knots with longitudinal surgeries to S^3 and unbounded genus 2 bridge number. In contrast, our earlier work gives an a priori upper bound on the genus g bridge number of a knot with a non-longitudinal S^3 surgery.

## Combinatorics Seminar

### Weighted Partition Posets

Tuesday, October 2, 2012, 5:00pm
Ungar Room 402

Abstract: V. V. Dotsenko and A. S. Khoroshkin introduced an analog of the lattice of set partitions where each block in a partition is provided with a nonnegative number less than its cardinality. These partitions are said to be weighted and the poset is called the poset of weighted partitions. We will discuss some of the interesting algebraic and homological properties of this poset.

## Geometry and Physics Seminar

### Lefschetz Fibrations on Adjoint Orbits

Monday, October 1, 2012, 2:00pm
Ungar Room 402

Abstract: I will explain how to obtain structures of Lefschetz fibrations on the contangent bundles of flag manifolds using Lie theory. This is joint work with L. Grama and L. San Martin.

## Geometry and Physics Seminar

### SU(N) Casson-Lin Invariants for Links

Friday, September 28, 2012, 3:45pm
Ungar Room 402

Abstract: We will introduce a family of invariants of links in the 3-sphere using projective SU(N) representations and braid theory. We will give some examples of computations, including the theorem that the SU(2) invariant of a two-component link is the linking number between its components.

## Geometry and Physics Seminar

### Modeling Wave Dark Matter in Dwarf Spheroidal Galaxies

Wednesday, September 26, 2012, 4:00pm
Ungar Room 402

Abstract: Many dwarf spheroidal galaxies are some of the most dark matter dominated galaxies known. As such they are excellent test beds for dark matter theories. There has been some recent work by astrophysicists attempting to match velocity dispersion profiles predicted by particle dark matter models to the observed velocity dispersions in dwarf spheroidal galaxies with reasonable success. We compare these models to those predicted by static spherically symmetric wave dark matter and use these comparisons to obtain an estimate on the value of the constant, Upsilon, which is a fundamental component of the wave dark matter model.

## Geometry and Physics Seminar

### Phantoms in Geometry

Wednesday, September 19, 2012, 4:00pm
Ungar Room 402

Abstract: Recently a new notion of phantom category was introduced. We give examples and discuss possible geometric applications.

## Geometry and Physics Seminar

### Global Regularity of Reflector Problem

Wednesday, September 12, 2012, 4:00pm
Ungar Room 402

Abstract: In this talk we study a reflector system which consists of a point light source, a reflecting surface and an object to be illuminated. Due to its practical applications in optics, electro-magnetics, and acoustic, it has been extensively studied during the last half century. This problem involves a fully nonlinear partial differential equation of Monge-Ampere type, subject to a nonlinear second boundary condition. In the far field case, it is related to the reflector antenna design problem and optimal transportation problem. Therefore, the regularity results of optimal transportation can be applied. However, in the general case, the reflector problem is not an optimal transportation problem and the regularity is an extremely complicated issue. In this talk, we give necessary and sufficient conditions for the global regularity and briefly discuss their connection with the Ma-Trudinger-Wang condition in optimal transportation. This is a joint work with Neil Trudinger.

## Combinatorics Seminar

### Laplacian Ideals, Arrangements, and Resolutions

Tuesday, September 11, 2012, 5:00pm
Ungar Room 402

Abstract: The lattice ideal of the Laplacian matrix of a graph G provides an algebraic perspective on the combinatorial dynamics of the Abelian Sandpile Model and the more general Riemann-Roch theory of G. The generators of this ideal form a Groebner bases with respect to a certain term order, and the associated initial ideals have nice connections to G-parking functions. We study resolutions of these initial ideals and show that, at least under certain conditions on G, a minimal free resolution is supported on the bounded subcomplex of a hyperplane section of the graphical arrangement of G. It is conjectured that these complexes also support resolutions for the Laplacian lattice ideal itself. This generalizes constructions from Postnikov and Shaprio (for the case of the complete graph) and connects to work of Manjunath and Sturmfels, and Perkinson on the commutative algebra of Sandpiles. Time permitting we will discuss some connections to the topology of generalized partition posets. This is joint work with Raman Sanyal.

## Geometry and Physics Seminar

### On the Shape of Representable Integral Homology Classes in a Riemannian Manifold

Wednesday, May 16, 2012, 4:00pm
Ungar Room 402

Abstract: Let (M,g) be a closed Riemannian manifold, D be a class in Hk(M;Z). If D can be represented by an embedded submanifold, we endow such a representative with the induced metric from that on M, and consider the space of all embedded representatives of D. On it we define the functional given by the squared L2-norm of the second fundamental form of the embedding. Its minimum of smallest volume, should it exists, provide for the canonical shape of D. We discuss this problem and some of its details in general, and focus on the particulars of the cases of P2 with the Fubini-Study metric, or Sp(2) with suitable left-invariant metrics adapted to the fibration S3 Sp(2) → S7.

## Geometry and Physics Seminar

### On the Positive Mass Theorem for Manifolds with Corners

Friday, May 4, 2012, 4:00pm
Ungar Room 402

Abstract: A problem originally studied by P. Miao is whether the positive mass theorem holds on manifolds with certain singularities along a hypersurface. I will discuss an approach to this problem which uses the Ricci flow to smooth out the metric, so that one can apply the usual positive mass theorem. This allows for extending the rigidity statement in the zero mass case to higher dimensions, which was only known in the 3 dimensional case previously. This is joint work with D. McFeron.

## Geometry and Physics Seminar

### An Axiomatic Approach to Quasi-local Mass in General Relativity

Wednesday, April 11, 2012, 4:00pm
Ungar Room 402

Abstract: In general relativity, the notion of quasi-local mass seeks to answer the question: "how much mass is contained in a bounded region in a spacelike slice of a spacetime"? We propose a definition that is motivated by axioms and designed to be as simple as possible. This definition is related to the existence of solutions to a boundary value problem for metrics of nonnegative scalar curvature. Interestingly, it tends to vanish on static vacuum regions. Finally, we recognize this quasi-local mass as a type of product of two well-known other definitions.

## Combinatorics Seminar

### Cluster Combinatorics

Tuesday, April 10, 2012, 5:00pm
Ungar Room 402

Abstract: Cluster algebras were invented around 2000 by Fomin and Zelevinsky through their study of total positivity in algebraic groups, and have since become extremely popular. The algebraic structure of cluster algebras can be largely reduced to a combinatorial structure called a "cluster complex", which generalizes the classical associahedron. Via this reduction, one can show that cluster algebras of "finite type" are parametrized by Dynkin diagrams (again). I will bring two zome models of associahedra to the talk, which you may inspect. (Some people think that it is impossible to build associahedra from zome tools. I discovered today that this is quite false.)

## Combinatorics Seminar

### Probabilistic and Topological Bounds on Chromatic Number

Tuesday, April 3, 2012, 5:00pm
Ungar Room 402

Abstract: A graph G is called "bipartite" if can be properly colored with two colors (the chromatic number of G is at most 2). Consider two alternative characterizations for this notion:

Probabilistic: G is bipartite if for any random walk W on G, the position of W at arbitrarily large time t restricts the possible starting position of W.

Topological: G is bipartite if in the "space of directed edges" of G, there's no way to walk from an edge E to the same edge with the reverse orientation.

Brightwell and Winkler introduced a generalization of the first property which they called the "warmth" of a graph, and showed that warmth is a lower bound for chromatic number. This was somewhat surprising since the warmth a graph G is defined in terms of homomorphisms from some fixed graph (whereas chromatic number is about homomorphisms intocomplete graphs). In this sense warmth also looks like the Hom complexes first introduced by Lovasz (the objects I discussed in the Grad student seminar last month), where the topological connectivity of a "space of directed edges" provides a lower bound for chromatic number. In addition, warmth and Hom complexes behave similarly with respect to certain graph operations including "foldings". However, no direct connection has been established (as far as I know). We'll discuss these concepts and present a conjectural inequality relating warmth and the topology of the Hom complexes.

## Geometry and Physics Seminar

### Essential Surfaces and Dehn Filling

Wednesday, March 28, 2012, 4:00pm
Ungar Room 402

Abstract: A driving question in 3-dimensional topology is how the structure of a 3-manifold M with torus boundary can change under Dehn filling. For example, if M contains an essential surface, the surface will usually remain essential after Dehn filling, the exceptions occurring for a restricted set of filling slopes. Is the same true for the entire set of essential surfaces in M? In other words, will this set usually be preserved under Dehn filling except for a restricted set of filling slopes? We will show that the answer is yes. The techniques involved use a kind of thin position argument and some normal surface theory. This is joint work with Dave Bachman and Eric Sedgwick.

## Geometry and Physics Seminar

### Curvature vs. Curvature Operator

Wednesday, March 21, 2012, 4:00pm
Ungar Room 402

Abstract: The talk will deal with recent results and open questions in the global geometry and topology of manifolds with nonnegative and almost nonnegative curvature and curvature operator, resp., and, in particular, describe how to distinguish these spaces from each other.

