### A Geometric Analysis View on Spacetimes:Stability and Singularities

Thursday, November 14, 2019, 5:00pm
Ungar Room 402

Abstract: I will overview recent mathematical advances on the Einstein equations of general relativity, especially for solutions with low decay or low regularity. We will offer some answers to the following questions.

Are self-gravitating matter fields nonlinearly stable near Minkowski spacetime? What happens when two gravitational waves collide? Can we extend a spacetime beyond a singularity hypersurface?

Blog: http://philippelefloch.org

### From Numbers to Spaces in Floer Theory

IMSA Colloquium
Thursday, September 5, 2019, 5:00pm
Ungar Room 528B

Abstract: I will describe the development of Floer theory over the last 30 years as a progression of refined invariants starting with numbers, and rising to categories stably enriched in spaces. At each step, I will introduce some geometric question whose answer is made possible by the additional structure at hand.

### Some Results on an Evolutionary-epidemic Problem Arising in Plant Disease

Thursday, May 16, 2019, 5:00pm
Ungar Room 402

Abstract: In this talk we discuss various properties of an evolutionary-epidemic system modelling plant disease epidemic and incorporating the ability of the pathogen to adapt to the environment by mutation. The resulting problem consists in an intregro-differential system of equations that typically depends on a small parameter $\varepsilon>0$ that describes the dispersion of the pathogen in the phenotype trait space.

In the first part of this talk, we show that the system asymptotically stabilizes toward its unique endemic equilibrium and we describe, using a small parameter ($\varepsilon$) asymptotic, a possibly long transient behaviour before reaching the endemic equilibrium.

In the second part, the above problem is extended to the case where the populations are also structured with respect to physical space and where the infection is able to disperse. In that setting, we discuss the spatio-temporal evolution of the disease by studying some properties of the travelling wave solutions for this system, that models the spatial spread of the disease.

### Existence of Wave Trains for the Gurtin-McCamy Equation

Tuesday, May 14, 2019, 5:00pm
Ungar Room 402

Abstract: This work is mainly motivated by the study of periodic wave train solutions for the so-called Gurtin-McCamy equation. To that aim we construct a smooth center manifold for a rather general class of abstract second order semi-linear differential equations involving non-densely defined operators. We revisit results on commutative sums of linear operators using the integrated semigroup theory. These results are used to reformulate the notion of the weak solutions of the problem. We also derive a suitable fixed point formulation for the graph of the local center manifold that allows us to conclude to the existence and smoothness of such a local invariant manifold. Then we derive a Hopf bifurcation theorem for second order semi-linear equations. This result is applied to study the existence of periodic wave trains for the Gurtin-McCamy problem, that is for a class of non-local age structured equations with diffusion.

### Analytic and Combinatorial Aspects of Finite Point Configurations

Thursday, April 18, 2019, 5:00pm
Ungar Room 402

Abstract: We are going to discuss the following basic question. How large does a subset of a vector space need to be to ensure that it determines a positive proportion of all possible point configurations of a given type, where the notion of large depends on the structure of the underlying field. We shall discuss the analytic and combinatorial aspects of this problem, describe some recent results and applications to problems in classical analysis involving the existence and non-existence of exponential bases and frames.

### Towards Constructing a Mathematically Rigorous Framework for Modelling Evolutionary Fitness

Thursday, April 11, 2019, 5:00pm
Ungar Room 402

Abstract: In modelling biological evolution, a major mathematical challenge consists in an adequate quantification of selective advantages of species. Current approaches to modelling natural section are often based on the idea of maximization of a certain prescribed criterion – evolutionary fitness. This paradigm was inspired by the seminal Darwin's idea of the 'survival of the fittest'. However, the concept of evolutionary fitness is still somewhat vague, intuitive and is often subjective. On the other hand, by using different definitions of fitness one can predict conflicting evolutionary outcomes, which is obviously unfortunate. In this talk, I present a novel axiomatic approach to model natural selection in dynamical systems with inheritance in an arbitrary function space. For a generic self-replication system, I introduce a ranking order of inherited units following the underlying measure density dynamics. Using such ranking, it becomes possible to derive a generalized fitness function which maximization will predict long-term evolutionary outcome. The approach justifies the variational principle of determining evolutionarily stable behavioural strategies. I demonstrate a new technique allowing to derive evolutionary fitness for population models with structuring (e.g. in models with time delay) which was so far a mathematical challenge. Finally, I show how the method can be applied to a von Foerster continuous stage population model.

CANCELLED

### Poincaré Inequalities on Hamming Cubeand Related Combinatorial and Probabilistic Problems

Thursday, March 28, 2019, 5:00pm
Ungar Room 402

Abstract: Geometric inequalities on Hamming cube imply corresponding isoperimetric inequalities in Gaussian spaces. Inequalities in discrete setting (on Hamming cube) are usually more difficult and more deep. In particular, Poincaré inequalities on Hamming cube give sharp lower estimates for the product measure of the boundaries of arbitrary sets of Hamming cube. Such estimates were used by Margulis in his famous network connectivity theorem. We will survey such estimates obtained by Margulis, Bobkov, Ledoux, Lust-Piquard. Recently the constant in L1 discrete Poincaré inequality was improved. The sharp constant remains unknown (unlike the Gaussian case, where it was found by Maurey–Pisier and then Ledoux), but we will show the idea of the improvement.

### On Projective Invariants of k-tuples of Torsion Points on Elliptic Curves

Tuesday, March 19, 2019, 5:00pm
Ungar Room 402

Abstract: Every complex elliptic curve $E$ has a natural (so called hyperlliptic) involution $\theta$ (as an abelian group and algebraic curve) with $4$ stable points and $P^1$ as quotient $E/\theta$. The stable points of $\theta$ can be identified with a subgroup of points of order $2$ on $E$ with a torsion subgroup $Q/Z+ Q/Z\subset E$.

The problem which I am going to consider is about the variation of the images of collections of points from $Q/Z+ Q/Z$ in $P^1$ which occur under the variation of elliptic curve $E_t$.

We are conidering more precisely the variation of projective invariants of such collections. Thus the problem becomes interesting when the number of corresponding points in $P^1$ is $\geq 4$.

In our joint work with Yuri Tschinkel and Hang Fu we formulated several conjecture conecerning the behavior of such sets.

In our recent article with Hang Fu we managed to describe all collections of $4$-tuples for which the variation of projective invariants is trivial.

I will discuss the proof and description of such $4$-tuples in my talk and several other general concepts and results related to the subject.

### Morsifications and Mutations

Friday, March 1, 2019, 5:00pm
Ungar Room 402

Abstract: I will discuss a new and somewhat mysterious connection between singularity theory and cluster algebras, more specifically between the topology of isolated singularities of plane curves and the mutation equivalence of quivers associated with their morsifications. This is joint work with Pavlo Pylyavskyy, Eugenii Shustin, and Dylan Thurston.

### Diophantine and Tropical Geometry

Thursday, February 21, 2019, 5:00pm
Ungar Room 402

Abstract: Diophantine geometry is the study of integral solutions to a polynomial equation. For instance, for integers a,b,c ≥ 2 satisfying 1/a + 1/b + 1/c > 1, Darmon and Granville proved that the individual generalized Fermat equation x^a + y^b = z^c has only finitely many coprime integer solutions. Conjecturally something stronger is true: for a,b,c ≥ 3 there are no non-trivial solutions.

I'll discuss various other Diophantine problems, with a focus on the underlying intuition and conjectural framework. I will especially focus on the uniformity conjecture, and will explain new ideas from tropical geometry and our recent partial proof of the uniformity conjecture.

### Residual Intersections, Old and New

Friday, February 15, 2019, 5:00pm
Ungar Room 402

Abstract: Two general quadric hypersurface in complex 3-space that contain a line, intersect in the line and also a curve of degree 3, the "residual intersection". I'll describe the 19th-century motivations and origins of the theory of residual intersections, and also some recent work in the area.

### Homological Knot Invariants, Relations and Applications

Wednesday, December 12, 2018, 3:00pm
Ungar Room 402

Abstract: Knot theory is about studying knots i.e. image of a smooth injective map from circle to R^3. In this talk, we will start by sketching some problems in knot theory. Then we will discuss two knot invariants, Khovanov homology and knot Floer homology, and we will explain how they can be used to answer some of these questions.

Khovanov homology and knot Floer homology are algebraic knot invariants that are defined combinatorially and analytically, respectively. Despite their very different definitions, the two invariants seem to contain a great deal of the same information and are conjectured to be related. In parallel, we will discuss some of their similarities. This talk is based on joint works with Nathan Dowlin and Eaman Eftekhary.

### Generalized Square Knots and Homotopy 4-spheres

Monday, December 10, 2018, 5:00pm
Ungar Room 402

Abstract: Perhaps the most elusive open problem in low-dimensional topology is the smooth 4-dimensional Poincare Conjecture, which asserts that any 4-manifold with the homotopy type of the 4-sphere is diffeomorphic to the 4-sphere. For the last forty years, potential counterexamples to this conjecture have been constructed, illustrated, and subsequently standardized. Many of these examples are geometrically simply connected, meaning they can be built without 1-handles. If such a homotopy 4-sphere is built with only one 2-handle, then it must be the 4-sphere; this is a consequence of David Gabai's solution to the Property R Conjecture.

