Calendar of Events

Vanderbilt Mathematics

Throughout the semester, we will be looking at semi-group theory with applications to age-dependent models.

Denis Osin, Vanderbilt University
Asymptotic dimension and type functions of finitely generated groups

Location: Room 1432

To each metric space of finite asymptotic dimension, one associates a collection of invariants called type functions. These functions are closely related to Hilbert space compression rate, dimension of asymptotic cones, and other asymptotic invariants.  I will review some results about type functions of connected Lie groups, lattices, and relatively hyperbolic groups. We will also discuss the question which functions can be realized as type functions of finitely generated groups.

Vandy Math Club
Presentations of Spring 2010 Math Courses

Location: Room 1431

Unsure of what math courses to take next semester?  Confused by the descriptions in the Undergraduate Catalog? These presentations will give you a preview of the courses offered so that you can make informed decisions when you register for courses.  Each professor will give a 5-10 minute talk describing the content and format of the course they are teaching in the Spring.  The math courses being presented are:

221 Theory of Numbers - Ratcliffe
229 Advanced Engineering Mathematics - Ahner
234 Methods for Initial and Boundary-Value Problems - Foldes
242 Topology of Surfaces - Hoehn
250 Introduction to Mathematical Logic - Bova
256 Mathematical Modeling in Economics - Tschantz
262 Mathematical Modeling in Biology - Webb
287 Nonlinear Optimization - Ellingham

Refreshments will be offered before the event in SC 1425 at 4:45.

Derek Bruff, Vanderbilt University
Cryptography: Nby Bcmnils uhx Gunb iz Wixym uhx Wcjbylm

Location: Room 1206

Cryptography is the science of secret writing—using codes and ciphers to conceal the meaning of messages and also cracking those codes and ciphers to discover the meaning of messages one is not intended to read.  The history of cryptography is a game of one-upmanship between code makers and code breakers, a game that has played out in military settings (such as the cracking of the encoded Zimmerman Telegram that led to the entry of the United States in World War One), information security (such as the data encryption used to protect financial information online), and popular culture (such as the codes and ciphers used in Sherlock Holmes’ stories, books and movies like The Da Vinci Code, and online alternate reality games such as ones used to promote the TV show Lost). In this talk, we’ll take a quick tour through the history of cryptography and take a look at a few classical cipher systems, as well as some of the mathematics that makes them work. FREE PIZZA!

John Ratcliffe, Vanderbilt University
JSJ Decompositions of Coxeter Groups over FA subgroups

Location: Room 1310

A group G has property FA if G fixes a point of every tree on which G acts without inversions.  A Coxeter group W, with finitely many Coxeter generators S, has property FA if and only if the product of any two elements of S has finite order in W.  A visual subgroup of a Coxeter system (W,S) is a subgroup of W generated by a subset of S.  A graph of groups decomposition of a Coxeter system (W,S) is said to be visual if every vertex and edge group is visual.  We prove that every Coxeter system of finite rank has a visual JSJ graph of groups decomposition with edge groups having property FA.  As an application, we reduce the twist conjecture to Coxeter systems that are indecomposable with respect to amalgamated products over visual subgroups with property FA. 

Weekly Math Club Meeting

Location: Room 1425 

Viktor Ginzburg, University of California, Santa Cruz
The Conley conjecture: infinitely many periodic points of Hamiltonian diffeomorphisms 

Location: Room 5211

One distinguishing feature of Hamiltonian dynamical systems is that such systems, with very few exceptions, tend to have numerous fixed and periodic points.  In 1984 Conley conjectured that a Hamiltonian diffeomorphism (i.e., the time-one map of a Hamiltonian flow) of a torus has infinitely many periodic points or, more precisely, such a diffeomorphism with finitely many fixed points has simple periodic points of arbitrarily large period. This fact was proved by Hingston some twenty years later, in 2004. Similar results for Hamiltonian diffeomorphisms of surfaces of positive genus were also established by Franks and Handel. Of course, one can expect the Conley conjecture to to hold for a much broader class of closed manifolds and this is indeed the case. For instance, by now, the conjecture has been proved for the so-called closed, symplectically aspherical manifolds (including tori and surfaces of positive genus) and the Calabi-Yau manifolds using symplectic topological techniques. In this talk we will discuss the underlying reasons for the existence of periodic orbits for Hamiltonian flows and maps and outline a proof of the Conley conjecture.  Tea at 3:30 in SC 1425.

Zhian Wang, Vanderbilt University
Micro and macro models for chemotaxis

Location: Room 1307

This talk is focused on two questions of chemotaxis modeling. One is how to establish the communications between microscopic and macroscopic chemotaxis models. The other is how information in the microscopic model is passed to the macroscopic model. For the first question, I use a novel approach to derive the macroscopic limits and express the microscopic quantities in terms of macroscopic quantities with the preservation of energy law. For the second question, I investigate the traveling waves of both microscopic and macroscopic models from which we see how traveling waves in the microscopic model are retained, lost or created during the transition from the microscopic to macroscopic models. Biological implications will be discussed along the talk.