## Combinatorics Seminar

### A Conjectured Combinatorial Interpretation for Induced Sign Characters of the Hecke Algebra

Tuesday, March 6, 2012, 5:00pm
Ungar Room 402

Abstract: Many combinatorial formulas for computations in the symmetric group Sn can be modified appropriately to describe computations in the Hecke algebra Hn(q), a deformation of C[Sn]. Unfortunately, the known formulas for induced sign characters of Sn are not among these. For induced sign characters of Hn(q), we conjecture formulas which specialize at q=1 to formulas for induced sign characters of Sn. We will discuss evidence in favor of the conjecture, and relations to the chromatic quasi-symmetric functions of Shareshian and Wachs.

This is joint work with Brittany Shelton of Lehigh University.

## Combinatorics Seminar

### Promotion and Rowmotion

Friday, March 2, 2012, 5:00pm
Ungar Room 506

Abstract: We present an equivariant bijection between two actions--promotion and rowmotion--on order ideals in certain posets. This bijection simultaneously generalizes a result of R. Stanley concerning promotion on the linear extensions of two disjoint chains and the type A case of work of D. Armstrong, C. Stump, and H. Thomas on noncrossing and nonnesting partitions. We apply this bijection to several classes of posets, obtaining equivariant bijections to various known objects under rotation.

## Applied Math Seminar

### Optimal Prepayment of Mortgages

Friday, March 2, 2012, 5:00pm
Ungar Room 402

Abstract: The optimal strategy for the prepayment of fixed rate mortgages is modeled mathematically as a free boundary problem for a parabolic PDE. Basic existence and uniqueness results are summarized. Non-linear integral equations are then developed for the location of the free boundary (the risk-free interest rate below which the mortgage should be prepaid). They are used to derive a fast and accurate numerical scheme for calculating the early prepayment boundary. Finally, a simple, easily implemented analytic approximation for this boundary is obtained using asymptotic analysis. (Joint work with Xinfu Chen (Pittsburgh) and Dejun Xie (Suzhou))

## Combinatorics Seminar

### Fiber Polytopes

Tuesday, February 14, 2012, 5:00pm
Ungar Room 402

Abstract: Given a linear projection $P\to Q$ of convex polytopes, Billera and Sturmfels defined a third polytope $\Sigma(P\to Q)$ called the {\bf fiber polytope} of the projection. If P is an $(n-1)$-dimensional simplex and $Q$ is a convex $n$-gon then $\Sigma(P\to Q)$ is the associahedron. We will discuss the basics of this theory, and maybe more.

## Combinatorics Seminar

### Clusters, Generating Functions and Asymptotics

Friday, February 10, 2012, 5:00pm
Ungar Room 402

Abstract: A permutation p avoids a consecutive pattern q if no subsequence of adjacent entries of p is in the same relative order as the entries of q. For example, alternating permutations are those that avoid the consecutive patterns 123 and 321.

I will discuss some old and new results on the enumeration of permutations that avoid consecutive patterns. One of the main tools is the cluster method of Goulden and Jackson, based on inclusion-exclusion, which reduces the enumeration of these permutations to counting linear extensions of certain posets. For several patterns of arbitrary length, we obtain differential equations for the generating functions counting occurrences of the consecutive patterns.

I will also show that among consecutive patterns of any fixed length, the monotone pattern is easier to avoid than any non-overlapping pattern.

## Combinatorics Seminar

### The Monotone Column Permanent Conjecture and Multivariate Eulerian Polynomials

Monday, December 19, 2011, 3:00pm
Ungar Room 402

Abstract: Let B be an n by n matrix of real numbers, weakly increasing down columns. The Monotone Column Permanent Conjecture says that the permanent, of the matrix whose ij-th entry is (B)ij +z, has only real zeros, as a polynomial in z. In this talk we discuss the recent proof of this conjecture by Branden, Visontai, Wagner and the speaker. Our proof is based on the theory of stable polynomials, which are multivariate polynomials which are non-vanishing if all the variables have positive imaginary part. As a by-product of our work we obtain mutivariate stable versions of Eulerian polynomials.

## Geometry and Physics Seminar

### SHS and Automorphic Forms

Tuesday, November 29, 2011, 5:00pm
Ungar Room 506

## Combinatorics Seminar

### Eulerian Numbers, Chromatic Quasisymmetric Functions and Hessenberg Varieties

Tuesday, November 22, 2011, 5:00pm
Ungar Room 402

Abstract: We consider three distinct topics of independent interest; one in enumerative combinatorics, one in symmetric function theory, and one in algebraic geometry. The topic in enumerative combinatorics concerns a q-analog of a generalization of the Eulerian numbers, the one in symmetric function theory deals with a refinement of Stanley's chromatic symmetric functions, and the one in algebraic geometry deals with a representation of the symmetric group on the cohomology of the regular semisimple Hessenberg variety of type A. Our purpose is to explore some connections between these topics and consequences of these connections. This talk is based on joint work with John Shareshian.

## Applied Math Seminar

### Synchrony in Metapopulations:

#### The Role of Dispersal

Monday, November 21, 2011, 4:30pm
Ungar Room 402

Abstract: Many plant and animal populations have been shown to synchronize over large areas. In this talk, I will discuss synchrony in networks of food chains, composed of resource, consumer and predator populations (for example, algae, zooplankton and fish). Each community is described by the Rosenzweig-MacArthur tritrophic food chain model, and the communities interact through a network, composed of patches and migration corridors. I will present a general method to determine global stability of synchronization in ecological networks with any coupling topology. I will also demonstrate that, if only one species can migrate, the dispersal of the consumer (i.e., the intermediate trophic level) is the most effective mechanism for promoting synchronization.

## Applied Math Seminar

### A Problem in Transitioning from Spatial to Landscape Ecology:Perspectives from Several Modeling Formulations:Part 2- Integro-difference and Average Dispersal Success Matrix Approaches

Friday, November 18, 2011, 5:00pm
Ungar Room 402

Abstract: In this talk we compare and contrast the predictions of some spatially explicit and implicit models in the context of a thought problem at the interface of spatial and landscape ecology. The situation we envision is a one-dimensional spatial universe of infinite extent in which there are two disjoint focal patches of a habitat type that is favorable to some specified species. We assume that neither patch is large enough by itself to sustain the species in question indefinitely, but that a single patch of size equal to the combined sizes of the two focal patches provides enough contiguous favorable territory to sustain the given species indefinitely. When the two patches are separated by a patch of unfavorable matrix habitat, the natural expectation is that the species should persist indefinitely if the two patches are close enough to each other but should go extinct over time when the patches are far enough apart. Our focus here is to examine how different mathematical regimes may be employed to model this situation, with an eye toward exploring the trade-off between the mathematical tractability of the model on one hand and the suitability of its predictions on the other. In particular, we are interested in seeing how precisely the predictions of mathematically rich spatially explicit regimes (reaction-diffusion models, integro-difference models) can be matched by those of ostensibly mathematically simpler spatially implicit patch approximations (discrete-diffusion models, average dispersal success matrix models).

Joint work with Chris Cosner (University of Miami) and William Fagan (University of Maryland).

## Geometry and Physics Seminar

### Some Simple Triangulations

Friday, November 11, 2011, 1:30pm
Ungar Room 506

Abstract: I'll describe the story of how Thurston observed some very simple triangulations of knot and link complements in the 3-sphere. This allowed for a relatively simple way to find hyperbolic structures on such manifolds, and was a key inspiration for the Geometrization Conjecture of 3-manifolds. Ben Burton and I have recently been studying 4-dimensional triangulations and we came across an analogous triangulation for the complement of an embedded 2-sphere in the 4-sphere. While this does not lead to an amazing conjecture like Geometrization, it does lead to an interesting insight into things called Cappell-Shaneson knots, which are historically related to the smooth 4-dimensional Poincare conjecture. This is joint work with Ben Burton and Jonathan Hillman.

## Geometry and Physics Seminar

### Generalized Langlands Correspondences

Wednesday, November 9, 2011, 4:00pm
Ungar Room 506

## Combinatorics Seminar

### Parking Spaces

Tuesday, November 8, 2011, 5:00pm
Ungar Room 402

Abstract: There is a program called, say, "Catalan Combinatorics" that seeks to unify various kinds of combinatorics (parking functions, noncrossing/nonnesting partitions, cluster complexes/associahedra, Shi arrangements, core partitions, etc.) under the theory of reflection groups. Today I will talk about the role of parking functions in this project. The classical parking functions are well known. Given a Weyl group W with root lattice Q and Coxeter number h, Haiman generalized parking functions to the finite torus Q/(h+1)Q. In joint work with Vic Reiner and Brendon Rhoades, we have now generalized Q/(h+1)Q in two directions for any real (and maybe complex) reflection group W. We call these the "noncrossing parking space" and the "algebraic parking space". These new parking spaces are actually W × C-modules, where C is the cyclic group generated by a Coxeter element. Our Main Conjecture says that the NC parking space and the algebraic parking space are W × C-isomorphic. A uniform proof of this conjecture would solve several open problems in the subject.

