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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
Professor Santiago Simanca
University of New Mexico
will present
On the Shape of Representable Integral Homology Classes in a Riemannian Manifold
Wednesday, May 16, 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
Professor Gabor Szekelyhidi
University of Notre Dame
will present
On the Positive Mass Theorem for Manifolds with Corners
Friday, May 4, 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.
Geometry and Physics Seminar
Dr. Jeff Jauregui
University of Pennsylvania
will present
An Axiomatic Approach to Quasi-local Mass in General Relativity
Wednesday, April 11, 2012, 4:00pm
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
Drew Armstrong
University of Miami
will present
Cluster Combinatorics
Tuesday, April 10, 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:
Combinatorics Seminar
Anton Dochtermann
University of Miami
will present
Probabilistic and Topological Bounds on Chromatic Number
Tuesday, April 3, 2012, 5:00pm
Ungar Room 402
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 into complete 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.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
Professor Ryan Derby-Talbot
Quest University
will present
Essential Surfaces and Dehn Filling
Wednesday, March 28, 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.
Geometry and Physics Seminar
Dr. Wilderich Tuschmann
Karlsruhe Institute of Technology
will present
Curvature vs. Curvature Operator
Wednesday, March 21, 2012, 4:00pm
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.
Combinatorics Seminar
Mark Skandera
Lehigh University
will present
A Conjectured Combinatorial Interpretation for Induced Sign Characters of the Hecke Algebra
Tuesday, March 6, 2012, 5:00pm
Ungar Room 402
This is joint work with Brittany Shelton of Lehigh University.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.
Combinatorics Seminar
Nathan Williams
University of Minnesota
will present
Promotion and Rowmotion
Friday, March 2, 2012, 5:00pm
Ungar Room 506
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))
Applied Math Seminar
John Chadam
Department of Mathematics
University of Pittsburgh
will present
Optimal Prepayment of Mortgages
Friday, March 2, 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
Gabriel Kerr
University of Miami
will present
Fiber Polytopes
Tuesday, February 14, 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.
Combinatorics Seminar
Sergi Elizalde
Dartmouth University
will present
Consecutive Patterns in Permutations:
Clusters, Generating Functions and Asymptotics
Friday, February 10, 2012, 5:00pm
Ungar Room 402
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.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.
Combinatorics Seminar
Jim Haglund
University of Pennsylvania
will present
The Monotone Column Permanent Conjecture and Multivariate Eulerian Polynomials
Monday, December 19, 2011, 3:00pm
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.
Geometry and Physics Seminar
L. Katzarkov
University of Miami
will present
SHS and Automorphic Forms
Tuesday, November 29, 2011, 5:00pm
Ungar Room 506
Combinatorics Seminar
Michelle Wachs
University of Miami
will present
Eulerian Numbers, Chromatic Quasisymmetric Functions and Hessenberg Varieties
Tuesday, November 22, 2011, 5:00pm
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
Professor Igor Belykh
Georgia State University
will present
Synchrony in Metapopulations:
The Role of Dispersal
Monday, November 21, 2011, 4:30pm
Ungar Room 402
Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.
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).
Applied Math Seminar
Steve Cantrell
University of Miami
will present
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
Joint work with Chris Cosner (University of Miami) and William Fagan (University of Maryland).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
Ryan Budney
University of Victoria
will present
Some Simple Triangulations
Friday, November 11, 2011, 1:30pm
Ungar Room 506
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.
Geometry and Physics Seminar
R. Fedorov
Kansas State University
will present
Generalized Langlands Correspondences
Wednesday, November 9, 2011, 4:00pm
Ungar Room 506
Combinatorics Seminar
Drew Armstrong
University of Miami
will present
Parking Spaces
Tuesday, November 8, 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).
Applied Math Seminar
Steve Cantrell
University of Miami
will present
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
Joint work with Chris Cosner (University of Miami) and William Fagan (University of Maryland).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.
Geometry and Physics Seminar
Professor Albert Fathi
Ecole Normale Supérieure de Lyon
will present
Smooth Time Functions for Stably Causal and Non-stably Causal Manifolds
Wednesday, November 2, 2011, 4:00pm
Ungar Room 506
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.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.