Shanks Workshop: "A combustible mixture: finite algebras, graphs, and the dichotomy conjecture for constraint satisfaction problems" (November 9 - November 18).

Room 1310

Miklos Maroti will talk today and tomorrow, laying the foundation for more advanced talks on algebraic CSP. These two talks are titled "Graph homomorphisms, algorithmic complexity and universal algebra." A complete schedule of all workshop talks is available at http://www.math.vanderbilt.edu/shanks_csp.

Throughout the semester, we will be looking at semi-group theory with applications to age-dependent models.

Matjaz Konvalinka, Vanderbilt University
A weighted analog of complementary hook length formula

Location: Room 1403

The branching rule for hook lengths, used in the proof of the hook length formula, has a complementary version that can be used to prove that the sum of squares of dimensions of irreducible representations of S_n equals n!. In this talk, I will present a weighted analog of this formula, in the spirit of weighted hook length formulas I presented in my previous talk. I will also present some applications to weighted hook walks. 

Marius Junge, University of Illinois Urbana-Champaign
Operator algebra techniques in Quantum information

Location: room 1432

By now there are a number of connections between operator algebras and Quantum Information Theory (QI) beside the obvious one that completely positive maps are important in both areas. In this talk I will focus on violations of Bell inequalities for tripartite and bipartite systems, and show how free groups, free probability, and, of course, tensor norm techniques from operator space yield (almost) optimal bounds and examples for violations. (Joint work with Perez-Garcia, Villuneavo, Wolf, Villuneavo and Palazuelos)

Shanks Workshop: "A combustible mixture: finite algebras, graphs, and the dichotomy conjecture for constraint satisfaction problems" (November 9 - November 18).

Room 1312

This second talk by Miklos Maroti on "Graph homomorphisms, algorithmic complexity and universal algebra" will finish laying a foundation for the remaining talks in the workshop. A complete schedule of all workshop talks is available at http://www.math.vanderbilt.edu/shanks_csp.

Mikil Taylor, Vanderbilt University
The Development of Concept of the Number

Location: Room 1206 

Contrary to popular belief, the concept of number has not been static throughout history.  Humans started in the most humble of conditions, counting on fingers, unable to write or even conceive of many numbers above 2.  As time passed, the idea expanded to include written forms, both well and terribly suited for calculation, and other ideas of numbers, from fractions to irrationals to negatives, all the way to the imaginary.  Join Vanderbilt Undergraduate Mikil Taylor on this exciting journey through the ever-changing ideas of what exactly constitutes a number. Free pizza.

Shanks Workshop: "A combustible mixture: finite algebras, graphs, and the dichotomy conjecture for constraint satisfaction problems" (November 9 - November 18).

Room 1312

Matthew Valeriote will speak on "Maltsev conditions associated with tame congruence theory and the constraint satisfaction problem." A complete schedule of all workshop talks is available at http://www.math.vanderbilt.edu/shanks_csp.

Mark Sapir, Vanderbilt University
Non-linear matrix groups

Location: Room 1310 

Joint with Martin Kassabov. We prove that the group EL_n(R) is not linear for many rings R (including the free ring) and n>2.

Mark Sapir, Vanderbilt University
Non-linear matrix groups

Room 1310

Joint with Martin Kassabov. We prove that the group EL_n(R) is not linear for many rings R (including the free ring) and n>2.

Math Club Meeting

Room 1425

Divisibility Tests. We will investigate tricks to determining whether an integer is divisible by n.

Doron Lubinsky, Georgia Tech
de Branges spaces, universality limits, and orthogonal polynomials

Location: SC 5211

Universality limits for random matrices describe the spacings between successive eigenvalues of random matrices, and their distribution. In the unitary case, one way to establish such universality limits is via the the theory of entire functions of exponential type, and de Branges spaces. We shall discuss the method and some recent results. No background on de Branges spaces, or universality is assumed. Tea at 3:30 p.m. in SC 1425.

Shanks Workshop: "A combustible mixture: finite algebras, graphs, and the dichotomy conjecture for constraint satisfaction problems" (November 9 - November 18).

Room 1312

Matthew Valeriote will speak on "The complexity of equivalence and isomorphism of primitive positive formulas."  A complete schedule of all workshop talks is available at http://www.math.vanderbilt.edu/shanks_csp.

Leonardo Marazzi, Western Kentucky University
Scattering and inverse scattering on some classes of conformally compact manifolds

Location: Room 1307

We study scattering theory on Asymptotically Hyperbolic (AH) manifolds and its generalizations to conformally compact manifolds and AH Einstein manifolds. Some examples of AH manifolds are the de Sitter-Schwarzschild model of the exterior of a black hole, which can be viewed as an AH manifold with two ends; and the Schwarzschild model of the exterior of a black hole, for which one of the two ends is a AH manifold, and the other end is an Asymptotically Euclidean manifold. Other examples of AH manifolds are given by quotients of the hyperbolic space by particular groups of motion. We also discuss some open problems in this area.