## Applied Math Seminar

### A Problem in Transitioning from Spatial to Landscape Ecology:Perspectives from Several Modeling Formulations:Part 1- Reaction-diffusion, Discrete-diffusion and Metapopulation Approaches

Friday, November 4, 2011, 5:00pm
Ungar Room 402

Abstract: In this talk we compare and contrast the predictions of some spatially explicit and implicit models in the context of a thought problem at the interface of spatial and landscape ecology. The situation we envision is a one-dimensional spatial universe of infinite extent in which there are two disjoint focal patches of a habitat type that is favorable to some specified species. We assume that neither patch is large enough by itself to sustain the species in question indefinitely, but that a single patch of size equal to the combined sizes of the two focal patches provides enough contiguous favorable territory to sustain the given species indefinitely. When the two patches are separated by a patch of unfavorable matrix habitat, the natural expectation is that the species should persist indefinitely if the two patches are close enough to each other but should go extinct over time when the patches are far enough apart. Our focus here is to examine how different mathematical regimes may be employed to model this situation, with an eye toward exploring the trade-off between the mathematical tractability of the model on one hand and the suitability of its predictions on the other. In particular, we are interested in seeing how precisely the predictions of mathematically rich spatially explicit regimes (reaction-diffusion models, integro-difference models) can be matched by those of ostensibly mathematically simpler spatially implicit patch approximations (discrete-diffusion models, average dispersal success matrix models).

Joint work with Chris Cosner (University of Miami) and William Fagan (University of Maryland).

## Geometry and Physics Seminar

### Smooth Time Functions for Stably Causal and Non-stably Causal Manifolds

Wednesday, November 2, 2011, 4:00pm
Ungar Room 506

Abstract: Existence of smooth time functions on stably causal Lorentzian manifolds has finally been established 5/6 years ago by Bernal and Sanchez, About the same time with Antonio Siconolfi we obtained a proof that is also valid for cone structures on manifolds. Our approach uses ideas that we have developed to construct smooth subsolutions of the Hamilton-Jacobi Equation. We will first explain the ideas of this approach.

Time permitting in a second part, we will explain the current development, with Antonio Siconolfi and Pierre Pageault, where we have been able to understand how to obtain degenerate smooth time functions for a general cone structure on a manifold that gives a genuine time function on the stably causal part.

## Geometry and Physics Seminar

### Flexible Varieties

Wednesday, October 19, 2011, 4:00pm
Ungar Room 506

## Geometry and Physics Seminar

### Hypersurfaces with Nonnegative Scalar Curvature and the Positive Mass Theorem

Wednesday, October 12, 2011, 4:00pm
Ungar Room 506

Abstract: Since the time of Gauss, geometers have been interested in the interplay between the intrinsic metric structure of hypersurfaces and their extrinsic geometry from the ambient space. For example, a result of Sacksteder tells us that if a complete hypersurface has non-negative sectional curvature, then its second fundamental form in Euclidean space must be positive semi-definite.

In a recent joint work with Damin Wu, we study hypersurfaces under a much weaker curvature condition. We prove that a hypersurface with nonnegative scalar curvature which is either closed or complete of finite many regular ends must be weakly mean convex. This result is optimal in the sense that the scalar curvature cannot be replaced by other k-th mean curvatures. The result and argument have applications to the mean curvature flow, positive mass theorem, and rigidity theorems.

## Applied Math Seminar

### Within-host Dynamics of Malaria Infection with Immune Responses -with an Introduction of Research by Some Nobel Laureates in Medicine or Physiology

Friday, October 7, 2011, 5:00pm
Ungar Room 402

Abstract: On Monday (October 3), three scientists won this year's Nobel Prizes in Medicine or Physiology for their discoveries on how the innate and adaptive phases of the immune response are activated and thereby provided novel insights into disease mechanisms. Their work has opened up new avenues for the development of prevention and therapy against infections, cancer, and inflammatory diseases. In this talk I'll introduce how both innate immunity and adaptive immunity fight again malaria infection and model the within-host dynamics of malaria infection with immune response. I will show that synchronization with regular periodic oscillations (of period 48 h) occurs in blood-stage malaria infection.

## Applied Math Seminar

### Modeling the Evolution of Conditional Dispersal in Spatially Heterogeneous Environments

Friday, September 23, 2011, 5:00pm
Ungar Room 402

Abstract: Mathematical models predict that in environments that are heterogeneous in space but constant in time, there will be selection for slower rates of unconditional dispersal, including specifically random dispersal by diffusion. However, some types of unconditional dispersal may be unavoidable for some organisms, and some organisms may disperse in ways that depend on environmental conditions. In some cases, models predict that certain types of conditional dispersal strategies may be evolutionarily stable within a given class of strategies. For environments that vary in space but not in time those strategies are often the ones that lead to an ideal free distribution of the population using them, provided that such strategies are available within the class of feasible strategies.

Problems in the evolution of dispersal have been addressed from two complementary mathematical viewpoints, namely game theory and mathematical population dynamics. This talk will describe some results and open problems from the viewpoint of spatially explicit models in population dynamics, specifically reaction-diffusion-advection models. Some of the results and problems are related to the evolutionary stability of dispersal strategies leading to an ideal free distribution and the mechanisms that might allow organisms to realize such strategies.

## Geometry and Physics Seminar

### Quasi-Einstein Metrics and Conformal Geometry

Thursday, September 1, 2011, 4:00pm
Ungar Room 402

Abstract: Quasi-Einstein Metrics are an important class of metrics which include Einstein metrics, static metrics, and gradient Ricci solitons. Except for gradient Ricci solitons, these metrics all admit a natural formulation in conformal geometry. Moreover, this conformal formulation is reflected in many aspects of the study of gradient Ricci solitons. We will introduce and describe this conformal formulation in two contexts. First, we will use it to prove a precompactness theorem for compact quasi-Einstein metrics, yielding in particular convergence to gradient Ricci solitons. Second, we will use it to introduce the tractor calculus to quasi-Einstein metrics, which will yield some insights into the basic structure of quasi-Einstein metrics.

## Geometry and Physics Seminar

### Categorifications of the Polynomial Ring Z[x]

Wednesday, August 31, 2011, 4:00pm
Ungar Room 506

Abstract: We develop a diagrammatic categorification of the polynomial ring Z[x], based on a geometrically defined graded algebra, and show how to lift various operations on polynomials to the categorified setting. Our categorification satisfies a version of Bernstein-Gelfand-Gelfand reciprocity property with the indecomposable projective modules corresponding to xn and standard modules to (x-1)n in the Grothendieck ring. Generalization of this approach leads to categorification of the Chebyshev, Hermite, and other orthogonal polynomials. This is joint work with M. Khovanov.

## Combinatorics Seminar

### Abacus Models for Parabolic Quotients of Affine Weyl Groups

Wednesday, August 3, 2011, 4:45pm
Ungar Room 402

Abstract: The cosets of a finite Weyl group inside the corresponding affine Weyl group have remarkable structure with connections to various objects in algebra and geometry. The abacus is a versatile combinatorial model for these cosets that originates in the work of James and Kerber for the symmetric group. We describe generalizations of this model for the affine types B, C, and D.

## Workshop: Topics in Mathematical Relativity

### Second Variation of Wang-Yau Quasi-local Energy

Thursday, July 28, 2011, 4:00pm
Ungar Room 411

Abstract: Recently Wang and Yau have introduced a new concept of quasi-local energy associated to an admissible function on a closed spacelike two-surface. The Wang-Yau quasi-local mass is then defined as the infimum of the quasi-local energy over all admissible functions. In this talk, we provide some remarks on the second variation of this quasi-local functional.

## Workshop: Topics in Mathematical Relativity

### On the Bartnik Splitting Conjecture

Thursday, July 28, 2011, 3:00pm
Ungar Room 411

## Workshop: Topics in Mathematical Relativity

### Periodic Geodesics on Compact Lorentzian Manifolds with a Killing Vector Field

Tuesday, July 26, 2011, 3:00pm
Ungar Room 411

Abstract: In this talk we prove a compactness result for subgroups of the isometry group of a compact Lorentzian manifold with a Killing vector field which is timelike somewhere. As a consequence, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is never vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics.

## Workshop: Topics in Mathematical Relativity

### Local Flatness in Asymptotically Flat Spacetimes

Thursday, July 21, 2011, 3:00pm
Ungar Room 411

Abstract: It is known that an asymptotically simple solution to the vacuum Einstein equations having a null line has to be isometric to Minkowski space. Here we present a result that guarantees the local flatness and 1-connectedness of vacuum solutions having a null line for the broader class of asymptotically flat and globally hyperbolic spacetimes.