In this talk, I will discuss work to understand geometrically simply connected homotopy 4-spheres that are built with two 2-handles. In the case that one of the 2-handles is attached along a fibered knot, we obtain strong results about the nature of the second component. Building on this, we use the beautiful periodic structure of torus knots to classify the attaching curve of the second component when the first component is a generalized square knot (a torus knot summed with its mirror). Finally, we prove that for an infinite family of such links, the corresponding homotopy 4-sphere is the standard one, proving the Poincare Conjecture in this setting. We also give intriguing new potential counterexamples to the Poincare Conjecture coming from these families. This talk is based on joint work with Alex Zupan.

### Modeling HIV Dynamics under Treatment

Thursday, December 6, 2018, 5:00pm
Ungar Room 402

Abstract: Highly active antiretroviral therapy has successfully controlled HIV replication in many patients. The treatment effectiveness depends on many factors, such as pharmacokinetics/pharmacodynamics of drugs and the intracellular stages of the viral replication cycle inhibited by antiretroviral drugs. In this talk, I will present some recent work on studying HIV dynamics under treatment. Using multi-stage models, I will show that drugs from different classes have different influence on HIV decay dynamics. Using models that combine pharmacodynamics and virus dynamics, I will show that pharmacodynamic profiles of drugs can significantly affect the outcome of either early or late treatment of HIV infection.

### Phase Transitions and Minimal Hypersurfaces

Monday, December 3, 2018, 5:00pm
Ungar Room 402

Abstract: Long standing questions in the theory of minimal hypersurfaces have been solved in the past few years. This progress can be explained and enriched through a strong analogy with the theory of phase transitions. I will present the current state of these ideas, discuss my contributions to the subject and share directions for future developments.

### Instantons and Lattices of Smooth 4-manifolds with Boundary

Thursday, November 29, 2018, 5:00pm
Ungar Room 402

Abstract: A classical invariant associated to a 4-manifold is the intersection form on its second homology group, which is an integral lattice. A famous result of Donaldson from the early 1980s says that a definite lattice of a smooth compact 4-manifold without boundary is diagonalizable over the integers. What if there is non-empty boundary? This talk surveys recent advances on this problem which have been obtained using Yang-Mills instanton Floer theory.

### Isometric Embedding and Quasi-local Mass

Tuesday, November 27, 2018, 5:00pm
Ungar Room 402

Abstract: In this talk, we will first review the classic result of isometric embedding of (S^2,g) into 3-dimensional Euclidean space by Nirenberg and Pogorelov. We will then discuss how to apply it to define quasi-local mass in general relativity. In particular, the positivity of Brown-York quasi-local mass proved by Shi-Tam is equivalent to the Riemannian Positive mass theorem by Schoen-Yau and Witten.

We will then discuss the recent progress in isometric embedding of (S^2,g) into general Riemannian manifold. We will also discuss the recent work on a localized Riemannian Penrose inequality, which is equivalent to the Riemannian Penrose inequality.

CANCELLED

### Diophantine and Tropical Geometry

Thursday, November 8, 2018, 5:00pm
Ungar Room 402

Abstract: Diophantine geometry is the study of integral solutions to a polynomial equation. For instance, for integers a,b,c ≥ 2 satisfying 1/a + 1/b + 1/c > 1, Darmon and Granville proved that the individual generalized Fermat equation x^a + y^b = z^c has only finitely many coprime integer solutions. Conjecturally something stronger is true: for a,b,c ≥ 3 there are no non-trivial solutions.

I'll discuss various other Diophantine problems, with a focus on the underlying intuition and conjectural framework. I will especially focus on the uniformity conjecture, and will explain new ideas from tropical geometry and our recent partial proof of the uniformity conjecture.

### Ergodicity and Lyapunov Functions for Langevin Dynamics with Singular Potentials

Thursday, September 27, 2018, 5:00pm
Ungar Room 402

Abstract: We discuss Langevin dynamics of N particles on R^d interacting through a singular repulsive potential, e.g. the well-known Lennard-Jones type, and show that the system converges to the unique invariant Gibbs measure exponentially fast in a weighted total variation distance. The proof of the result turns on an explicit construction of a Lyapunov function. In contrast to previous results for such systems, our result implies geometric convergence to equilibrium starting from an essentially optimal family of initial distributions.

### Hyperbolicity and Jet Differentials

Tuesday, June 5, 2018, 2:30pm
Ungar Room 402

Abstract: In the 60's, Kobayashi introduced on any complex manifold X an intrinsic pseudo-metric generalizing the Poincaré metric on the unit disc. When this pseudo metric is in fact a metric, the manifold X is said to be hyperbolic in the sense of Kobayashi. By a result of Brody, when X is compact, then it is hyperbolic if and only if it does not contain any entire curve (a non constant holomorphic map from the complex plane to X).

A fruitful way to study hyperbolicity problems is to use jet differential equations. Those objects, generalizing symmetric differential forms, provide obstructions to the existence of entire curves and can be used in some particular situation to prove that some given varieties is hyperbolic.

The purpose of this talk, aimed at a general audience, is to give an overview on hyperbolicity and the theory of jet differentials.

### The Circle Method in Algebraic Geometry

Monday, May 7, 2018, 5:00pm
Ungar Room 402

Abstract: Certain counting problems in group theory can be formulated either in terms of varieties over finite fields, or (dually) in terms of irreducible character values. By comparing the two points of view, one can either use geometry to give character estimates or (what I will mostly talk about) character estimates to prove theorems in geometry.

### Cohomology Theories and Topological Hochschild Homology

Monday, May 7, 2018, 3:30pm
Ungar Room 402

Abstract: Cohomology theories assign to every space a graded abelian group, satisfying appropriate axioms. Particularly nice ones have a product, that is: they assign to every space a graded ring. It turns out that such theories can be described in terms of ring spectra. Modern constructions let us define actual products that are both associative and unital on the ring spectra that correspond to multiplicative cohomology theories, which is harder than one would think due to some "flabbiness" in the definition of the ring spectra.  Using these products, though, we can look at the ring spectra of discrete rings (corresponding to the usual cohomology of a space with coefficients in that ring) and replicate homological algebra constructions on the discrete rings with topological constructions on the corresponding ring spectra. The integers are an initial object in the category of discrete unital rings; in the category of ring spectra, that role is played by the sphere spectrum. So even if we are only interested in understanding discrete rings, doing homological algebra with their ring spectra over the sphere spectrum turns out to give new and interesting constructions.

Topological Hochschild homology is the ring spectrum version of Hochschild homology, with tensor products taken over the sphere spectrum rather than over the integers. It is a finer invariant of discrete rings, and the Dennis trace map from algebraic K-theory to Hochschild homology can be applied to ring spectra and thus factors through Topological Hochschild homology. I will discuss the topological Hochschild homology of rings including number rings (joint with Ib Madsen) and maximal orders in simple algebras over the rationals (joint with Henry Chan).

### Reflexive Polytopes

Monday, April 30, 2018, 5:00pm
Ungar Room 402

Abstract: A lattice polytope P of dimension d in the d-dimensional euclidean space is called reflexive if the origin is contained in the interior of P and if the dual polytope of P is again a lattice polytope. For example, the triangle in the euclidean plane with the vertices (-1,-1), (-1,2) and (2,-1) is reflexive. It turns out that reflexive polytopes play an important role in various areas of mathematics. One of the questions in combinatorics is how to construct reflexive polytopes in natural ways. In my talk, in the frame of Gröbner bases, a technique to yield reflexive polytopes will be discussed. No special knowledge will be required to understand my talk.

### Symmetries of Polynomials

Thursday, April 26, 2018, 5:00pm
Ungar Room 402

Abstract: Symmetries of polynomials are closely connected to the geometry of the variety of zeroes of the polynomials. Varieties come in three types and all three types have very different symmetry groups. We review some results, both old and new, which place bounds on the size of the symmetry group.

### Conformal Geometry and Topology of Manifolds

Thursday, April 19, 2018, 5:00pm
Ungar Room 402

Abstract: I will start with basics on conformal geometry, we will discuss the Einstein-Hilbert functional and Yamabe problem. Then I plan to discuss the problem of existence of metrics with positive scalar curvature for simply connected spin manifolds. At the end I would like to describe some recent results on the space of metrics with positive scalar curvature.