Shanks Workshop: "A combustible mixture: finite algebras, graphs, and the dichotomy conjecture for constraint satisfaction problems" (November 9 - November 18).

Room 1310

Miklos Maroti will speak on "Algebraic characterization of CSP problems of bounded width." A complete schedule of all workshop talks is available at http://www.math.vanderbilt.edu/shanks_csp.

Throughout the semester, we will be looking at semi-group theory with applications to age-dependent models.

Mark Ellingham, Vanderbilt University
Connectivity and P_4-linkage

Location: Room 1403

Given graphs G and H, we say G is H-linked if for every injective mapping f:V(H) -> V(G) we can find a subgraph H' of G which is a subdivision of H, with f(v) being the vertex of H' corresponding to each vertex v of H.  Xingxing Yu characterized certain graphs related to P_4-linkage.  We use his characterization to show that every 7-connected graph is P_4-linked, and to construct 6-connected graphs that are not P_4-linked.  This is joint work with Mike Plummer and Gexin Yu.

Richard Burstein, Vanderbilt University
Central sequences and s-McDuff factors, I

Room 1308

I will discuss a recent paper of Popa, in which he constructs examples of subfactors with trivial chi invariant which are not s-McDuff. My first talk will deal primarily with the technical machinery needed to describe these examples, namely ultrafilters and central sequences.

Mrinal Raghupathi, Vanderbilt  University
Toeplitz corona theorems for some algebras of holomorphic functions

Location: Room 1432 

In this talk I will describe the operator theoretic version of the classical corona problem. I will explain the connection to the tangential version of the Nevanlinna-Pick interpolation problem and an associated distance problem.  I will show how the well-known distance-duality method can be used to prove a new Toeplitz corona theorem for Riemann surfaces and also recover known results on the ball, polydisk, and Drury-Arveson space. (This is joint work with Brett D. Wick)

Vandy Math Club Meeting
Topics to be Explored: Quaternions/Hypercomplex numbers, Platonic Solids 

Location: Room  1425

Jeremy LeCrone, Vanderbilt University
IF You Believe That, THEN You Won't Believe This (Subtitle: LOGIC: The Science of Argument)

Location: Room 1206

If Mathematics is the language of science, as most people would agree, then the language of Mathematics is Logic. In this talk I will introduce you to some of the basics of this not-so-foreign language. As rational, educated human beings, you will see that learning logic is not like learning another language, but rather like formally defining the language you have been speaking since birth. Recognizing proper logic and knowing how to apply it can benefit you in all arenas of life. Not only will you see how logic is used in mathematics to prove and disprove claims, but also how it is used -- and abused -- in television, politics, the court system, philosophy, cartoons, movies, games, business, etc...  Free Pizza!

Shanks Workshop: "A combustible mixture: finite algebras, graphs, and the dichotomy conjecture for constraint satisfaction problems" (November 9 - November 18).

Room 1312

The speaker will be either Miklos Maroti or Ralph McKenzie. Title to be announced. A complete schedule of all workshop talks is available at http://www.math.vanderbilt.edu/shanks_csp.

Dan Ramras, New Mexico State University
Chern-Weil theory, representation theory, and topological K-theory

Location: Room 1310

In a number of computations, it has been observed that below the (rational) cohomological dimension of a group G, there is a discrepancy between the representation theory of G, which is captured by the spaces Hom(G, U(n)), and the topological K-theory of the classifying space of G. This discrepancy is closely analogous to Beilinson-Quillen-Lichtenbaum conjectures relating algebraic K-theory and Galois cohomology. Focusing on the case in which G is the fundamental group of an aspherical, closed manifold, I'll describe joint work with Tom Baird, which explains this phenomenon. The main ingredients are Chern-Weil theory and the homotopical relationship between flat connections, representations, and K-theory. I'll discuss how these results fit with particular examples, including products of surfaces and certain crystallographic groups.

Hanfeng Li, SUNY Buffalo and Vanderbilt University
Combinatorial independence in dynamics

Location: Room 5211

Combinatorial independence originated from the work of Rosenthal on characterization of Banach spaces containing l_1 isomorphically. Based on joint work with Wen Huang, David Kerr, and Xiangdong Ye, I will discuss how it leads to unified combinatorial and functional-analytic approaches to the study of various mixing properties in dynamics. Tea at 3:30 pm in SC 1425.

Thanksgiving Break - No Speakers.

Throughout the semester, we will be looking at semi-group theory with applications to age-dependent models.

Richard Burstein, Vanderbilt University
TBA

Location: Room 1308