## Dissertation Defense

### Quasisymmetric Functions and Permutation Statistics for Coxeter Groups and Wreath Product Groups

Wednesday, July 20, 2011, 10:00am
Ungar Room 301

## Dissertation Defense

### Fitness Dependent Dispersal in Intraguild Predation Communities

Tuesday, July 12, 2011, 10:00am
Ungar Room 301

## Geometry and Physics Seminar

### Cycles and Sheaves: A 45 Minute Intro

Wednesday, April 27, 2011, 4:00pm
Ungar Room 506

## Combinatorics Seminar

### Partial Matroid Decomposition Posets

Monday, April 25, 2011, 5:00pm
Ungar Room 402

Abstract: The partition lattices are a fundamental class in the theory of posets, exhibiting an array of nontrivial properties. The study of q-analogues of these lattices has yielded large families of interesting posets, though often without some degree of "niceness". In this talk, we introduce a new q-analogue and some of its properties, including a generalization to a larger family of posets.

## Geometry and Physics Seminar

### An Index Theorem for End-periodic Operators

Wednesday, April 13, 2011, 4:00pm
Ungar Room 506

Abstract: I will present a new index theorem which generalizes to manifolds with periodic ends the index theorem of Atiyah, Patodi and Singer. This is a joint project with Tomasz Mrowka and Daniel Ruberman.

## Combinatorics Seminar

### Critical Groups of Simplicial Complexes

Monday, April 11, 2011, 5:00pm
Ungar Room 402

Abstract: The critical group of a graph G is a finite abelian group K(G) whose order is the number of spanning trees of G. We generalize the definition of the critical group from graphs to simplicial complexes. Specifically, given a simplicial complex X of dimension d, we define a family of finite abelian groups K_0(X), ..., K_{d-1}(X) in terms of combinatorial Laplacian operators, generalizing the construction of K(G). We show how to compute the groups K_i(X) explicitly as cokernels of reduced Laplacians, and prove that they are finite, with orders given by weighted enumerators of simplicial spanning trees. We describe the groups completely for the cases that X is (a) a simplicial sphere or (b) a skeleton of a simplex; the latter result uses work of M. Maxwell. If time permits, I will talk about how to interpret the critical groups in terms of higher-dimensional analogues of flows in graphs, and/or another potential interpretation as discrete analogues of Chow groups. This is joint work with Art Duval and Carly Klivans.

## Combinatorics Seminar

### Unimodality of q-Eulerian Numbers and p,q-Eulerian Numbers, Part II

Monday, April 4, 2011, 5:00pm
Ungar Room 402

Abstract: This talk is a continuation of a seminar talk I gave last fall. First I will review the previous talk, which focused on my work with Shareshian on unimodality of q-Eulerian polynomials, and then I will present my more recent work with Henderson on the cycle type refinement of the unimodality result.

The Eulerian numbers enumerate permutations in the symmetric group S_n by their number of excedances or by their number of descents. It is well known that they form a unimodal palindromic sequence of integers. In this talk, which is based on joint work with John Shareshian and Anthony Henderson, we consider the q-analog of the Eulerian numbers obtained by considering the joint distribution of the major index and the excedance number, and the p,q-analog of the Eulerian numbers obtained by considering the multivariate distribution of the major index, descent number and excedance number. We show that the q-Eulerian numbers form a unimodal palindromic sequence of polynomials in q and the p,q-Eulerian numbers refined by cycle type form a unimodal palindromic sequence of polynomials in p and q. The proofs of these results rely on the Eulerian quasisymmetric functions introduced by Shareshian and Wachs, on symmetric and quasisymmetric function theory, and on representation theory of the symmetric group.

## Geometry and Physics Seminar

### Braid Monodromy, Floer Combinatorics and Fukaya Category

Monday, April 4, 2011, 4:00pm
Ungar Room 411

## Geometry and Physics Seminar

### Combinatorics of Multiplier Ideal Sheaves III

Wednesday, March 30, 2011, 4:00pm
Ungar Room 506

## Geometry and Physics Seminar

### Combinatorics of Multiplier Ideal Sheaves II

Thursday, March 24, 2011, 4:00pm
Ungar Room 411

## Geometry and Physics Seminar

### Strong Form of the Grothendieck Section Conjecture in Functional Case

Wednesday, March 16, 2011, 4:00pm
Ungar Room 506

Abstract: In the talk I will give a proof of the Grothendieck section conjecture in the following form. Let $f : X\to Y$ be a surjective map of projective manifolds with an irreducible generic fiber and $f_a : G_a(X)\to G_a(Y)$ the corresponding map between pro-$l$-abelian Galois groups of the algebraic closures of the fields $k(X),k(Y)$ respectively, i.e. if we denote the Galois group $Gal(\bar k(X)/k(X))$ as $G_X$ then $G_a(X)= (G_X/[G_X,G_x])_l$ where $l$ stand for maximal pro-$l$- quotient and $l\neq k(X)$If there is a rational section $s: Y\to X$ then there are associated group sections $s^a : G_a(Y)\to G_a(X)$ (usually nonunique) with $f_a s_a = id$. The problem we are trying to solve is what conditions have to be imposed on $s^a$ so that it is associated to a rational section. It is clear that geometric section $s$ provides with a possiblity to lift the group section $s^a$ to the section of for $s^g : G_Y\to G_X$ for the surjective map of the Galois groups $f_g : G_X\to G_Y$. Since we are dealing with $l$-quotient only we will also consider geometric $p$-section. The latter correspond to the sections for induced maps $f^F :X^F\to Y^F$ where $Y^F$ is model of a purely inseparable extension of $k(Y)$ and $f_F,X^F$ are induced from $f,X$ by the map $Y^F\to Y$ (which is geometrically identical map).

Theorem: Assume that the ground field $k= \bar F_p,p\neq l$ and $dim Y \geq 2$. Let $s^a : G_a(Y)\to G_a(X)$ be a group section which image is a closed subgroup with additional property: for any pair $x,y\in G_a(Y)$ such that preimages $\tiled x,\tilde y\in (G_Y/[[G_Y,G_Y] G_Y])_l$ commute the images $s^a(x),s_a(y)$ have the same property with respect to $(G_X/[[G_X,G_X] G_X])_l$. Then there is a rational $p$-section $s : Y^F\to X^F$ some $Y^F$ such that $s^a$ is associated to $s$.

Note that since $G_Y/[[G_Y,G_Y] G_Y])_l$ is central extesnion of $(G_X/[[G_X,G_X])_l= G_a(Y)$ the property that $\tiled x,\tilde y$ commute in $G_Y/[[G_Y,G_Y] G_Y])_l$ does not depend on $x,y$.

The proof in general functional case with $k$-algebraically closed is similar but is technically more invloved and hence is not yet completed. The initial Grothedieck conjecture states a similar correspondence for a Galois group $s': G_Y\to G_X$ and in our approach we derive the result from minimal noncommutative quotients : $(G_Y/[[G_Y,G_Y] G_Y])_l$ and $(G_X/[[G_X,G_X] G_X])_l$. We hope that the result and the method ( after some modifications) can be extended to the case of arbitrary field $k$. and may be even to the case when $Y$ is a curve over arithmetic field.

It is a joint work with Yuri Tschinkel. In essence it is a corollary of the description of commuting pairs of elements in $G_Y/[[G_Y,G_Y] G_Y])_l$ which was obtained some time ago.

## Geometry and Physics Seminar

### Hidden Symmetries, Exceptional Surgeries, and Commensurability

Tuesday, March 9, 2011, 4:00pm
Ungar Room 506

Abstract: Two manifolds are in the same commensurability class if they share a common finite sheeted cover. Commensurability classes of hyperbolic 3-manifolds have infinitely many elements, so it is appealing to find types of manifolds that are rare in a commensurability class, eg knot complements. In 2006, Reid and Walsh conjectured that there are at most three hyperbolic knot complements in a given commensurability class. Recently, Boileau, Boyer, Cebanu, and Walsh announced that the conjecture holds in the case of no hidden symmetries. After providing some of the necessary background, I will talk about obstructions to knot complements admitting hidden symmetries.

## Combinatorics Seminar

### Path Tableaux and Combinatorial Interpretations for S_n-class Functions

Monday, March 7, 2011, 5:00pm
Ungar Room 402

Abstract: Around 1991, Goulden-Jackson, Greene, Haiman, Stanley, and Stembridge studied the evaluation of S_n class functions on generating functions in Z[S_n] which are products of Kazhdan-Lusztig basis elements. This led Stembridge to prove algebraically that irreducible S_n-characters evaluate nonnegatively on the Z[S_n] generating functions, and to conjecture that related "monomial virtual characters" have the same property. We point out that the analogous result for induced sign characters, which follows from the earlier Littlewood-Merris-Watkins identity, has a nice combinatorial interpretation. Using this interpretation, we combinatorially prove special cases of the Stembridge result and conjecture. We also conjecture a combinatorial interpretation for a known q-analog of the Littlewood-Merris-Watkins identity, and relate this to Haimans q-analogs of Stembridge's result and conjecture.

This is joint work with Brittany Shelton and Sam Clearman of Lehigh University.