Geometry and Physics Seminar
Professor S. Kaliman
University of Miami
will present
Flexible Varieties
Wednesday, October 19, 2011, 4:00pm
Ungar Room 506
Geometry and Physics Seminar
Professor Lan-Hsuan Huang
Columbia University
will present
Hypersurfaces with Nonnegative Scalar Curvature and the Positive Mass Theorem
Wednesday, October 12, 2011, 4:00pm
Ungar Room 506
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.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
Shigui Ruan
University of Miami
will present
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: 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.
Applied Math Seminar
Chris Cosner
University of Miami
will present
Modeling the Evolution of Conditional Dispersal in Spatially Heterogeneous Environments
Friday, September 23, 2011, 5:00pm
Ungar Room 402
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.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
Dr. Jeffrey Case
Princeton University
will present
Quasi-Einstein Metrics and Conformal Geometry
Thursday, September 1, 2011, 4:00pm
Ungar Room 402
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.
Geometry and Physics Seminar
R. Sazdanovic
University of Pennsylvania
will present
Categorifications of the Polynomial Ring Z[x]
Wednesday, August 31, 2011, 4:00pm
Ungar Room 506
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.
Combinatorics Seminar
Brant Jones
James Madison University
will present
Abacus Models for Parabolic Quotients of Affine Weyl Groups
Wednesday, August 3, 2011, 4:45pm
Ungar Room 402
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
Pengzi Miao
University of Miami
will present
Second Variation of Wang-Yau Quasi-local Energy
Thursday, July 28, 2011, 4:00pm
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
Carlos Vega
University of Miami
will present
On the Bartnik Splitting Conjecture
Thursday, July 28, 2011, 3:00pm
Ungar Room 411
Workshop: Topics in Mathematical Relativity
José Luis Flores
Universidad de Málaga
will present
Periodic Geodesics on Compact Lorentzian Manifolds with a Killing Vector Field
Tuesday, July 26, 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.
Workshop: Topics in Mathematical Relativity
Dr. Didier Solis
Universidad Autónoma de Yucatán
(UM PhD, 2006)
will present
Local Flatness in Asymptotically Flat Spacetimes
Thursday, July 21, 2011, 3:00pm
Ungar Room 411
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.
Dissertation Defense
Matthew Hyatt
University of Miami
will present
Quasisymmetric Functions and Permutation Statistics for Coxeter Groups and Wreath Product Groups
Wednesday, July 20, 2011, 10:00am
Ungar Room 301
Refreshments at 9:30am in Ungar Room 521
Dissertation Defense
Daniel P. Ryan
University of Miami
will present
Fitness Dependent Dispersal in Intraguild Predation Communities
Tuesday, July 12, 2011, 10:00am
Ungar Room 301
Refreshments at 9:30am in Ungar Room 521
Geometry and Physics Seminar
Yijia Liu
University of Miami
will present
Cycles and Sheaves: A 45 Minute Intro
Wednesday, April 27, 2011, 4:00pm
Ungar Room 506
Combinatorics Seminar
Julian Moorehead
University of Miami
will present
Partial Matroid Decomposition Posets
Monday, April 25, 2011, 5:00pm
Ungar Room 402
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.
Geometry and Physics Seminar
Professor N. Saveliev
University of Miami
will present
An Index Theorem for End-periodic Operators
Wednesday, April 13, 2011, 4:00pm
Ungar Room 506
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
Jeremy Martin
University of Kansas
will present
Critical Groups of Simplicial Complexes
Monday, April 11, 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.
Combinatorics Seminar
Michelle Wachs Galloway
University of Miami
will present
Unimodality of q-Eulerian Numbers and p,q-Eulerian Numbers, Part II
Monday, April 4, 2011, 5:00pm
Ungar Room 402
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.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).
Geometry and Physics Seminar
Professor L. Katzarkov
University of Miami
will present
Braid Monodromy, Floer Combinatorics and Fukaya Category
Monday, April 4, 2011, 4:00pm
Ungar Room 411
Geometry and Physics Seminar
Professor C. Diemer
University of Miami
will present
Combinatorics of Multiplier Ideal Sheaves III
Wednesday, March 30, 2011, 4:00pm
Ungar Room 506
Geometry and Physics Seminar
Professor C. Diemer
University of Miami
will present
Combinatorics of Multiplier Ideal Sheaves II
Thursday, March 24, 2011, 4:00pm
Ungar Room 411
Geometry and Physics Seminar
Professor Fedor Bogomolov
New York University
will present
Strong Form of the Grothendieck Section Conjecture in Functional Case
Wednesday, March 16, 2011, 4:00pm
Ungar Room 506
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.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.