### Using Mathematical Modeling to Understand the Role of Diacylglycerol (DAG) as a Second Messenger

Thursday, April 12, 2018, 5:00pm
Ungar Room 402

Abstract: Diacylgylcerol (DAG) plays a key role in cellular signaling as a second messenger. In particular, it regulates a variety of cellular processes and the breakdown of the signaling pathway that involves DAG contributes to the development of a variety of diseases, including cancer. A mathematical model of the G-protein signaling pathway in RAW 264.7 macrophages downstream of P2Y6 activation by the ubiquitous signaling nucleotide uridine 5'-diphosphate is presented. The primary goal is to better understand the role of diacylglycerol in the signaling pathway and the underlying biological dynamics that cannot always be easily measured experimentally. The model is based on time-course measurements of P2Y6 surface receptors, inositol trisphosphate, cytosolic calcium, and with a particular focus on differential dynamics of multiple species of diacylglycerol. When using the canonical representation, the model predicted that key interactions were missing from the current pathway structure. Indeed, the model suggested that to accurately depict experimental observations, an additional branch to the signaling pathway was needed, whereby an intracellular pool of diacylglycerol is immediately phosphorylated upon stimulation of an extracellular receptor for uridine 5'-diphosphate and subsequently used to aid replenishment of phosphatidylinositol. As a result of sensitivity analysis of the model parameters, key predictions can be made regarding which of these parameters are the most sensitive to perturbations and are therefore most responsible for output uncertainty. (Joint work with Hannah Callender, University of Portland, and the H. Alex Brown Lab, Vanderbilt.)

### Sloshing, Steklov, and Corners

Thursday, April 5, 2018, 5:00pm
Ungar Room 402

Abstract: The sloshing problem is a Steklov type eigenvalue problem describing small oscillations of an ideal fluid. We will give an overview of some latest advances in the study of Steklov and sloshing spectral asymptotics, highlighting the effects arising from corners, which appear naturally in the context of sloshing. In particular, we will discuss the proofs of the conjectures posed by Fox and Kuttler back in 1983 on the asymptotics of sloshing frequencies in two dimensions. We will also outline an approach towards obtaining sharp asymptotics for Steklov eigenvalues on polygons. The talk is based on a joint work with M. Levitin, L. Parnovski and D. Sher.

### Ehrhart Positivity

Tuesday, April 3, 2018, 5:00pm
Ungar Room 402

Abstract: The Ehrhart polynomial counts the number of lattice points inside dilation of an integral polytope, that is, a polytope whose vertices are lattice points. We say a polytope is Ehrhart positive if its Ehrhart polynomial has positive coefficients. In the literature, different families of polytopes have been shown to be Ehrhart positive using different techniques. We will survey these results in the first part of the talk, after giving a brief introduction to polytopes and Ehrhart polynomials.

Through work of Danilov/McMullen, there is an interpretation of Ehrhart coefficients relating to the normalized volumes of faces. In the second part of the talk, I will discuss joint work with Castillo in which we try to make this relation more explicit in the case of regular permutohedra. The motivation is to prove Ehrhart positivity for generalized permutohedra. If time permits, I will also discuss some other related questions.

### On Self-Dual Curves

Friday, March 2, 2018, 5:00pm
Fieldhouse at the Watsco Center

Abstract: An algebraic curve in the projective plane (or, more generally in a higher dimensional projective space) is said to be 'self-dual' if it is projectively equivalent to its dual curve (after, possibly, an automorphism of the curve). Familiar examples are the nonsingular conics (or, more generally, rational normal curves in higher dimensions) and the 'binomial curves' y^a = x^b, but there are many more such curves, even in the plane.

I'll survey some of the literature on these curves, particularly in the plane and 3-space, and some of what is known about their classification and moduli, including their connection with contact curves in certain contact 3-folds, some of which are singular. I'll also provide what appear to be some new examples of these curves.

### Nonidentifiability Issue in Parameter Estimation of Differential Equation Models

Thursday, February 22, 2018, 2:00pm
Ungar Room 402

Abstract: Transmission models for infectious diseases using differential equations are frequently confronted with disease data. For instance, these type of models are used to analyze HIV surveillance data, for the assessment of HIV epidemics and estimation of true burden of HIV in terms of incidence, prevalence and the size of undiagnosed HIV positive population. Parameter estimation when fitting the model to data is a key step of the modelling approach, and can often be complicated by the the presence of nonidentifiable parameters. Nonidentifiability results in multiple, often infinitely many, parameter values for which the model fit the data equally well, while different choices of these best-fit parameter values can produce very different model predictions for unobserved quantities of public health interest.

In this talk, I will begin by discussing various notions of nonidentifiability: structural vs practical,local vs global, etc, and different methods to detect and diagnose nonidentifiability. Then I will present a new mathematical approach to study the issue of nonidentifiability based on singular value decomposition and variance decomposition. I will then illustrate our approach with a case study of HIV estimation using an ordinary differential equation model. Some open questions for infinite dimensional models such as delay differential equation and partial differential equation models will also be discussed.

### Curved Version of the Radon Inversion Formula

Thursday, February 15, 2018, 5:00pm
Ungar Room 402

Abstract: 100 years ago Radon published his famous formula for the reconstruction of functions on the plane through their integrals along lines. Is it possible, to replace in this construction lines with different curves? There are only known just a few such examples, such as geodesicsor horocycles on the hyperbolic plane. These formulas usually look similar to the Radon's formula. We give a universal reconstruction formula, as a closed differential form on the manifold of all curves, whose restriction on different cycles of curves gives specific examples of inversion formulas for curves. It is possible to interpret this construction as areal Cauchy integral formula.

### Probing the Cosmic Expansion with Large Scale Galaxy Surveys:Prospects and Challenges

Tuesday, January 30, 2018, 5:00pm
Ungar Room 402

Abstract: The Universe is found to be expanding in an accelerating phase. Mysterious dark energy is one possible solution to explain the acceleration, and modifying the gravity theory is another. In order to find out what is driving the cosmic accelerating expansion, one has to first answer the question whether the General Relativity is still valid on the cosmic scales. Astronomers are carrying out this test by observing many millions galaxies. I will introduce the basic information of this observation and summarize the current status. I will also introduce the future observations, and finally outline theoretical and observational challenges.

### Filling Groovy:The Goodness of Generic Projections

Thursday, December 14, 2017, 5:00pm
Ungar Room 402

Abstract: The classical process of projection amounts to selecting a linear subspace W from a given vector space V of functions on a geometric object X. This is analogous to vision, where W is the 2-dimensional space of coordinate functions on a retina. What information is lost by passing from V to W when W is selected randomly among subspaces of given dimension?

We will describe some progress on a version of this problem in complex algebraic geometry.

### The Structure of Semi-hyperbolic Projective Algebraic Manifolds

Thursday, April 27, 2017, 5:00pm
Ungar Room 402

Abstract: S. Kobayashi coined the term hyperbolic for a compact complex manifold M without nontrivial holomorphic images of C and conjectured the positivity of the canonical bundle of M. In particular M would be projective if true. But the conjecture is still wide open for projective manifolds beyond dimension two.

A spectacular advance in this direction is the resolution in the projective case by D. Wu-S.T. Yau (Invent. 2016) of the differential geometric analog of the conjecture, due to S.T. Yau. The analog pertains to compact Kähler manifolds with negative holomorphic curvature and the said advance resolves in particular the abundance conjecture, a key conjecture for the classification of algebraic varieties, for such a manifold.

In this talk, I will mainly focus on a recent joint paper with G. Heier, B. Wong and F.Y. Zheng that offers a structure theorem for projective Kähler manifolds with negative holomorphic curvature, assuming the abundance conjecture. The analysis involves a careful study of the rank of the said curvature, and offers relationships to the global abundance problem.

### On the Obstruction and Propagation of Entire Solutions to a Non-local Reaction Diffusion Equation with a Gap

Thursday, April 13, 2017, 5:00pm
Ungar Room 402

Abstract: In this talk I will discuss the propagation properties of a bistable spatially heterogeneous reaction-diffusion equation where the diffusion is generated by a jump process. Here the spatial heterogeneity is due to a small region with decay. First, I will focus on the existence and uniqueness of a "generalized transition front". Then I will give some partial results about propagation and obstruction of the transition front. Throughout the talk I will point out many interesting differences between the non-local and local reaction-diffusion equations.

### Isometric Embeddings, Geometric Inequalities and Nonlinear PDEs

Wednesday, April 12, 2017, 5:00pm
Ungar Room 402

Abstract: In 1950s, Nirenberg and Pogorelov solved the classical Weyl problem regarding the isometric embedding of positively curved compact surface in to R^3. Solution to Weyl's problem is crucial to the definition of the Brown-York quasi local mass in general relativity in 1990s, and also play key role in the recent works of Liu-Yau and Wang-Yau. The development brings renewed focus on the Weyl problem of isometric embedding of surfaces to general 3D ambient space. We will discuss elliptic PDEs involved the problem and recent work on this type of nonlinear equations. We also discuss new proof of isoperimetric type of inequalities using parabolic PDEs. The talk is try to illustrate some beautiful interaction of nonlinear PDE, differential geometry and general relativity. It is accessible for general mathematical audience.