## Geometry and Physics Seminar

### Combinatorics of Multiplier Ideal Sheaves

Wednesday, March 2, 2011, 4:00pm
Ungar Room 506

## Geometry and Physics Seminar

### The Riemannian Positive Mass and Penrose Inequalities for Graphs over R^n

Tuesday, March 1, 2011, 5:00pm
Ungar Room 402

Abstract: The Riemannian positive mass theorem asserts that an asymptotically flat Riemannian manifold M with nonnegative scalar curvature R has nonnegative ADM mass, and that the mass is strictly positive unless M is isometric to flat Euclidean space. If M contains an area outer minimizing horizon, the Riemannian Penrose inequality gives a positive lower bound to the ADM mass in terms of the area of the horizon. For manifolds that are graphs over R^n, we are able to prove stronger versions of the above inequalities by bounding the ADM mass from below with an integral of the product of R and a nonnegative potential function. I will give an overview of some previously known results before discussing our approach.

## Geometry and Physics Seminar

### Spectra as Cohomology Theory II

Wednesday, February 23, 2011, 3:30pm
Ungar Room 506

## Geometry and Physics Seminar

### Orbifolds, Ghost Loop Spaces and Twisted Sectors

Wednesday, February 23, 2011, 2:00pm
Ungar Room 411

## Geometry and Physics Seminar

### Non-compact Topological Field Theories and Frobenius Structures

Tuesday, February 22, 2011, 4:00pm
Ungar Room 411

## Geometry and Physics Seminar

### Regular Value Theorem in a Fractal Setting

Monday, February 21, 2011, 5:00pm
Ungar Room 506

Abstract: The classical regular value theorem says that if $f: X \to Y$ is an immersion, where $X,Y$ are smooth manifolds of dimension $n,m$, $n>m$, respectively, then the set $\{x \in X: f(x)=y \}$ is either empty or is an $n-m$ dimensional sub-manifold of $X$. We shall see that a suitable analog of this result is available if a manifold $X$ is replaced by a set of sufficiently large Hausdorff dimension and the function $f$ satisfies a "rotational curvature" condition. Regularity of generalized Radon transforms plays a key role. Sharpness results are based on an interplay between ideas from discrete geometry and number theory.

## Geometry and Physics Seminar

### Relativistic Fluids in 1+1 Dimensions with a Vacuum Boundary

Wednesday, February 16, 2011, 5:00pm
Ungar Room 402

Abstract: Relativistic isentropic fluids are characterized by their density, velocity, and pressure. The evolution of these fluids is governed by the relativistic Euler equations. In regions where the density is bounded away from zero, it is known how to write the Euler equations as a symmetric hyperbolic system. This allows for the use of standard theory to guarantee the well-posedness (i.e. local existence and uniqueness of solutions) of the Euler equations. However, fluids with compact support for which the pressure and density vanish simultaneously at the boundary between the fluid and the vacuum region, the known symmetric hyperbolic formulations of the Euler equations become degenerate at the vacuum boundary, and consequently, standard existence theory no-longer applies.

Until very recently, it was a long standing open problem to prove the existence of solutions to the Euler equations with a vacuum boundary that have non-zero fluid acceleration at the boundary. Physically, these type of solutions represent bodies such as stars that can be either static, expanding, or collapsing. In 2009, first in 1+1 spacetime dimensions and subsequently 3+1 dimensions, the existence of solutions to the non-relativistic Euler equations with non-zero acceleration at the fluid vacuum boundary was established by two different groups using non-standard energy estimates combined with suitable approximation techniques. The arguments used to establish existence are technical, involved, highly original, and quite different from one another.

In this talk, I will, after first providing a introduction to the relativistic and non-relativistic equations, describe the history of the problem and describe the major developments leading up the breakthrough existence results of 2009. I will also outline a new method for establishing the existence of solutions to relativistic Euler equations that have non-zero acceleration at the vacuum boundary. In contrast to the previous existence results, mine are rather straightforward, relying only on routine computation, some elementary geometry, and standard hyperbolic theory for initial boundary value problems, while, at the same time, producing very explicit representations of the solutions that are applicable to both the relativistic and non-relativistic settings.

## Geometry and Physics Seminar

### Free Boundary Problem for Embedded Minimal Surfaces

Tuesday, February 1, 2011, 5:00pm
Ungar Room 402

Abstract: For any smooth compact Riemannian 3-manifold with boundary, we prove that there always exists a smooth, embedded minimal surface with (possibly empty) free boundary. We also obtain a priori upper bound on the genus of such minimal surfaces in terms of the Heegard genus of the ambient compact 3-manifold. An interesting note is that no convexity assumption on the boundary is required. In this talk, we will describe the min-max construction for the free boundary problem, and then we will sketch a proof of the existence part of the theory.

## Geometry and Physics Seminar

### Spectra as Cohomology Theory

Wednesday, January 19, 2011, 4:00pm
Ungar Room 402

## Combinatorics Seminar

### Moon Polyominoes, Pipe Dreams and Simplicial Spheres

Monday, November 29, 2010, 5:00pm
Ungar Room 402

Abstract: We exhibit a canonical connection between maximal (0,1)-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable, and thus shellable, sphere. In particular, this implies a positivity result for Schubert polynomials.

## Applied Math Seminar

### A Mathematical Study

Friday, November 12, 2010, 4:00pm
Ungar Room 402

Abstract: Here we have considered a three-component model consisting of non-toxic phytoplankton (NTP), toxin producing phytoplankton (TPP) and zooplankton (Z), where the growth of zooplankton species reduce due to toxic chemicals released by phytoplankton species. We have taken into account the competition between TPP and NTP and tried to observe its effect on the marine ecosystem, both in the presence and absence of the environmental fluctuation. We observe that competition helps in the coexistence of the species, but if the effect of competition is very high on the TPP population, it results in the planktonic bloom.

Next we have proposed a three component model consisting of dissolved limiting nutrient (N) supplied at constant rate and partially recycled after the death of plankton by bacterial decomposition, phytoplankton (P) and zooplankton (Z), where the growth of zooplankton species reduce due to toxic chemicals released by phytoplankton species. Our analysis leads to different thresholds which are expressible in terms of model parameters and determine the existence and stability of various states of the system.

On combining the above two models we have studied a third one consisting of nutrient, non-toxic phytoplankton, toxin producing phytoplankton and their predator zooplankton population in open marine system. It is observed that nutrient- phytoplankton-zooplankton interactions are very complex and situation specific. Different exciting results, ranging from stable situation to cyclic blooms may occur under different favorable conditions, which may give some insights for predictive management.

## Geometry and Physics Seminar

### Inverse Spectra Problem in Algebraic Geometry

Tuesday, November 9, 2010, 5:00pm
Ungar Room 402

## Applied Math Seminar

### Time Periodic Traveling Wave Solutions of Reaction-diffusion Systems

Friday, November 5, 2010, 4:00pm
Ungar Room 402

Abstract: The study of traveling wave solutions for parabolic equations and systems is an area of great interest, not only in the applications of the waves themselves but also in their use in gaining a better understanding of phenomena in large domains. Typically, traveling wave solutions arise from a competition between two equilibria and describe the transition processes that appear in many areas of biology, chemistry and physics. Over the past three decades, there have been many interesting studies on (stationary) traveling wave solutions to reaction-diffusion systems for which the corresponding kinetic systems are autonomous. In this presentation, I will talk about our recent work concerning time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion. I will focus on basic problems of wave existence, uniqueness of waves, and stability of waves.

## Combinatorics Seminar

### Unimodality of q-Eulerian Numbers and p,q-Eulerian Numbers

Monday, November 1, 2010, 5:00pm
Ungar Room 402

Abstract: The Eulerian numbers enumerate permutations in the symmetric group S_n by their number of excedances or by their number of descents. It is well known that they form a unimodal palindromic sequence of integers. In this talk, which is based on joint work with John Shareshian and Anthony Henderson, we consider the q-analog of the Eulerian numbers obtained by considering the joint distribution of the major index and the excedance number, and the p,q-analog of the Eulerian numbers obtained by considering the multivariate distribution of the major index, descent number and excedance number. We show that the q-Eulerian numbers form a unimodal palindromic sequence of polynomials in q and the p,q-Eulerian numbers refined by cycle type form a unimodal palindromic sequence of polynomials in p and q. The proofs of these results rely on the Eulerian quasisymmetric functions introduced by Shareshian and Wachs, on symmetric and quasisymmetric function theory, and on representation theory of the symmetric group.

## Applied Math Seminar

### Reaction-diffusion-advection Models for the Effects and Evolution of Dispersal

Friday, October 29, 2010, 4:00pm
Ungar Room 402

Abstract: The dispersal of organisms is an important ecological process that can often be described mathematically in terms of diffusion and advection. The dispersal strategy that a species uses can affect its population dynamics and interactions with other species, and those in turn can impose selective pressure on dispersal strategies. Reaction-diffusion-advection models can be used to study the effects and evolution of dispersal strategies. One way to compare dispersal strategies is to construct and analyze models for competing populations that are the same in all ecological respects except their dispersal strategies. In the context of reaction-diffusion-advection models for dispersal in environments that are variable in space but constant in time this approach suggests that the effects of a given dispersal strategy depend on how well it allows a population to match its spatial distribution to the distribution of its resources. A way to understand which dispersal strategies are most likely to evolve is to study the models from the viewpoint of evolutionary stability. (A strategy is evolutionarily stable relative to a given class of strategies if a population using it cannot be invaded by any small population using any other strategy in the class.) There is evidence that evolution favors strategies that let a population match its resources perfectly. This talk will review a number of results and open questions related to those ideas.