Geometry and Physics Seminar
Professor Neil Hoffman
University of Texas
will present
Hidden Symmetries, Exceptional Surgeries, and Commensurability
Tuesday, March 9, 2011, 4:00pm
Ungar Room 506
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.
Combinatorics Seminar
Professor Mark Skandera
Lehigh University
will present
Path Tableaux and Combinatorial Interpretations for S_n-class Functions
Monday, March 7, 2011, 5:00pm
Ungar Room 402
This is joint work with Brittany Shelton and Sam Clearman of Lehigh University.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
Professor C. Diemer
University of Miami
will present
Combinatorics of Multiplier Ideal Sheaves
Wednesday, March 2, 2011, 4:00pm
Ungar Room 506
Geometry and Physics Seminar
George Lam
Duke University
will present
The Riemannian Positive Mass and Penrose Inequalities for Graphs over R^n
Tuesday, March 1, 2011, 5:00pm
Ungar Room 402
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
Professor G. Kerr
University of Miami
will present
Spectra as Cohomology Theory II
Wednesday, February 23, 2011, 3:30pm
Ungar Room 506
Geometry and Physics Seminar
Professor Ernesto Lupercio
Research and Advanced Studies Center of the National Polytechnic Institute of Mexico (Cinvestav - IPN)
Winner of the 2009 Srinivasa Ramanujan Prize
will present
Orbifolds, Ghost Loop Spaces and Twisted Sectors
Wednesday, February 23, 2011, 2:00pm
Ungar Room 411
Geometry and Physics Seminar
Professor Ernesto Lupercio
Research and Advanced Studies Center of the National Polytechnic Institute of Mexico (Cinvestav - IPN)
Winner of the 2009 Srinivasa Ramanujan Prize
will present
Non-compact Topological Field Theories and Frobenius Structures
Tuesday, February 22, 2011, 4:00pm
Ungar Room 411
Geometry and Physics Seminar
Professor Alex Iosevich
University of Rochester
will present
Regular Value Theorem in a Fractal Setting
Monday, February 21, 2011, 5:00pm
Ungar Room 506
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.
Geometry and Physics Seminar
Dr. Todd Oliynyk
Monash University
will present
Relativistic Fluids in 1+1 Dimensions with a Vacuum Boundary
Wednesday, February 16, 2011, 5:00pm
Ungar Room 402
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.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
Dr. Martin Li
Stanford University
will present
Free Boundary Problem for Embedded Minimal Surfaces
Tuesday, February 1, 2011, 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.
Geometry and Physics Seminar
Professor G. Kerr
University of Miami
will present
Spectra as Cohomology Theory
Wednesday, January 19, 2011, 4:00pm
Ungar Room 402
Combinatorics Seminar
Christian Stump
Centre de Recherches Mathématiques
Université de Montréal
and
Laboratoire de Combinatoire et d'Informatique Mathématique
Université du Québec à Montréal
will present
Moon Polyominoes, Pipe Dreams and Simplicial Spheres
Monday, November 29, 2010, 5:00pm
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.
Applied Math Seminar
Samares Pal
Georgia Institute of Technology
and
University of Kalyani
will present
Nutrient, Phytoplankton, Zooplankton Interaction in an Open Marine System –
A Mathematical Study
Friday, November 12, 2010, 4:00pm
Ungar Room 402
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.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.
Geometry and Physics Seminar
L. Katzarkov
University of Miami
will present
Inverse Spectra Problem in Algebraic Geometry
Tuesday, November 9, 2010, 5:00pm
Ungar Room 402
Applied Math Seminar
Guangyu Zhao
Department of Mathematics
University of Miami
will present
Time Periodic Traveling Wave Solutions of Reaction-diffusion Systems
Friday, November 5, 2010, 4:00pm
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.
Combinatorics Seminar
Michelle Wachs Galloway
University of Miami
will present
Unimodality of q-Eulerian Numbers and p,q-Eulerian Numbers
Monday, November 1, 2010, 5:00pm
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.
Applied Math Seminar
Chris Cosner
Department of Mathematics
University of Miami
will present
Reaction-diffusion-advection Models for the Effects and Evolution of Dispersal
Friday, October 29, 2010, 4:00pm
Ungar Room 402
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.
Combinatorics Seminar
Matthew Hyatt
University of Miami
will present
Double Feature
Monday, October 25, 2010, 5:00pm
Ungar Room 506
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).