### Enhancement of Biological Reactions by Chemotaxis

Thursday, April 6, 2017, 5:00pm
Ungar Room 402

Abstract: Many reactions and processes in nature take place in fluid and in presence of both fluid flow and chemotaxis - directed motion of cells or species guided by attractive (or repulsive) chemical. One example of such process is broadcast spawning by corals, the way corals reproduce. Models of this process based on pure reaction-diffusion tend to dramatically underestimate the fertilization success rate. I will discuss a simplified 2D single equation model which incorporates fluid flow and chemotaxis effects. In the framework of this model built on the basis of the well known Keller-Segel equation, the role of chemotaxis turns out to be crucial. In the presence of a sufficiently strong chemotaxis, even weakly coupled reaction can lead to high fertilization rate on a fixed time scale.

If time permits, I will discuss some progress in a more sophisticated model which involves a system of two equations. Novel mathematical tools used in this work involve sharp convergence to equilibrium estimates for a class of Fokker-Planck operators with logarithmic-type potentials.

### Around Codimension One Embeddings

Tuesday, March 7, 2017, 5:00pm
Ungar Room 402

Abstract: Being drawable in the plane without intersecting edges is a very important and much studied graph property. Euler observed in 1752 that planarity implies a linear upper bound on the number of edges of a graph (which otherwise is quadratic in the number of vertices). Several ways of characterizing planar graphs have been given during the previous century.

Planarity is, of course, a special case of a general notion of embedding a simplicial d-complex into real k-space. The k=d+1 and k=2d cases are of particular interest in higher dimensions, since they both generalize planarity. Embedding a space into some manifold is a much studied question in geometry/topology. For instance, van Kampen showed that in the k=2d case there is a very useful cohomological obstruction to embeddability.

Higher-dimensional embeddability has been studied also from the combinatorial point of view, in a tradition inspired by Euler. In this talk I will survey a few topics from the combinatorial study of embeddings, such as bounds for the number of maximal faces and algorithmic questions. I will end with mention of some joint work with A. Goodarzi concerning an obstruction to k=d+1 embeddings.

The talk will not presuppose previous familiarity with the topic.

### Collective Motion, Collective Decision-making, and Collective Action

Friday, March 3, 2017, 4:00pm
Ungar Room 402

Abstract: There exists a rich history of research on the mathematical modeling of animal populations. The classical literature, however, is inadequate to explain observed spatial patterning, or foraging and anti-predator behavior, because animals actively aggregate. This lecture will begin from models of animal aggregation, the role of leadership in collective motion and the evolution of collective behavior, and move from there to implications for decision-making in human societies. Ecological and economic systems are alike in that individual agents compete for limited resources, evolve their behaviors in response to interactions with others, and form exploitative as well as cooperative interactions as a result. In these complex-adaptive systems, macroscopic properties like the flow patterns of resources like nutrients and capital emerge from large numbers of microscopic interactions, and feedback to affect individual behaviors. I will explore common features of these systems, especially as they involve the evolution of cooperation in dealing with public goods, common pool resources and collective movement across systems; Examples and lessons will range from bacteria and slime molds to groups to insurance arrangements in human societies and international agreements on environmental issues.

### Cyclic Descents, Toric Schur Functions and Gromov-Witten Invariants

Tuesday, February 28, 2017, 5:00pm
Ungar Room 402

Abstract: Descents of permutations have been studied since Euler. This notion has been vastly generalized in several directions, and in particular to the context of standard Young tableaux (SYT). More recently, cyclic descents of permutations were introduced by Cellini and further studied by Dilks, Petersen and Stembridge. Looking for a corresponding notion for SYT, Rhoades found a very elegant solution for rectangular shapes.

In an attempt to extend this concept, explicit combinatorial definitions for two-row and certain other shapes have been found, implying the Schur-positivity of various quasi-symmetric functions. In all cases, the cyclic descent set admits a cyclic group action and restricts to the usual descent set when the letter $n$ is ignored. Consequently, the existence of a cyclic descent set with these properties was conjectured for all shapes, even the skew ones.

This talk will report on the surprising resolution of this conjecture: Cyclic descent sets do exist for nearly all skew shapes, with an interesting small set of exceptions. The proof applies nonnegativity properties of Postnikov's toric Schur polynomials and a new combinatorial interpretation of certain Gromov-Witten invariants. We shall also comment on issues of uniqueness.

Joint with Sergi Elizalde, Vic Reiner and Yuval Roichman.

### Hyperplane Sections of K3 Surfaces

Friday, February 3, 2017, 5:00pm
Ungar Room 402

Abstract: K3 surfaces and their hyperplane sections play a central role in algebraic geometry. This is a survey of the work done during the past five years to characterize which smooth curves lie on a K3 surface. Related topics will be discussed. These are joint works with a combination of the following authors: Andrea Bruno, Gavril Farkas, Edoardo Sernesi and Giulia Saccà.

### Arithmetic and Geometry of Fano Varieties

Thursday, February 2, 2017, 5:00pm
Ungar Room 402

Abstract: I will discuss recent advances in the theory of Fano varieties over nonclosed fields (joint with B. Hassett, A. Kresch, and A. Pirutka).

### Left-orderability and Three-manifolds

Thursday, January 26, 2017, 5:00pm
Ungar Room 402

Abstract: A group is called left-orderable if it can be given a left-invariant total order. We will discuss the question of when the fundamental group of a three-manifold Y is left-orderable. Orderability is known to be related to certain topological aspects of Y, such as the surfaces which sit inside it. We will discuss a conjectural relationship between left-orderability and the solutions to a certain nonlinear PDE on Y.

Joint Math-Physics Colloquium

### The Bannai-Ito Algebra in Many Guises

Thursday, January 26, 2017, 4:00pm
Physics Conference Room, 3rd Floor

Abstract: This talk will offer a review of the Bannai-Ito algebra and of its higher rank extension. It will first be explained that it is in a Schur-Weyl duality with the super algebra osp(1,2). Its occurrence as the symmetry algebra of the Dirac-Dunkl equation will be discussed and its relation to orthogonal polynomials will also be presented.

### Spin Manifolds of Dimensions at Most 4

Wednesday, January 25, 2017, 5:00pm
Ungar Room 402

Abstract: Low dimensional spin manifolds are interesting objects with close connections to quadratic forms. The first part of the talk will provide an overview of these manifolds and their invariants. The second part of the talk will use the understanding of low dimensional spin manifolds to give a detailed description of invariants that for each topological space X detect all 3- and 4-dimensional spin manifolds mapping to X up to spin bordism.

### Final Size of an Epidemic for a Two Group SIR Model

Thursday, December 15, 2016, 4:00pm
Ungar Room 402

Abstract: In this talk we consider a two-group SIR epidemic model. We study the final size of the epidemic for each sub-population. The qualitative behavior of the infected classes at the earlier stage of the epidemic is described with respect to the basic reproduction number. Numerical simulations are also preformed to illustrate our results.

### Enhancement of Fisher-KPP Propagation through Lines and Strips of Fast Diffusion

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

Abstract: In this talk I will present some systems of reaction-diffusion equations which take into account the presence of diffusion (and/or reaction) heterogeneities in some regions of the domain. The motivation is the modelling of the invasion of an environment by a species whose individuals can move (and/or reproduce) faster in some regions of the habitat. I will describe the dynamics of such systems and focus on the qualitative properties of the propagation speed, showing in particular when it is larger than in the case of a homogeneous environment. These are joint works with H. Berestycki, L. Rossi and E. Valdinoci.

### Zika Outbreaks in a Highly Heterogeneous Environment: Insights from Dynamical Modelling

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

Abstract: Zika virus is in the family of Flaviviridae, and is often transmitted to human by Aedes aegypti, a common vector for transmitting several tropical fevers, including dengue and chikungunya. The environmental heterogeneity and intervention strategies of Zika spread also involve seasonality, co-circulation of other vector-borne diseases, and demographic structures of the mosquito population. We have been developing a variety of dynamical models to understand the transmission dynamics with focus on different aspects of environmental heterogeneities. We first consider the co-infection and co-circulation of dengue and Zika and their implication of dengue vaccination program for Zika control in the presence of experimentally reported antibody-dependent enhancement. We then consider the impact of heterogeneity of vector demographics on the initial outbreak rate and outbreak potential using age-structured partial differential equation systems and calculating the relevant threshold using non-linear semigroup theory and spectral theory. We also examine both numerically and analytically the mechanisms for potential nonlinear oscillations using the global bifurcation theory in delay differential equations.

### Modeling Invasive Dispersal at Multiple Scales

Thursday, December 8, 5:00pm
Ungar Room 402

Abstract: Biological invasions represent an interesting challenge to model mathematically. Landscape heterogeneity, non-local and temporally dependent spreading mechanisms, coarse data, and the presence of long-distance transportation connections are but a few of the complications that can greatly affect our understanding of invasive spread. In this talk, I will look at dispersal from a multi-scale perspective in an attempt to address some of these challenges.