## Combinatorics Seminar

### Double Feature

Monday, October 25, 2010, 5:00pm
Ungar Room 506

Abstract: The first part of the talk will be a review of the connection between representation theory and symmetric functions. In the second part we consider a colored analog of Eulerian quasisymmetric functions. Eulerian quasisymmetric functions were introduced by Shareshian and Wachs in order to obtain a $q$-analog of Euler's exponential generating function formula for the Eulerian numbers. They are defined via the symmetric group, and applying the stable and nonstable principal specializations yields formulas for joint distributions of permutation statistics. We consider the group of colored permutations and use this group to introduce colored Eulerian quasisymmetric functions. We derive a formula for the generating function of these colored Eulerian quasisymmetric functions, and show that applying the stable and nonstable principal specializations yields formulas for joint distributions of colored permutation statistics, generalizing Euler's formula.

## Geometry and Physics Seminar

### Structure Theorem for Symmetric Differentials of Rank 1

Wednesday, October 20, 2010, 4:00pm
Ungar Room 402

Abstract: The presence of holomorphic 1-forms on a compact kahler manifold $X$ implies topological properties of $X$. Moreover, from their presence also follows the existence of a holomorphic map from $X$ into a complex torus from which all the holomorphic 1-forms of $X$ are induced from. The talk gives a complete extension of this result to symmetric differentials of rank 1. This result belongs to the program whose aim is to understand the class of symmetric differentials that have a close to topological nature (symmetric differentials of rank 1 will be shown to be closed symmetric differentials).

## Combinatorics Seminar

### An Explicit Derivation of the Möbius Function for Bruhat Order

Monday, October 18, 2010, 5:00pm
Ungar Room 506

Abstract: We give an explicit non-recursive complete matching for the Hasse diagram of the strong Bruhat order of any interval in any Coxeter group. This yields a new derivation of the Möbius function, recovering a classical result due to Verma. The matching is given in terms of combinatorial objects called masks that arise in Deodhar's formula for the Kazhdan–Lusztig polynomials, and has connections to Armstrong's sorting order on Coxeter groups.

## Geometry and Physics Seminar

### Spectra, Gaps and Chow Groups

Friday, October 15, 2010, 4:00pm
Ungar Room 506

## Geometry and Physics Seminar

### On a Localized Riemannian Penrose Inequality

Wednesday, October 13, 2010, 4:00pm
Ungar Room 402

Abstract: Given a compact, orientable, three dimensional Riemannian manifold with boundary, we call it "a body surrounding horizons" if its boundary is the disjoint union of two pieces: the outer boundary and the horizon boundary, where the outer boundary is a topological 2-sphere and the horizon boundary is the unique minimal surface in the manifold. Such a manifold can be thought as a bounded region, surrounding the outermost apparent horizons of black holes, in a time-symmetric slice of a space-time in the context of general relativity. Physically, one expects that there exists a geometric quantity computed from the area and the mean curvature of the outer boundary that can be estimated from below by the area of the horizon boundary. In the case that the manifold is non-compact whose outer boundary is replaced by an asymptotically flat end, such an expectation then leads to the Riemannian Penrose Inequality. In this talk, we establish an inequality of this type on a body surrounding horizons whose outer boundary is metrically a round sphere. Its potential role in suggesting the right concept of quasi-local mass will also be discussed.

## Combinatorics Seminar

### Cellular Resolutions of Hypergraph Edge Ideals

Monday, October 4, 2010, 5:00pm
Ungar Room 506

Abstract: Given an ideal I in the polynomial ring S = k[x_1,...,x_n], a basic problem in commutative algebra is to describe a (minimal) free resolution of I. One particularly geometric method is through the construction of a 'cellular resolution', where the syzygies of I are encoded by the faces of a polyhedral (or more general CW) complex. If our ideal is square-free and generated in a fixed degree d, then its generators can be thought of as the edges of a (hyper)graph G on n vertices; this defines the edge ideal I_G. In this talk we construct a polyhedral complex X_G whose vertices encode the directed edges of G. Using basic methods of combinatorial topology we show that X_G supports a minimal cellular resolution of I_G whenever G is what we call 'cointerval'. This class of graphs corresponds to the complement of interval graphs (when d = 2), and in general strictly contains the class of hypergraphs corresponding to pure shifted complexes whose resolutions were described by Nagel and Reiner. Furthermore, the complexes X_G admit natural embeddings into certain 'mixed subdivisions' of dilated simplices, allowing us to draw some nice pictures. This is joint work with Alex Engstrom (UC Berkeley).

## Combinatorics Seminar

### Geometry of Arrangements of Hyperplanes and Cohomology of Orlik-Solomon Algebras

Monday, September 27, 2010, 5:00pm
Ungar Room 506

Abstract: The Orlik-Solomon algebra is a combinatorial invariant of an arrangement of hyperplanes. Part of its multiplicative structure is encoded in an invariant called the resonance variety, which has unexpected linearity properties. Calculation of resonance varieties leads to questions about non-linear geometry involving arrangements. I will survey general properties of Orlik-Solomon algebras and more recent results and open problems on resonance varities.

## Applied Math Seminar

### Modeling the Transmission Dynamics and Control of Hepatitis B Virus

Friday, September 24, 2010, 4:00pm
Ungar Room 402

Abstract: Hepatitis B is a potentially life-threatening liver infection caused by the hepatitis B virus (HBV) and is a major global health problem. HBV is the most common serious viral infection and a leading cause of death in mainland China. Around 130 million people in China are carriers of HBV, almost a third of the people infected with HBV worldwide and about 10% of the general population in the country; among them 30 million are chronically infected. Every year, 300,000 people die from HBV-related diseases in China, accounting for 40-50% of HBV-related deaths worldwide. Despite an effective vaccination program for newborn babies since the 1990s, which has reduced chronic HBV infection in children, the incidence of hepatitis B is still increasing in China. Based on the HBV data from China, we first propose an ordinary differential equation model to describe the transmission dynamics and prevalence of HBV infection. The model provides an approximate estimate of the basic reproduction number is 2.406 in China which indicates that hepatitis B is endemic in China and is approaching its equilibrium with the current immunization program and control measures. Taking the fact that age structure is one of the characteristics of HBV transmission, we also propose an age-structured model. By determining the basic reproduction number, we study the existence and stability of the disease-free and endemic steady state solutions of the model and explore optimal strategies for controlling the transmission of HBV.

## Combinatorics Seminar

### The Ish Arrangement of Hyperplanes

Monday, September 20, 2010, 5:00pm
Ungar Room 506

Abstract: The Shi arrangement of hyperplanes plays an important role in the representation theory of affine Weyl groups. In type A, this arrangement is Shi(n)=\{x_i-x_j=0, x_i-x_j=1 : 1\leq i<j\leq n\}. The arrangement Shi(n) divides \R^n into (n+1)^{n-1} regions --- an interesting number, yes? --- and it has beautiful combinatorics. In this talk I will define a new hyperplane arrangement Ish(n), which I call the Ish arrangement. You will like this hyperplane arrangement. (Some of this is joint work with Brendon Rhoades.)

## Geometry and Physics Seminar

### Tropical Geometry, Compactifications, and Birational Geometry

Wednesday, August 25, 2010, 4:30pm
Ungar Room 402

Abstract: Tropical geometry is a collection of methods which replace algebro-geometric objects with certain polyhedral complexes and aims to give combinatorial interpretations of constructions in algebraic geometry. In this talk I'll give a (friendly) survey of the foundations of tropical geometry and some of its applications. Particular attention will be given to the "geometric tropicalization" approach of Hacking, Keel, and Tevelev which relates tropical geometry to the boundary structure of compactifications of varieties, and in turn gives combinatorial manifestations of some constructions in (log) birational geometry.

## Geometry and Physics Seminar

### Alexander Invariants of Fundamental Groups of the Complements to Plane Algebraic Curves

Tuesday, August 24, 2010, 4:00pm
Ungar Room 402

Abstract: This will be an introductory talk to the role and properties of Alexander polynomials and their generalizations in the context of Algebraic geometry. I also will discuss main problems and conjectures in this theory.

## Combinatorics Seminar

### Triangulations, the Associahedron and Gamma-vectors for Planar Lattices

Wednesday, July 7, 2010, 4:00pm
Ungar Room 506

Abstract: The talk is divided into two parts. In the first part we consider posets given as the product of two chains $C_k \times C_{n-k}$. We construct a special reverse lexicographic triangulation of the order polytope of $C_2 \times C_{n-2}$ which is abstractly isomorphic to the join of a simplex with the associahedron. It remains open if there is a meaningful generalization of this result to general k. In the second part of the talk we focus on Gal's conjecture in the special setup of planar lattices. It was already shown by Bränden that Gal's conjecture holds for $C_2\times C_{n-2}$. Being the lattice of order ideals of this poset planar or equivalently being the poset of width 2 we ask if the conjecture is true in this greater generality. We are able to answer this question in the affirmative and give some hints how one could proceed for posets of width at least 3. This is joint work with Kathrin Vorwerk.