Geometry and Physics Seminar
Professor B. de Oliveira
University of Miami
will present
Structure Theorem for Symmetric Differentials of Rank 1
Wednesday, October 20, 2010, 4:00pm
Ungar Room 402
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.
Combinatorics Seminar
Brant Jones
James Madison University
will present
An Explicit Derivation of the Möbius Function for Bruhat Order
Monday, October 18, 2010, 5:00pm
Ungar Room 506
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.
Geometry and Physics Seminar
L. Katzarkov
University of Miami
will present
Spectra, Gaps and Chow Groups
Friday, October 15, 2010, 4:00pm
Ungar Room 506
Geometry and Physics Seminar
Professor P. Miao
University of Miami
will present
On a Localized Riemannian Penrose Inequality
Wednesday, October 13, 2010, 4:00pm
Ungar Room 402
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
Anton Dochtermann
Stanford University
will present
Cellular Resolutions of Hypergraph Edge Ideals
Monday, October 4, 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.
Combinatorics Seminar
Anatoly Libgober
University of Miami
will present
Geometry of Arrangements of Hyperplanes and Cohomology of Orlik-Solomon Algebras
Monday, September 27, 2010, 5:00pm
Ungar Room 506
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.
Applied Math Seminar
Shigui Ruan
Department of Mathematics
University of Miami
will present
Modeling the Transmission Dynamics and Control of Hepatitis B Virus
Friday, September 24, 2010, 4:00pm
Ungar Room 402
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.)
Combinatorics Seminar
Drew Armstrong
University of Miami
will present
The Ish Arrangement of Hyperplanes
Monday, September 20, 2010, 5:00pm
Ungar Room 506
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
Professor C. Diemer
University of Pennsylvania
will present
Tropical Geometry, Compactifications, and Birational Geometry
Wednesday, August 25, 2010, 4:30pm
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.
Geometry and Physics Seminar
Professor A. Libgober
University of Miami
will present
Alexander Invariants of Fundamental Groups of the Complements to Plane Algebraic Curves
Tuesday, August 24, 2010, 4:00pm
Ungar Room 402
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.
Combinatorics Seminar
Dr. Martina Kubitzke
Reykjavik University
will present
Triangulations, the Associahedron and Gamma-vectors for Planar Lattices
Wednesday, July 7, 2010, 4:00pm
Ungar Room 506
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.
Geometry and Physics Seminar
L. Katzarkov
University of Miami
will present
How Not to Learn Minimal Model Program
Thursday, April 29, 2010, 4:00pm
Ungar Room 402
Geometry and Physics Seminar
Professor Leonid Parnovski
University College London
will present
Integrated Density of States of Schröedinger Operators with Periodic and Almost-periodic Potentials
Wednesday, April 14, 2010, 4:00pm
Ungar Room 402
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.
Combinatorics Seminar
Jay Schweig
University of Kansas
will present
On Lattice Path Matroids and Polymatroids
Friday, April 9, 2010, 5:00pm
Ungar Room 506
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.
Geometry and Physics Seminar
Professor Gordon Heier
University of Houston
will present
On Uniformly Effective Boundedness of Shafarevich Conjecture-type
Wednesday, April 7, 2010, 4:00pm
Ungar Room 402
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.
Combinatorics Seminar
Julian Moorehead
University of Miami
will present
On q-analogs of the k-equal Partition Lattice
Tuesday, April 6, 2010, 5:00pm
Ungar Room 506
Abstract: We discuss a theory of completely integrable algebraic (resp. holomorphic) vector fields on smooth affine algebraic varieties.
Geometry and Physics Seminar
Professor Shulim Kaliman
University of Miami
will present
On the Present State of the Andersen-Lempert Theory
Wednesday, March 24, 2010, 4:00pm
Ungar Room 402
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.
Combinatorics Seminar
Brendon Rhoades
Massachusetts Institute of Technology
will present
Cyclic Sieving and Polygon Multidissection Enumeration
Tuesday, March 23, 2010, 5:00pm
Ungar Room 506
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 $f^*(b)= a$.
Geometry and Physics Seminar
Professor Fedor Bogomolov
New York University
will present
Galois Groups and Birational Invariants of Functional Fields
Wednesday, March 17, 2010, 4:00pm
Ungar Room 402
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
Matthew Hyatt
University of Miami
will present
Signed Eulerian Quasisymmetric Functions
Tuesday, March 9, 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.