To begin, I will introduce a generalization of Mollison’s stochastic contact birth process (J R Stat Soc 39(3):283, 1977) which is robust to non-local distribution kernels and heterogeneity in the landscape. By interpreting the quantity of interest as species occurrence probability rather than population size, I will describe how this process may also be approximated and simulated deterministically, using niche modeling tools to characterize landscape heterogeneity. Adding to this is a method for considering the effects of a disease-vector transportation network, which can unwittingly transport a biological invader to distant sites. Finally, I will shift focus to the intial stages of an invasion and concentrate on the local and mesoscale by considering the intentional release of a parasitoid wasp biocontrol agent. Results indicate that the fluid physics of air above the landscape likely plays a critical role in the dispersal process.

Numerical results will be included throughout the talk, including simulations for the cheatgrass (bromus tectorum) invasion in Rocky Mountain National Park and the initial spread of parasitoid wasps (Eretmocerus hayati) during a biocontrol introduction.

### Strong Cosmic Censorship and the Wave Equation in the Interior of Black Holes

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

Abstract: The Einstein equations admit a locally well-posed initial value problem. However, there are explicit black hole solutions of the Einstein equations for which global uniqueness fails. The strong cosmic censorship conjecture in general relativity states that global uniqueness should hold generically -- which implies the expectation that small perturbations of the initial data for the above black hole solutions should give rise to a spacetime which is globally uniquely determined. In this talk I will explain how this motivates the study of the wave equation in the interior of black holes, present an overview of previous results, and discuss a recent instability result, obtained in collaboration with Jonathan Luk, in more detail.

### Stochastic Homogenization for Reaction-Diffusion Equations

Monday, December 5, 2016, 5:00pm
Ungar Room 402

Abstract: One way of modeling phenomena in "typical" physical settings is to study PDEs in random environments. The subject of stochastic homogenization is concerned with identifying the asymptotic behavior of solutions to PDEs with random coefficients. Specifically, we are interested in the following: if the random effects are microscopic compared to the lengthscale at which we observe the phenomena, can we predict the behavior which takes place on average? For certain models of PDEs and under suitable hypotheses on the environment, the answer is affirmative. In this talk, I will focus on the stochastic homogenization for reaction-diffusion equations with both KPP and ignition nonlinearities. In the large-scale-large-time limit, the behavior of typical solutions is governed by a simple deterministic Hamilton-Jacobi equation modeling front propagation. Such models are relevant for predicting the evolution of a population or the spread of a fire in a heterogeneous medium. This talk is based on joint work with Andrej Zlatos.

### Systems of Partial Differential Equations Arising from Population Dynamics and Neuroscience:Free Boundary Problems as a Result of Segregation

Friday, December 2, 2016, 5:00pm
Ungar Room 402

Abstract: In this talk we will motivate and present our recent results on two phase free boundary problems arising from population dynamics. We will focus on systems with fully nonlinear diffusion and local interaction, and linear systems with a (nonlocal) long-range interaction. In the long-range model, the growth of a population at a point is inhibited by the other populations in a full area surrounding that point. This will force the populations to stay at distance one from each other in the limit configuration. So for the first time is obtained a free boundary problem with a gap of no-man's land between the regions where the populations exist. This is a joint work with Luis Caffarelli and Stefania Patrizi.

We will also present briefly some models of differential equations arising from neuroscience and share our current research on propagation of activity in the brain. We will motivate the need to incorporate the "volume" conductivity as well as of the neurons on a model. This is a joint work with Aaron Yip, Zoltan Nadasdy and Silvia Barbeiro.

### Global Flows with Invariant Measures for a Family of Almost Inviscid SQG Equations

Thursday, December 1, 2016, 5:00pm
Ungar Room 402

Abstract: We present a new result, joint with Andrea Nahmod, Natasa Pavlovic, and Gigliola Staffilani, in which very low regularity flows are constructed globally in time almost surely for a family of modified SQG equations using a Gibbs measure; the resulting flows leave this Gibbs measure invariant. The family of equations we treat is formed by adding a small amount of smoothing to the active scalar of the standard inviscid SQG equation. We find that global solutions can be constructed almost surely for any nonzero amount of smoothing.

### Deterministic and Stochastic Aspects of Fluid Mixing

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

Abstract: The process of mixing of a scalar quantity into a homogeneous fluid is a familiar physical phenomenon that we experience daily. In applied mathematics, it is also relevant to the theory of hydrodynamic stability at high Reynolds numbers - a theory that dates back to the 1830's and yet only recently developed in a rigorous mathematical setting. In this context, mixing acts to enhance, in certain senses, the dissipative forces. Moreover, there is also a transfer of information from large length-scales to small length-scales vaguely analogous to, but much simpler than, that which occurs in turbulence. In this talk, we focus on the study of the implications of these fundamental processes in linear settings, with particular emphasis on the long-time dynamics of deterministic systems (in terms of sharp decay estimates) and their stochastic perturbations (in terms of invariant measures).

CANCELLED

### Left-orderability and Three-manifolds

Friday, October 7, 2016, 4:00pm
Ungar Room 402

Abstract: A group is called left-orderable if it can be given a left-invariant total order. We will discuss the question of when the fundamental group of a three-manifold Y is left-orderable. Orderability is known to be related to certain topological aspects of Y, such as the surfaces which sit inside it. We will discuss a conjectural relationship between left-orderability and the solutions to a certain nonlinear PDE on Y.

CANCELLED

### On the Topology of Steel

Thursday, October 6, 2016, 5:00pm
Ungar Room 402

Abstract: Polycrystalline materials, such as metals, are composed of crystal grains of varying size and shape. Typically, the occurring grain cells have the combinatorial types of 3-dimensional simple polytopes, and together they tile 3-dimensional space.

We will see that some of the occurring grain types are substantially more frequent than others - where the frequent types turn out to be "combinatorially round". Here, the classification of grain types gives us, as an application of combinatorial low-dimensional topology, a new starting point for a topological microstructure analysis of steel.

### Partial Differential Equations in Finance

Thursday, August 25, 2016, 5:00pm
Ungar Room 402

Abstract: Starting with the Black & Scholes equation, I will give a tour of partial differential equations arising in various financial models. My goal is to emphasize what is mathematically interesting about each equation and how solutions are used in applications.

### The Importance of Synchrony in Mass Drug Administration

Friday, July 8, 2016, 4:30pm
Ungar Room 402

Abstract: Mass drug administration (MDA), a strategy in which all individuals in a population are subject to treatment without individual diagnosis, has been recommended by the World Health Organization for controlling and eliminating several neglected tropical diseases. In this talk, I will present some results arising from mass treatment of trachoma with azithromycin. In the first part, we compare three typical drug distribution strategies (regardless of health status): constant treatment, impulsive synchronized MDA, and impulsive non-synchronized treatment. We show that synchronized and constant strategies are respectively the most and least effective treatments in disease control. Elimination through synchronized treatment is always possible when adequate drug efficacy and coverage is fulfilled and sustained. In the second part, the optimal seasonal timing of mass administration of azithromycin for maximum antimalarial benefit has been established. This is joint work with Thomas M. Lietman and Travis C. Porco.

### Nonlocal Effects and Nonlocal Dispersal

Thursday, July 7, 2016, 4:30pm
Ungar Room 402

Abstract: This talk is concerned with some aspects of nonlocal dispersal equations. It consists of three parts. In part one, I will present some relations between local (random) and nonlocal dispersal problems. In part two, I will report our recent results on traveling waves and entire solutions of nonlocal dispersal equations. Part III is devoted to some problems on traveling waves and entire solutions of nonlocal dispersal equations.

### Traveling Wave Solutions of Evolutionary Models without Monotonicity

Monday, May 9, 2016, 4:00pm
Ungar Room 402

Abstract: This talk is concerned with the traveling wave solutions of evolutionary systems including delayed reaction-diffusion systems and integrodifference equations. Even if the general monotone conditions fail, the existence of traveling wave solutions is studied by generalized upper and lower solutions. The asymptotic behavior is established by the idea of contracting rectangles. In the study of classical Lotka-Volterra competitive systems, we obtain the existence of nonmonotone traveling wave solutions, which weakly confirms the conjecture by Tang and Fife [ARMA, 1980].

### Spectral Asymptotics for First Order Systems

Thursday, April 14, 2016, 5:00pm
Ungar Room 402

Abstract: In layman's terms a typical problem in this subject area is formulated as follows. Suppose that our universe has finite size but does not have a boundary. An example of such a situation would be a universe in the shape of a 3-dimensional sphere embedded in 4-dimensional Euclidean space. And imagine now that there is only one particle living in this universe, say, a massless neutrino. Then one can address a number of mathematical questions. How does the neutrino field (solution of the massless Dirac equation) propagate as a function of time? What are the eigenvalues (energy levels) of the particle? Are there nontrivial (i.e. without obvious symmetries) special cases when the eigenvalues can be evaluated explicitly? What is the difference between the neutrino (positive energy) and the antineutrino (negative energy)? What is the nature of spin? Why do neutrinos propagate with the speed of light? Why are neutrinos and photons (solutions of the Maxwell system) so different and, yet, so similar?