## Geometry and Physics Seminar

### How Not to Learn Minimal Model Program

Thursday, April 29, 2010, 4:00pm
Ungar Room 402

## Geometry and Physics Seminar

### Integrated Density of States of Schröedinger Operators with Periodic and Almost-periodic Potentials

Wednesday, April 14, 2010, 4:00pm
Ungar Room 402

Abstract: I will discuss new results (joint with R. Shterenberg) on the asymptotic behaviour of the integrated density of states of a Schrödinger operator $H=-\Delta+b$ acting in $\R^d$ when the potential $b$ is either smooth periodic, or generic quasi-periodic (finite linear combination of exponentials), or belongs to a wide class of almost-periodic functions.

## Combinatorics Seminar

### On Lattice Path Matroids and Polymatroids

Friday, April 9, 2010, 5:00pm
Ungar Room 506

Abstract: Lattice path matroids are an especially tractable class of transversal matroids whose bases are in correspondence with planar lattice paths. We discuss some enumerative properties of these matroids, one of which leads naturally to a related class of discrete polymatroids. We then examine these polymatroids and their toric ideals. Finally, we provide generating sets and Gröbner bases for these ideals, and discuss many possible directions for future research. No previous knowledge of matroid theory or toric ideals will be assumed.

## Geometry and Physics Seminar

### On Uniformly Effective Boundedness of Shafarevich Conjecture-type

Wednesday, April 7, 2010, 4:00pm
Ungar Room 402

Abstract: The talk deals with uniformly effective versions of the classical Shafarevich Conjecture over function fields (aka Parshin-Arakelov Theorem). We will discuss the speaker's effective solution to the classical case and his recent extension to the case where the fibers are canonically polarized compact complex manifolds. In the proofs, Chow varieties play a key role.

## Combinatorics Seminar

### On q-analogs of the k-equal Partition Lattice

Tuesday, April 6, 2010, 5:00pm
Ungar Room 506

Abstract: The ordinary k-equal partition lattice served as the original motivating example for Björner and Wachs to extend the notion of lexicographic shellability of posets from the pure case to more general nonpure cases. In this talk, we discuss the construction of a family of q-analogs to this lattice and a common edge labeling which indicates that each is a shellable poset. We also describe methods for counting falling chains in these lattices, as well as conjectures for certain special cases which greatly improve the computational time necessary to determine the total number of chains.

## Geometry and Physics Seminar

### On the Present State of the Andersen-Lempert Theory

Wednesday, March 24, 2010, 4:00pm
Ungar Room 402

Abstract: We discuss a theory of completely integrable algebraic (resp. holomorphic) vector fields on smooth affine algebraic varieties.

## Combinatorics Seminar

### Cyclic Sieving and Polygon Multidissection Enumeration

Tuesday, March 23, 2010, 5:00pm
Ungar Room 506

Abstract: Let X be a finite set, C = \langle c \rangle be a finite cyclic group acting on X, \zeta be a root of unity of multiplicative order |C|, and X(q) \in \mathbb{Z}[q] be a polynomial with integer coefficients. Following Reiner, Stanton, and White, we say the triple (X, C, X(q)) exhibits the cyclic sieving phenomenon (CSP) if for any d \geq 0, the fixed point set cardinality |X^{c^d}| equals the polynomial evaluation X(\zeta^d). We prove a collection of CSPs related to the action of rotation on multidissections of polygons, i.e., dissections where edges can occur with multiplicity and boundary edges may or may not be included. Our proofs involve modelling the action of rotation via general linear group representations and use geometric realizations of finite type cluster algebras due to Fomin and Zelevinsky.

## Geometry and Physics Seminar

### Galois Groups and Birational Invariants of Functional Fields

Wednesday, March 17, 2010, 4:00pm

Ungar Room 402

Abstract: I want to discuss our joint results with Yuri Tschinkel Bloch-Kato conjecture implies that any element in tht cohomology of algebraic variety with finite coefficients after restriction to some open subariety can be induced from abelian quotient of the fundamantal group of the latter. Our theorem on the structure of the Galois groups of functional fields implies a similar result for nonramified cohomology. Namely for any element $a$ of nonramified cohomology $H^i_{nr}(Gal(\bar K/ K, Z_{l^n}, i\geq 2, K=\bar F_p(X), p\neq l)$ there is a finite topological quotient $G^c$ of $Gal(\bar K/ K)$ such that $a$ is induced from a nonramified element $b$ of $H^i_{nr}(G^c, Z_{l^n}$. Here $G^c$ is a finite group which is a central extension of an abelian group. It has the following geometric interpretation: there exists a rational map $f :X \to \prod P^i/ A$ and a nonramified element $b\in H^i_{nr}(G^c, Z_{l^n}$ such that $f^*(b)= a$.

## Combinatorics Seminar

### Signed Eulerian Quasisymmetric Functions

Tuesday, March 9, 2010, 5:00pm
Ungar Room 506

Abstract: We introduce signed Eulerian quasisymmetric functions, which are an extension of the Eulerian quasisymmetric functions introduced by Shareshian and Wachs. We define them via the hyperoctahedral group, or group of signed permutations, and we compute their generating function. A central part of this computation is a so called tri-colored necklace bijection, which is an extension of the bi-colored necklaces appearing the work of Shareshian and Wachs, which is in turn an extension of techniques introduced by Gessel and Reutenauer. By applying certain ring homomorphisms to our formula for the generating function, we obtain results for the joint distribution of certain signed permutation statistics. Some of these results are new, although one is a special case of a joint distribution previously computed by Foata and Han, but here an alternate proof is given.

## Combinatorics Seminar

### On the Half-plane Property and the Tutte-group of a Matroid

Tuesday, March 2, 2010, 5:00pm
Ungar Room 506

Abstract: A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. If the polynomial can be chosen with all nonzero coefficients equal to one then the matroid has the half-plane property (HPP). We describe a systematic method that allows us to reduce the WHPP to the HPP for large families of matroids. This method makes use of the Tutte-group of a matroid. We prove that no projective geometry has the WHPP and that a binary matroid has the WHPP if and only if it is regular.

## Geometry and Physics Seminar

### Axiomatic Thin Position and Applications

Thursday, February 18, 2010, 4:00pm
Ungar Room 402

Abstract: The notion of "thin position" has been a powerful tool for understanding surfaces in 3-manifolds and knot complement. However, it has been defined and applied in a number of different ways that are related more in spirit than in details. I will describe an axiomatic framework that allows one to define exactly what is meant by thin position, and which leads to a toolbox of methods that can be used in a number of different settings.

## Combinatorics Seminar

### Nowhere-Harmonic Colorings of Graphs

Wednesday, February 17, 2010, 5:00pm
Ungar Room 506

Abstract: Proper vertex colorings of a graph are related to its boundary map, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, a natural extension of the boundary map, leads us to introduce nowhere-harmonic colorings and analogues of the chromatic polynomial and Stanley's theorem relating negative evaluations of the chromatic polynomial to acyclic orientations. Further, we discuss some examples demonstrating that nowhere-harmonic colorings are more complicated from an enumerative perspective than proper colorings. Our primary tool for these investigations is the theory of "inside-out polytopes," developed by M. Beck and T. Zaslavsky, and the theory of Ehrhart quasi-polynomials for rational polytopes. This is joint work with Matthias Beck of San Francisco State University.

## Geometry and Physics Seminar

### Conjugation Spaces and 4-manifolds

Wednesday, February 17, 2010, 4:00pm
Ungar Room 402

Abstract: A conjugation space is a space X with involution, where the cohomology mod 2 of the fixed set is the same as the cohomology of the space after doubling dimensions. The first example is X = complex projective space, with the involution given by complex conjugation. In the talk I will describe the relation between smooth conjugation 4-manifolds and knotted surfaces in mod 2 homology 4-spheres. This is joint work with Jean-Claude Hausmann.

## Geometry and Physics Seminar

### On Real Determinantal Quartics

Wednesday, February 10, 2010, 4:00pm
Ungar Room 402

Abstract: Let A_0, A_1, A_2, and A_3 be real symmetric 4 x 4 matrices. One can associate to these four matrices a spectral surface in the three dimensional complex projective space CP^3 (the set of points (x_0 : x_1 : x_2 : x_3) in CP^3 such that the determinant of the matrix x_0 A_1 + x_1 A_1 + x_2 A_2 + x_3 A_3 is zero) and a spectrahedron in the three dimensional real projective space RP^3 (the set of points (x_0 : x_1 : x_2 : x_3) in RP^3 such that the matrix x_0 A_1 + x_1 A_1 + x_2 A_2 + x_3 A_3 is semidefinite).

In general, the spectral surface considered has 10 double points. We show that the boundary of the spectrahedron cannot contain more than 8 doubles points of the spectral surface. The proof is based on a study of period spaces of real K3-surfaces.