Combinatorics Seminar
Rafael S. Gonzalez D'Leon
University of Miami
will present
On the Half-plane Property and the Tutte-group of a Matroid
Tuesday, March 2, 2010, 5:00pm
Ungar Room 506
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.
Geometry and Physics Seminar
Jesse Johnson
Oklahoma State University
will present
Axiomatic Thin Position and Applications
Thursday, February 18, 2010, 4:00pm
Ungar Room 402
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.
Combinatorics Seminar
Benjamin Braun
University of Kentucky
will present
Nowhere-Harmonic Colorings of Graphs
Wednesday, February 17, 2010, 5:00pm
Ungar Room 506
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
Professor I. Hambleton
McMaster University
will present
Conjugation Spaces and 4-manifolds
Wednesday, February 17, 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).
Geometry and Physics Seminar
Professor I. Itenberg
Université de Strasbourg
will present
On Real Determinantal Quartics
Wednesday, February 10, 2010, 4:00pm
Ungar Room 402
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.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.
Combinatorics Seminar
Michelle Wachs
University of Miami
will present
Eulerian Quasisymmetric Functions and Cyclic Sieving
Tuesday, February 9, 2010, 5:00pm
Ungar Room 506
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.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*.
Geometry and Physics Seminar
Dima
University of Miami
will present
Motives
Thursday, February 4, 2010, 2:00pm
Ungar Room 547
Combinatorics Seminar
Drew Armstrong
University of Miami
will present
Reduced Decompositions of Permutations
Tuesday, February 2, 2010, 5:00pm
Ungar Room 506
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.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
Alexandr Usnich
Ludmil Katzarkov
Dima
University of Miami
will present
On the Work of Lurie
Tuesday, February 2, 2010, 4:00pm
Ungar Room 547
Geometry and Physics Seminar
Ludmil Katzarkov
University of Miami
will present
Hodge Structures and Spectra
Tuesday, February 2, 2010, 2:00pm
Ungar Room 547
Geometry and Physics Seminar
Alexandr Usnich
University of Miami
will present
A infty Categories
Monday, February 1, 2010, 11:00am
Ungar Room 547
NSF-CSMS Project Industry-Liaison Seminar
Michael Goldberg
President and CEO
Flamingo Software
will present
A Career in Software Development, Web-Based Systems
Wednesday, January 27, 2010, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
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
Professor A. Dvorsky
University of Miami
will present
Realizations of Minimal Representation of O(p,q)
Wednesday, December 9, 2009, 3:00pm
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
Jeremy Van Horn-Morris
American Institute of Mathematics
will present
Symplectic Fillings of Contact Manifolds
Friday, December 4, 2009, 3:30pm
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.
Geometry and Physics Seminar
Professor L. Katzarkov
University of Miami
will present
Spectra of Categories and Applications to Low Dimensional Topology
Tuesday, December 1, 2009, 5:00pm
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.
Applied Math Seminar
Dr. Orou Gaoue
ITME Post Doc
Department of Biology
University of Miami
will present
Modeling the Impact of Non-timber Forest Product Harvest in Variable Environments
Friday, November 20, 2009, 4:30pm
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.
Geometry and Physics Seminar
Professor N. Saveliev
University of Miami
will present
Seiberg-Witten Equations and End-periodic Dirac Operators
Wednesday, November 18, 2009, 4:00pm
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.
Applied Math Seminar
Professor Gaetano Zampieri
Dipartimento di Informatica
Universita di Verona
will present
A Class of Integrable Hamiltonian Systems and Weak Lyapunov Stability
Friday, November 13, 2009, 4:30pm
Ungar Room 402
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.
Combinatorics Seminar
Professor Eric Gottlieb
Rhodes College
will present
A Combinatorial Optimization Problem from Genomics
Friday, November 13, 2009, 3:00pm
Ungar Room 506
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.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.
Applied Math Seminar
Professor Neil Johnson
Department of Physics
University of Miami
will present
Insurgent Wars, Pandemics, Global Emissions and Market Crises:
One Model Fits All?
Friday, November 6, 2009, 4:30pm
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.
Geometry and Physics Seminar
Professor Bruno de Oliveira
University of Miami
will present
Closed Symmetric Differentials of Degree 2 and the Geometry of Complex Surfaces
Wednesday, November 4, 2009, 4:00pm
Ungar Room 402
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