The speaker will approach the study of first order systems of partial differential equations from the perspective of a spectral theorist using techniques of microlocal analysis and without involving geometry or physics. However, a fascinating feature of the subject is that this purely analytic approach inevitably leads to differential geometric constructions with a strong theoretical physics flavour.

### Rectifiability of Harmonic Measure

Wednesday, April 13, 2016, 4:00pm
Ungar Room 402

Abstract: In a recent multi-authored paper by J. Azzam, S. Hofmann, J.-M. Martell, S. Mayboroda, M. Mourgoglou, X. Tolsa and myself the following result is proved:

### Souls of Manifolds via Curvature and Surgery

Thursday, January 27, 2011, 5:00pm
Ungar Room 402

Abstract: Deep connections between topology and geometry will be discussed in the case of manifolds with non-negative (sectional) curvature. Historical perspective of these connections and new developments will be presented.

### A Mathematical Model of Chronic Wounds

Thursday, January 20, 2011, 5:00pm
Ungar Room 402

Abstract: Chronic wound healing is a staggering public health problem, affecting 6.5 million individuals annually in the U.S. Ischemia, caused primarily by peripheral artery diseases, represents a major complicating factor in the healing process. In this talk, I will present a mathematical model of chronic wounds that represents the wounded tissue as a quasi-stationary Maxwell material, and incorporates the major biological processes involved in the wound closure. The model was formulated in terms of a system of partial differential equations with the surface of the open wound as a free boundary. Simulations of the model demonstrate how oxygen deficiency caused by ischemia limit macrophage recruitment to the wound-site and impair wound closure. The results are in tight agreement with recent experimental findings in a porcine model. I will also show analytical results of the model on the large-time asymptotic behavior of the free boundary under different ischemic conditions of the wound.

### A Shape-based Method for Determining Protein Binding Sites in a Genome

Tuesday, January 18, 2011, 5:00pm
Ungar Room 402

Abstract: We present a new algorithm for the identification of bound regions from ChIP-Seq experiments. ChIP-Seq is a relatively new assay for measuring the interactions of proteins with DNA. The binding sites for a given protein in a genome are "peaks" in the data, which is given by an integer-valued height function defined on the genome. Our method for identifying statistically significant peaks is inspired by the notion of persistence in topological data analysis and provides a non-parametric approach that is robust to noise in experiments. Specifically, our method reduces the peak calling problem to the study of tree-based statistics derived from the data. The software T-PIC (Tree shape Peak Identification for ChIP-Seq) is available at http://math.berkeley.edu/~vhower/tpic.html and provides a fast and accurate solution for ChIP-Seq peak finding.

### Imatinib Dynamics and Cancer Vaccines: From Agent-Based Models to PDEs

Thursday, January 13, 2011, 5:00pm
Ungar Room 402

Abstract: Various models exist for the interaction between the drug imatinib and chronic myelogenous leukemia. However, the role of the immune response during imatinib treatment remains unclear. Based on experimental data, we hypothesize that imatinib gives rise to a brief anti-leukemia immune response as patients enter remission.

We propose that cancer vaccinations during imatinib treatment can boost the existing immune response and lead to a sustained remission or a potential cure. To examine this hypothesis, we take a model by Michor et al. and extend it to a delay differential equation (DDE) model by incorporating an anti-leukemia immune response. We show that properly-timed vaccines can sustain the immune response to potentially prolong remission or eliminate cancer.

For comparison, we analyze an agent-based model developed independently by Roeder et al. We develop a partial differential equation (PDE) model that captures the same behavior as the Roeder agent-based model and extend it by incorporating an immune response. We conclude that both the DDE and PDE models exhibit similar behaviors with regard to cancer remission, implying that anti-leukemia immune responses may play a role in leukemia treatment.

### Understanding Random Surfaces

Monday, December 20, 2010, 4:30pm
Ungar Room 402

Abstract: There is a bijection between a class of piece-wise linear surfaces and dimer configurations on planar graphs. A dimer configuration on a graph is a perfect matching on vertices connected by edges. Dimers are well known in biology, chemistry and statistical mechanics. For certain very natural probability measures on dimer configurations, important correlation functions can be computed as Pfaffians of N\times N matrices. This reduces the statistics of such special random surfaces to a reasonable problem in linear algebra. This allows to study such random surfaces corresponding tolarge graphs. The talk will outline this story and at the end the discussion will focus on the "continuum limit" of such random surfaces.

### Bifurcation Problems for Structured Population Dynamics Models

Monday, December 6, 2010, 4:30pm
Ungar Room 402

Abstract: This presentation is devoted to bifurcation problems for some classes of PDE arising in the context of population dynamics. The main difficulty in such a context is to understand the dynamical properties of a PDE with non-linear and non-local boundary conditions. A typical class of examples is the so called age structured models. Age structured models have been well understood in terms of existence, uniqueness, and stability of equilibria since the 80's. Nevertheless, up to recently, the bifurcation properties of the semiflow generated by such a system has been only poorly understood.

In this presentation, we will start with some results about existence and smoothness of the center mainfold, and we will present some general Hopf bifurcation results applying to age structured models. Then we will turn to normal theory in such a context. The point here is to obtain formula to compute the first order terms of the Taylor expansion of the reduced system.

### Controlling Mosquitoes by Classical or Transgenic Sterile Insect Techniques

Monday, November 22, 2010, 4:00pm
Ungar Room 402

Abstract: For centuries, humans have attempted to control insect populations. This is in part because of the significant mortality and morbidity burden associated with insect vector-borne diseases, but also due to the huge economic impact of insect pests leading to losses in global food production. The development of transgenic technologies, coupled with sterile insect techniques (SIT), is being explored in relation to new approaches for the biological control of insect pests.

In this talk, I explore the impact of two control strategies (classical SIT and transgenic late-acting bisex lethality) using a stage-structured mathematical model, which is parameterized for the mosquito Aedes aegypti, which can spread yellow fever, dengue fever and Chikungunya disease. Counter to the majority of studies, I use realistic pulsed release strategies and incorporate a fitness cost, which is manifested as a reduction in male mating competitiveness.

I will explore the timing of control release in constant and cyclic wild-type mosquito populations, and demonstrate that this timing is critical for effective pest management. Furthermore, I will incorporate these control strategies into an integrated pest management program (IPM) and find the optimal release strategy. Finally, I will extend the models to a spatial context, determining conditions for the prevention of mosquito invasion by the use of a barrier wall.

### The Euclidean Algorithm and Primitive Roots

Thursday, November 18, 2010, 5:00pm
Ungar Room 402

Abstract: Artin's famous primitive root conjecture states that if n is an integer other than -1 or a square, then there are infinitely many primes p such that n is a primitive root modulo p. Although this conjecture is not known to hold for any value of n, Hooley proved it to be true under the assumption of the generalized Riemann hypothesis (GRH). We will discuss a number field version of this conjecture and its connection to the following Euclidean algorithm problem. Let O be the ring of integers of a number field K. It is well-known that if O is a Euclidean domain, then O is a unique factorization domain. With the exception of the imaginary quadratic number fields, it is conjectured that the reverse implication is true. This was proven by Weinberger under the assumption of the GRH. We will discuss recent progress towards the unconditional resolution of the Euclidean algorithm problem and the related primitive root problem. This is joint work with M. Ram Murty.

### Bounded and Unbounded Motions for Asymmetric Oscillators at Resonance

Monday, November 8, 2010, 4:30pm
Ungar Room 402

Abstract: In this talk, we consider the boundedness and unboundedness of solutions for the asymmetric oscillator

x" + ax+ - bx- + g(x) = p(t),

where x+ = max{x,0},x- = max{-x,0}, a and b are two positive constants, p(t) is a 2π-periodic smooth function and g(x) satisfies lim|x|→+∞x-1g(x) = 0. We establish some sharp sufficient conditions concerning the boundedness of all the solutions and the existence of unbounded solutions. Unlike many existing results in the literature where the function g(x) is required to be a bounded function with asymptotic limits, here we allow g(x) be unbounded or oscillatory without asymptotic limits. Some critical cases will also be considered.

### Evolution Problem in General Relativity

Wednesday, November 3, 2010, 5:00pm
Ungar Room 402

Abstract: The talk will introduce basic mathematical concepts of General Relativity and review the progress, main challenges and open problems, viewed through the prism of the evolution problem. I will illustrate interaction of Geometry and PDE methods in the context of General Relativity on examples ranging from incompleteness theorems and formation of trapped surfaces to stability problems.

### The Black Hole Stability Problem

Friday, October 22, 2010, 4:00pm
Ungar Room 402

Abstract: The problem of nonlinear stability for the Kerr model of a rotating black hole is one of the central problems in general relativity. The analysis of linear fields on the Kerr spacetime is an important model problem for full nonlinear stability. In this talk, I will present recent work with Pieter Blue which makes use of the hidden symmetry related to the Carter constant to circumvent these difficulties and give a "physical space" approach to estimates for the wave equation, including energy bounds, trapping, and dispersive estimates. I will also discuss the field equations for higher spin fields including linearized gravity.