## Combinatorics Seminar

### Eulerian Quasisymmetric Functions and Cyclic Sieving

Tuesday, February 9, 2010, 5:00pm
Ungar Room 506

Abstract: Certain q-analogs of classical combinatorial numbers exhibit the curious phenomenon of evaluating to a positive integer when q is set equal to an nth root of unity. A stronger phenomenon called the cyclic sieving phenomenon (of Reiner, Stanton and White) is exhibited when these positive integers can be interpreted as the number of fixed points of an element of a cyclic group acting on a set whose size is equal to the classical combinatorial number.

In this talk I will present an instance of the cyclic sieving phenomenon involving a q-analog of the Eulerian numbers and their cycle type refinement. The main tool in proving this result is the Eulerian quasisymmetric functions introduced a few years ago in joint work with Shareshian.

This is joint work with Bruce Sagan and John Shareshian.

## DimaUniversity of Miami

### Motives

Thursday, February 4, 2010, 2:00pm
Ungar Room 547

## Combinatorics Seminar

### Reduced Decompositions of Permutations

Tuesday, February 2, 2010, 5:00pm
Ungar Room 506

Abstract: Consider the group of permutations of {1,2,\ldots, n}, which is generated as a Coxeter group by the adjacent transpositions (i,i+1). The reduced S-decompositions for a permutation \pi are the ways of writing \pi as a product of the fewest adjacent transpositions. A nice result gives a bijection from the (essentially different) reduced S-decompositions of the longest permutation to *rhombic tilings of a regular 2n-gon*.

We will describe an analogous result for the reduced T-decompositions of a permutation (using all transpositions, not just the adjacent ones). We will give a bijection from the (essentially different) reduced T-decompositions of the long cycle to *quadrangulations of a regular 2n-gon*.

We will note some striking similarities between these two results.

## Geometry and Physics Seminar

### On the Work of Lurie

Tuesday, February 2, 2010, 4:00pm
Ungar Room 547

## Geometry and Physics Seminar

### Hodge Structures and Spectra

Tuesday, February 2, 2010, 2:00pm
Ungar Room 547

## Geometry and Physics Seminar

### A infty Categories

Monday, February 1, 2010, 11:00am
Ungar Room 547

## NSF-CSMS Project Industry-Liaison Seminar

### A Career in Software Development, Web-Based Systems

Wednesday, January 27, 2010, 5:00pm
Ungar Room 402

Abstract: Over 40 years ago, Michael Goldberg started a Miami software company called FDP (Financial Data Planning), employing many UM mathematics and computer science students over the years and eventually becoming the leading provider of software for the insurance and pension industries. After selling FDP, Michael started another company 5 years ago called Flamingo Software, specializing in web-based systems for insurance and financial services companies. Find out what a career in software development can be like and what it takes to run a successful software company.

## Geometry and Physics Seminar

### Realizations of Minimal Representation of O(p,q)

Wednesday, December 9, 2009, 3:00pm
Ungar Room 402

Abstract: The "smallest" unitary representation of a non-compact simple Lie group is a surprisingly rich object with many interesting analytic properties. Unlike the metaplectic representation of Sp(2n), which was studied extensively, the minimal representation for the orthogonal groups O(p,q) has not been analyzed in comparable detail. We will discuss the explicit models for this representation, constructed recently by Kobayashi and Orsted, and introduce some applications of these models.

## Geometry and Physics Seminar

### Symplectic Fillings of Contact Manifolds

Friday, December 4, 2009, 3:30pm
Ungar Room 402

Abstract: Work of Loi and Piergalini as well as Akbulut and Ozbagci allows us to create symplectic fillings of contact 3-manifolds from factorizations of the monodromy of a compatible open book decomposition. Recent results of Wendl complete this correspondence: every symplectic filling comes from such a factorization. We'll explain this correspondence in more detail and give some example applications including the rational blowdown operation and the uniqueness of symplectic fillings of certain Lens spaces. Some of this work is joint with Tom Mark and Hasaaki Endo and some is joint with Olga Plamenevskaya.

## Geometry and Physics Seminar

### Spectra of Categories and Applications to Low Dimensional Topology

Tuesday, December 1, 2009, 5:00pm
Ungar Room 402

Abstract: In this talk we will define the notion of spectrum of category and will compute it on the example of some Fukaya categories. Other applications will be discussed.

## Applied Math Seminar

### Modeling the Impact of Non-timber Forest Product Harvest in Variable Environments

Friday, November 20, 2009, 4:30pm
Ungar Room 402

Abstract: Harvesting wild plants for non-timber forest products is an important source of income, food and medicine for millions of people around the world. Over-exploitation of these plant resources may lead to species extinction and impair their availability for future use by people who depend on them for their livelihoods. Yet, our knowledge of the way harvesting some non-timber forest products may affect population dynamics is still limited. I will use the case study of Khaya senegalensis (Meliaceae) foliage and bark harvest by indigenous Fulani people in Africa, to demonstrate that harvesting reduces population growth rate even further if environmental conditions vary stochastically. I will show how using harvest-specific elasticity analysis provides in-depth understanding of the management options that are available to mitigate the negative effects of harvest at the population level. I suggest that the temporal sequence of harvest intensity matters when modeling the impact of wild plant harvest.

## Geometry and Physics Seminar

### Seiberg-Witten Equations and End-periodic Dirac Operators

Wednesday, November 18, 2009, 4:00pm
Ungar Room 402

Abstract: Let X be a smooth spin 4-manifold with homology of S^1 x S^3. In our joint project with Tom Mrowka and Daniel Ruberman, we study the Seiberg-Witten equations on X. The count of their solutions, called the Seiberg-Witten invariant of X, depends on choices of Riemannian metric and perturbation. A similar dependency issue is resolved in dimension 3 by relating the jumps in the Seiberg-Witten invariant to the spectral flow of the Dirac operator; the resulting invariant is then the Casson invariant. In dimension 4, we use Taubes' theory of end-periodic operators to relate the jumps in the Seiberg-Witten invariant to the index theory of the Dirac operator on a manifold with periodic end modeled on the infinite cyclic cover of X. The resulting invariant is then a smooth invariant of X whose reduction is the Rohlin invariant. Some calculations and applications of this invariant will be discussed.

## Applied Math Seminar

### A Class of Integrable Hamiltonian Systems and Weak Lyapunov Stability

Friday, November 13, 2009, 4:30pm
Ungar Room 402

Abstract: The aim of the talk is to introduce a class of Hamiltonian autonomous systems which are completely integrable and their dynamics is described in all details. In particular we show explicit examples of Hamiltonian systems with an unstable equilibrium where the eigenvalues of the linearization are imaginary and no motion is asymptotic to the equilibrium in the past, namely no solution has the equilibrium as limit point as time goes to minus infinity.

## Combinatorics Seminar

### A Combinatorial Optimization Problem from Genomics

Friday, November 13, 2009, 3:00pm
Ungar Room 506

Abstract: Biologists often wish to locate the gene controlling for a specific feature in a given species. One approach is to use recombinant inbred lines (RILs) from that species. RILs are homozygous, with genetic material alternating between a parent having the trait in question and a parent not having the trait. The break points in the genetic contributions from the parents occur at different points in different RILs. Biologists typically select a (usually large) subset of the RILs that visually appears to have sufficiently varied break points to ensure that the location of the controlling gene can be resolved by comparing which RILs have the trait with the parental contribution at each gene.

Unfortunately, this subjective approach does not guarantee the ability of the selected subset to resolve the gene location as well as the full set of RILs. In addition, the experiments that must be performed to determine whether a given RIL has the trait in question can be intensive with respect to time, money, and laboratory space. For this reason, it is desirable to minimize the size of the set of RILs selected for analysis. The typical approach makes little or no effort to select a smallest set.

We describe a Mathematica program we have written to find sets of RILs that are as small as possible subject to the constraint of being able to resolve the location of any gene the full set of RILs can resolve.

This is joint work with Jonathan Fitz Gerald, Department of Biology, Rhodes College.

## Applied Math Seminar

### Insurgent Wars, Pandemics, Global Emissions and Market Crises:One Model Fits All?

Friday, November 6, 2009, 4:30pm
Ungar Room 402

Abstract: For complex real-world problems, it seems that there are (at least) as many models in the literature as there are researchers in the field. In this seminar, I will attempt the opposite approach: One model, stretched in various directions, to encompass four major issues. The model is a coalescence-fragmentation model in which clusters are continually playing the 'El Farol' bar attendance game. In certain limits, analytic solutions are obtainable which seem to capture the stylized statistical facts of each of these problems. Generalizations of the model, and their implications in each real-world scenario, are discussed.

## Geometry and Physics Seminar

### Closed Symmetric Differentials of Degree 2 and the Geometry of Complex Surfaces

Wednesday, November 4, 2009, 4:00pm
Ungar Room 402

Abstract: It is well understood how holomorphic differential p-forms reflect the topology of a given complex manifold. On the other hand, little is known about the relationship between the topology and the algebra of holomorphic symmetric differentials of complex manifolds. In this talk we will give results about the impact of the presence of closed symmetric 2-differentials on the topology and geometry of complex surfaces.