# COLLOQUIUM 2010-11

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Spring 2011
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Date Speaker & Title
January 20 John McCarthy (Washington University)  Host: B. MacCluer
Title: Operator Monotone Functions of Several Variables
February 3 Grigory Mikhalkin (Universite de Geneve)  Host: S. Krushkal
Title: Classical algebraic geometry in the tropical limit
February 10 Wojciech De Roeck (University of Heidelberg)  Host: A. Abdesselam
Title: Diffusion in Hamiltonian systems
February 17 David Nualart (KU)  Host: C. Gromoll
Title: Fractional Brownian Motion: Stochastic Calculus and Applications
February 24 David Saltman (UT@Austin & IDACCR)  Host: A. Rapinchuk
Title: Division Algebras, Cohomology, and Indices
February 25 Henri Gillet (UIC)  Host: B. Parshall
Title: Descent, differential algebra, and fields of definition of varieties in characteristic p
March 3 Bernhard Muehlherr (University of Giessen)  Host: P. Abramenko
Title: On the center conjecture for spherical buildings
March 10 No Colloquium
Title: Spring Break
March 15 Daniel Nakano (UGA)  Host: W. Wang
Title: Viewing representation theory through cohomology and geometry
March 17 Doug Ravenel (Rochester)  Host: M. Hill
Title: A solution to the Arf-Kervaire Invariant One Problem
March 24 Karen Vogtmann (Cornell)
Title: Outer spaces
March 31 David Kerr (Texas A&M)  Host: D. Sherman
Title: Entropy for actions of sofic groups
April 4 John McCarthy (Washington Univ. at St. Louis)
Title: Extending Bounded Holomorphic Functions
April 28 Jason Bell (Simon Fraser University)  Host: T. Melcher
Title: Linear recurrences, geometry, and finite-state automata

# Fall 2010

Date Speaker & Title
September 9 John Thompson (Cambridge University, University of Florida, Visiting UVA)   Host: L. Scott
Title: The zeta function and an elliptic curve
September 16 Mike Hill (UVA)   Host: M. Hill
Title: What is homology?
September 23 David Sherman (UVA)   Host: D. Sherman

Title: What is an ultrapower? The classical ultrapower is a fundamental construction in logic, where it plays a role in several striking, easy-to-understand theorems about algebraic structures. There are also versions of the ultrapower construction for analytic structures such as Banach spaces, but these do not satisfy the same theorems until the underlying logic is also made more analytic. This talk will overlap by about half with one I gave in the algebra seminar last spring. I presume no familiarity with logic.
September 30 Mikhail Ershov   Host: M. Ershov

Title: What is Kazhdan's property (T)? Property (T) is a representation-theoretic notion defined by Kazhdan in 1967. Originally introduced merely as a tool for studying lattices in Lie groups, property (T) quickly turned into a major research direction with many connections outside of group theory and representation theory. In this talk, I plan to define property (T), describe (in very general terms) basic techniques for proving / disproving property (T), and discuss some of its basic applications, mainly in combinatorics.
October 7 Weiqiang Wang   Host: W. Wang

Title: What is Boson-Fermion Correspondence? The answers depend on whom you talk to. The boson-fermion correspondence has roots in math physics. It can be turned into a new approach to the character tables and representations of the symmetric groups. It offers another view toward the combinatorics of partitions. It also provides a module theoretic framework [categorification] for the celebrated Jacobi triple product identity.
October 14 Malek Abdesselam (UVA)   Host: M. Abdesselam

Title: What is a Feynman Diagram?There are many viable answers to this question. Mine is: a finite combinatorial structure which encodes an algebraic expression given by contraction of indices, starting from a collection of elementary tensors. When seen in this way, Feynman diagrams turn out to be ubiquitous in mathematics and can shed light on questions in algebra as well as analysis. I will try to give a very selective sample of applications of this notion, starting with explicit series for solutions of systems of algebraic equations.
October 28 Jae-hoon Kwon (University of Seoul/UC Berkeley)   Host: W. Wang

Title: Crystal graphs and Young tableaux A crystal graph is a combinatorial object associated with an integrable highest weight module over the quantum group of a Kac-Moody algebra. The purpose of this talk is to give an exposition on the relation between the classical combinatorics of Young tableaux and the crystal graphs of integrable $U_q(\mathfrak{gl}_n)$-modules, as an introduction to crystal base theory for those who are not familiar with this area.
November 4 Karen Vogtmann (Cornell)   Host: S. Krushkal

Title: Maps into and out of automorphism groups of free groups The contemporary study of mapping class groups and outer automophism groups of free groups is heavily influenced by the analogy between these groups and lattices in semisimple Lie groups. It can be proved that these groups are not lattices, so the fact that they share many properties with lattices is intriguing and somewhat mysterious. In this talk I will focus on properties of automorphism groups of free groups which echo the strong rigidity properties enjoyed by lattices, i.e. properties which severely restrict the possibilities for homomorphisms between lattices.
November 25 Thanksgiving
December 2 Bridget Tenner (DePaul University)   Host: M. Hill

Title: Coxeter groups and Bruhat order: algebraic and topological structureCoxeter groups are classical objects of interest to mathematicians in a variety of disciplines. One aspect of these systems is the study of a particular partial ordering of their elements, known as the Bruhat order. In this talk we explore several different structural aspects of Coxeter groups and of the Bruhat order, from both algebraic and topological perspectives. The results discussed will involve reduced words, permutation patterns, zonotopal tilings, order ideals, and the homotopy type of a particular cell complex. For example, recent work has uncovered significant links between permutation patterns, the reduced words of a permutation, and the interval structure of the Bruhat order. Among other things, this connection implies that a certain class of 2n-gons can be tiled by convex centrally symmetric 2k-gons iff k is 2 or n.
December 9 Howard Masur (University of Chicago)   Host: A. Rapinchuk

Title: Dynamics and ergodic theory of billiards in rational angled polygonsAn appealing example of a dynamical system is given by billiards in a polygon in the plane. The billiard ball moves at unit speed in a straight line and reflects off the sides with angle of reflection equal to angle of incidence. In dynamical systems one typically wants to study the long term behavior of orbits. In ergodic theory one asks if typical orbits are equidistributed. The case when the vertex angles are rational multiples of pi has been extensively studied. There is an unfolding process which converts the table into what is called a translation surface, and the billiard flow into a flow by straight lines on the surface. The unfolding of a square gives a torus and the flow is by straight lines on the torus, a subject that goes back to Kronecker and Weyl. In general one gets flows on higher genus surfaces. I will survey some of what is known in this subject.