### Persistence of a Single Phytoplankton Species

Thursday, October 21, 2010, 4:30pm
Ungar Room 402

Abstract: Phytoplankton need light to grow. However, most of phytoplankton are heavier than water, so they sink. How can phytoplankton persist? We investigate a nonlocal reaction-diffusion-advection equation which models the growth of a single phytoplankton species in a water column where the species depends solely on light for its metabolism. We study the effect of sinking rate, water column depth and vertical turbulent diffusion rate on the persistence of a single phytoplankton species. This is based upon a joint work with Sze-Bi Hsu, National Tsing-Hua University.

### Macdonald Polynomials in Representation Theory and Combinatorics

Friday, October 8, 2010, 4:30pm
Ungar Room 402

Abstract: This talk surveys some recent work in algebraic combinatorics that illustrates surprising connections between representation theory and enumerative combinatorics. We describe how to calculate the Hilbert series of various spaces of polynomials (harmonics, diagonal harmonics, and Garsia-Haiman modules) using combinatorial statistics on permutations and parking functions. This leads to a discussion of the algebraic and combinatorial significance of the Macdonald polynomials, which have played a central role in the theory of symmetric functions for the past two decades.

### Modeling Approaches for Influenza and HIV

Wednesday, September 8, 2010, 4:00pm
Ungar Room 402

Abstract: In this talk, I will present a survey of research projects on different mathematical models for influenza and HIV. For influenza, I will discuss two different modeling approaches. In the first approach, I will present a multi-strain/multi-host (MSMH) model that tracks the spread of inter-species strains between birds, pigs and humans. In the MSMH model, pigs are "mixing vessels" between avian and human strains and are capable of producing super-strains as a consequence of genetic recombination of these strains. I will show how specific subtypes can cause an epidemic then virtually disappear for years or even decades before reemerging (e.g., the case of H1N1). In the second approach, I will present a model that tracks the spread of influenza within flight transmission. A plane flight is much shorter scale than influenza's infectious duration; hence, we use methods from microbial risk management to assess the number of potential infections. We show that the flight duration along with the compartment will ultimately determine the passenger's risk. For HIV, I will present cross-sectional data on HIV prevalence in Lesotho, a small sub-Saharan African nation with HIV prevalence at approximately 23%. I will present our current progress on data analysis from the Health and Demographic Survey (DHS) to develop risk maps by district based on prevalence and treatment, feasibility analysis of a clinical trial, and efficacy of male circumcision as prevention for HIV.

### A Non-neutral Theory of Dispersal-limited Community Dynamics

Thursday, April 22, 2010, 4:30pm
Ungar Room 402

Abstract: We introduce the first analytical model of a dispersal-limited, niche-structured community to yield Hubbell's neutral theory in the limit of functional equivalence among all species. Dynamics of the multivariate species abundance distribution (SAD) for an asymmetric local community are modeled explicitly as a dispersal-limited sampling of the surrounding metacommunity. Coexistence may arise either from approximate functional equivalence or a competition-colonization tradeoff. At equilibrium, these symmetric and asymmetric mechanisms both generate unimodal SADs. Multiple modes only arise in asymmetric communities and provide a strong indication of non-neutral dynamics. Although these stationary distributions must be calculated numerically in the general theory, we derive the first analytical sampling distribution for a nearly neutral community where symmetry is broken by a single species distinct in ecological fitness and dispersal ability. Novel asymptotic expansions of hypergeometric functions are developed to make evaluations of the sampling distribution tractable for large communities. In this regime, population fluctuations become negligible. A calculation of the macroscopic limits for the symmetric and asymmetric theories yields a new class of deterministic competition models for communities of fixed-size where rescue effects facilitate coexistence. For nearly neutral communities where the asymmetric species experiences linear density-dependence in ecological fitness, strong Allee-type effects emerge from a saddle-node bifurcation at a critical point in dispersal limitation. The bistable dynamics governing a canonical Allee effect are modified by a constant influx of migrants, which raises the lower stable fixed point above zero. In the stochastic theory, a saddle-node bifurcation corresponds to the development of bimodal stationary distributions and the emergence of inflection points in plots of mean first-time to extirpation as a function of abundance.

### The Role of Light and Nutrients in Aquatic Trophic Interactions

Friday, April 16, 2010, 4:30pm
Ungar Room 402

Abstract: Carbon (C), nitrogen (N), and phosphorus (P) are vital constituents in biomass: C supplies energy to cells, N is essential to build proteins, and P is an essential component of nucleic acids. The scarcity of any of these elements can severely restrict organism and population growth. Thus in nutrient deficient environments, the consideration of nutrient cycling, or stoichiometry, may be essential for population models. To show this idea, I will present two case studies in this talk.

We carried out a microcosm experiment evaluating competition of an invasive species Daphnia lumholtzi with a widespread native species, Daphnia pulex. We applied two light treatments to these two different microcosms and found strong context-dependent competitive exclusion in both treatments. To better understand these results we developed and tested a mechanistically formulated stoichiometric model. This model exhibits chaotic coexistence of the competing species of Daphnia. The rich dynamics of this model as well as the experiment allow us to suggest some plausible strategies to control the invasive species D. lumholtzi.

We modeled bacteria-algae interactions in the epilimnion with the explicit consideration of carbon (energy) and phosphorus (nutrient). We hypothesized that there are three dynamical scenarios determined by the basic reproductive numbers of bacteria and algae. Effects of key environmental conditions were examined through these scenarios. Competition of bacterial strains were modeled to examine Nishimura's hypothesis that in severely P-limited environments such as Lake Biwa, P-limitation exerts more severe constraints on the growth of bacterial groups with higher nucleic acid contents, which allows low nucleic acid bacteria to be competitive.

### Numerical Solutions to Noisy Systems

Thursday, April 15, 2010, 5:00pm
Ungar Room 402

Abstract: We study input-affine systems where input represents some bounded noise. The system can be rewritten as differential inclusion describing the evolution. Differential inclusions are a generalization of differential equations with multivalued right-hand side. They have applications in many areas of science, such as mechanics, electrical engineering, the theory of automatic control, economical, biological, and social macrosystems. A numerical method for rigorous over-approximation of a solution set of input-affine system will be presented. The method gives high order error for a single time step and a uniform bound on the error over the finite time interval. The approach is based on the approximations of inputs by piecewise linear functions.

### Ecological and Evolutionary Consequences of Dispersal in Multi-trophic Communities

Friday, March 12, 2010, 4:30pm
Ungar Room 402

Abstract: I investigate the effects of non-random dispersal strategies on coexistence and species distributions in multi-trophic communities with competition and predation. I conduct a comparative analysis of dispersal strategies with random and fitness-dependent dispersal at the extremes and two intermediate strategies that rely on cues (density and habitat quality) that serve as proxies for fitness. The most important finding is an asymmetry between consumer species in their dispersal effects. The dispersal strategy of inferior resource competitors that are less susceptible to predation have a large effect on both coexistence and species distributions, but the dispersal strategy of the superior resource competitor that is more susceptible to predation has little or no effect on dispersal. I explore the consequences of this asymmetry for the evolution of dispersal.

### How to Lose as Little as Possible

Thursday, March 11, 2010, 5:00pm
Ungar Room 402

Abstract: Suppose Alice has a coin with heads probability q and Bob has one with heads probability p > q. Now each of them will toss their coin n times, and Alice will win iff she gets more heads than Bob does. Of course, the game favors Bob, but for the given p, q, what is the choice, N(q,p), of n that maximizes Alice's chances of winning? The analysis uses the multivariate form of Zeilberger's algorithm, so a portion of the talk will be a review of the ideas underlying symbolic summation.

### Sobolev Inequalities on Probability Metric Spaces

Friday, March 5, 2010, 5:00pm
Ungar Room 402

Abstract: To formulate new Sobolev inequalities one needsto answer questions like: what is the role of dimension? What norms are appropriate to measure the integrabilitygains? Just to name a few...For example, in contrast to the Euclidean case, the integrability gains in Gaussian measure are logarithmic but dimension free (log Sobolev inequalities). So it is easy to understand the difficulties to derive a general theory. I will discuss some new methods to prove general Sobolev inequalities that unify the Euclidean and the Gaussian cases, as well as several important model manifolds.

CANCELLED

### Problems in Probability

Friday, February 26, 2010, 5:00pm
Ungar Room 402

Abstract: Several problems will be discussed: 1) What is the distribution of the empirical correlation coefficient of two (actually independent) Wiener processes? It is far from zero - correlation is induced by the arc sine law property of the sample paths. This is used by (bad) statisticians to show correlation between time series when none exists. It is a non-trivial calculation to find the actual distribution. 2) What is the relationship between the coefficients of a polynomial of degree n and the number of its real zeros? Descartes had something to say about it, but Mark Kac showed that probability theory can add a lot of insight. 3) An update on the situation discussed last year re the artificial pancreas project.

### Riemannian Manifolds of Constant Scalar Curvature

Tuesday, February 23, 2010, 5:00pm
Ungar Room 402

Abstract: The problem of constructing Riemannian metrics of constant scalar curvature is called the Yamabe problem. It is an important variational problem in conformal geometry, and also relates directly to the Einstein equations of general relativity. We will give a brief history and introduction to this problem and describe some new phenomena which have been discovered recently concerning issues of singular behavior and blow up of such metrics.

### Critical Metrics for the Volume Functional on Compact Manifolds with Boundary

Thursday, February 4, 2010, 5:00pm
Ungar Room 402

Abstract: It is known that, on closed manifolds, Einstein metrics of negative scalar curvature are critical points of the usual volume functional constrained to the space of metrics of constant scalar curvature. In this talk, I will discuss how this variational characterization of Einstein metrics can be localized to compact manifolds with boundary. I will derive the critical point equation and focus on geometric properties of its solutions. In particular, if a solution has zero scalar curvature and the boundary of the manifold can be isometrically embedded into the Euclidean space as a convex hypersurface, I will show that the volume of such a critical metric is always greater than or equal to the Euclidean volume enclosed by the image of the isometric embedding, and two volumes are the same if and only if the critical metric is isometric to the Euclidean metric on a round ball. I will also give a classification of all conformally flat critical metrics. This is joint work with Luen-Fai Tam.

### Einstein Metrics, the Bach Tensor, and Metric Degenerations

Monday, February 1, 2010, 5:00pm
Ungar Room 402

Abstract: One might search for "canonical metrics," such as Einstein metrics, on a manifold by trying to prove the convergence of a sequence of metrics that minimize some functional, although such a direct approach usually fails. In this talk we present an indirect approach which has been successful in some cases. A local obstruction to finding an Einstein metric in a conformal class is the non-vanishing of the Bach tensor, defined to be the gradient of the Weyl curvature functional $\int |W|^2$. On a Kaehler manifold there are no other obstructions, and any Bach-flat Kaehler metric is locally conformally Einsteinian. Additionally, the conformal factor is geometrically interesting and sometimes controllable. This talk will describe the results of a 2008 paper with X. Chen and C. LeBrun, where circumstances under which a Kaehler manifold is Bach-flat were established, and where it was shown that these conditions hold for a certain Kaehler metric on $CP^2 # 2\overline CP^2$ with non-zero conformal factor, establishing for the first time an Einstein metric on $CP^2 # 2\overline 2CP^2$.

### Metabelian SL(n,C) Representations of Knot Groups

Thursday, January 28, 2010, 5:00pm
Ungar Room 402

Abstract: In this talk, which represents joint work with Stefan Friedl, we will present a classification of irreducible metabelian SL(n,C) representations of knots groups. Under a mild hypothesis, we prove that such representations factor through a finite group, hence they are all conjugate to unitary representations, and we give a simple formula for the number of conjugacy classes. For knots with nontrivial Alexander polynomial, we discuss an existence result for irreducible metabelian representations. Given a knot group, its SL(n,C) character variety admits a natural action by the cyclic group of order n, and we show how to identify the fixed points of this action with characters of metabelian representations. If time permits, we will describe conditions under which such points are simple points in the character variety using a deformation argument of Abdelghani, Heusener, and Jebali.

### Backwards Uniqueness for the Ricci Flow and the Non-expansion of the Isometry Group

Monday, January 25, 2010, 5:00pm
Ungar Room 402

Abstract: One of the fundamental properties of the Ricci flow -- an evolution equation for Riemannian metrics -- is that of isometry preservation, namely, that an isometry of the initial metric remains an isometry of the solution, at least as long as the curvature remains bounded. In this talk, I will take up the complementary problem of isometry development under the flow. While the solution may acquire new isometries in the limit, one does not expect the flow to sponsor their generation within the lifetime of the solution. The impossibility of such a phenomenon is equivalent to a backwards uniqueness (or unique-continuation) property for the equation: two solutions which agree at some non-initial time must agree identically at all previous times. I will discuss recent work which establishes this property for complete solutions of bounded curvature, and prohibits, additionally, a solution from becoming Einstein or self-similar in finite time.

### Impulsive Delay Equation Models for the Control of Vector-borne Diseases

Friday, January 15, 2010, 4:00pm
Ungar Room 402

Abstract: Delay equation models for the control of a vector-borne disease such as West Nile virus will be presented. The models make it possible to compare the effectiveness of larvicides and adulticides in controlling mosquito populations. The models take the form of autonomous delay differential equations with impulses (if the adult insects are culled) or a system of nonautonomous delay differential equations where the time-varying coefficients are determined by the culling times and rates (in the case where the insect larvae are culled). Sufficient conditions can be derived which ensure eradication of the disease. Eradication of vector-borne diseases is possible by culling the vector at either the immature or the mature phase. Very infrequent culling can actually lead to the mean insect population being increased rather than decreased.

### Einstein Spacetimes with Bounded Curvature

Thursday, December 10, 2009, 4:30pm
Ungar Room 402

Abstract: I will present recent results on Einstein spacetimes of general relativity when the curvature is solely assumed to be bounded and no assumption on its derivatives is made. One such result, in a joint work with B.-L. Chen, concerns the optimal regularity of pointed spacetimes in which, by definition, an "observer" has been specified. Under geometric bounds on the curvature and injectivity radius near the observer, there exist a CMC (constant mean curvature) foliation as well as CMC--harmonic coordinates, which are defined in geodesic balls with definite size depending only on the assumed bounds, so that the components of the Lorentzian metric has optimal regularity in these coordinates. The proof combines geometric estimates (Jacobi field, comparison theorems) and quantitative estimates for nonlinear elliptic equations with low regularity.

CANCELLED

### Evolutionary Implications of Influenza Medication Strategies

Tuesday, November 24, 2009, 5:00pm
Ungar Room 402

Abstract: Patients at risk for complications of influenza are commonly treated with antiviral medications, which however also could be used to control outbreaks. The adamantanes and neuraminidase inhibitors are active against influenza A, but avian influenza (H5N1) is resistant to oseltamivir and swine influenza (H1N1) to the adamantanes (but see postscript). To explore influenza medication strategies (pre-exposure or prophylaxis, post-exposure/pre-symptom onset, and treatment at successive clinical stages) that may affect evolution of resistance (select for resistant strains within or facilitate their spread between hosts), we elaborated a published transmission model and chose parameters from the literature. Then we derived the reproduction numbers of sensitive and resistant strains, peak and final sizes, and time to peak. Finally, we made these results accessible via user-friendly Mathematica notebooks. (Joint work with Rongsong Liu, Dashun Xu, Yiding Yang, and John Glasser)

### Gauge Theory of Faddeev-Skyrme Functionals

Friday, November 20, 2009, 3:30pm
Ungar Room 402

Abstract: We study geometric variational problems for a class of nonlinear sigma-models in quantum field theory. Mathematically, one needs to minimize an energy functional on homotopy classes of maps from closed 3-manifolds into compact homogeneous spaces G/H, similar to the case of harmonic maps. The minimizers are known as Hopfions and exhibit localized knot-like structure. Our main results include proving existence of Hopfions as finite energy Sobolev maps in each (generalized) homotopy class when the target space is a symmetric space. For more general spaces we obtain a weaker result on existence of minimizers in each 2-homotopy class.

Our approach is based on representing maps into G/H by equivalence classes of flat connections. The equivalence is given by gauge symmetry on pullbacks of G-->G/H bundles. We work out a gauge calculus for connections under this symmetry, and use it to eliminate non-compactness from the minimization problem by fixing the gauge.

### Graph Theoretic Methods in Algebraic Statistics

Thursday, November 12, 2009, 5:00pm
Ungar Room 402

Abstract: First I will review how methods from commutative algebra, for example Gröbner bases and toric ideals, can be used in statistics. Then I will describe two applications of graph theoretic methods in this context: My proof of Sturmfels and Sullivant's conjecture on cut ideals; and the ideals of graph homomorphisms introduced together with Patrik Noren.

### Slice Knots and the Alexander Polynomial

Thursday, November 5, 2009, 5:00pm
Ungar Room 402

Abstract: A knot in the 3-sphere is slice if it bounds an embedded disk in the 4-ball. The disk may be topologically embedded, or we may require the stronger condition that it be smoothly embedded; the knot is said to be (respectively) topologically or smoothly slice. It has been known since the early 1980's that there are knots that are topologically slice, but not smoothly slice. These result from Freedman's proof that knots with trivial Alexander polynomial are topologically slice, combined with gauge-theory techniques originating with Donaldson. In joint work with C. Livingston and M. Hedden, we answer the natural question of whether Freedman's result is responsible for all topologically slice knots. We show that the group of topologically slice knots, modulo those with trivial Alexander polynomial, is infinitely generated. The proof uses Heegaard-Floer theory.