Graduate Colloquium Fall 2009

October 22, 2009

Additive Combinatorics and Random Matrices

Zhe Xu 

Additive combinatorics become a very powerful tool recent year. In this talk, I will talk about Tao and Vu's series works on Littlewood-Oord problem, which is in the spirit of Freiman inverse theorem on sumsets from additive combinatorics. This problem starts with the hypothesis that a concentration probability is large, and concludes that almost all of the v1; :::; vn are contained in an arithmetic progression. As an application it gives a new bound on the distribution of the least singular value of a random Bernoulli matrix, which play a key role in establishment of circular law.

October 29, 2009

Formulas for the Asymmetric Simple Exclusion Process

Prof. Harold Widom 

The lecture will describe joint work with Craig A. Tracy on the asymmetric simple exclusion process on the integers. In this process particles are initially at some integer sites. Each particle waits a random time (governed by an exponential law), then with probability p it tries to move one step to the right and with probability q=1-p it tries to move one step to the left. The particle moves only if the desired site is unoccupied. (Then its clock restarts.) For the totally asymmetric process (either p=1 or q=1, so particles may move in only one direction) there is a connection with random matrices, so the machinery of random matrix theory can be used, for example, to get asymptotic results. This is no longer the case for the general asymmetric process. Nevertheless it is possible to use an idea of Hans Bethe (the "Bethe Ansatz") to derive exact formulas for probabilities for finite systems, and to extend these to certain infinite systems. One such probability is expressible in terms of Fredholm determinants. This makes it possible, using operator theory, ultimately to obtain asymptotic results generalizing those for the totally asymmetric process.

November 5, 2009

Singular Maps and Cartan Prolongations

Alex Castro 

Jet spaces were first introduced by Ehresmann in an attempt to develop a theory of higher order differentials of maps between curved manifolds. They provide us with a very convenient model space to study differential equations from a geometric point-of-view, among other things. In this seminar I will introduce some examples of jet spaces and discuss my joint work with Prof. Montgomery where we propose a “compactification” procedure of these spaces which permits one to study singular objects, like curves and surfaces in this framework. The key-word here is “prolongation”, a technique first proposed by Cartan and helps one to introduce bundles encoding higher order data in a manifold. There is also an interesting problem in representation theory related to this work. The diffeomorphism group of $n$-space induces an action in these manifolds and I plan to discuss some results we obtained on the orbit structure of this action.

November 10, 2009

Expanders: from arithmetic to combinatorics and back

Prof. Alex Gamburd 

Expanders are highly-connected sparse graphs widely used in computer science. The optimal expanders -- Ramanujan graphs -- were constructed using deep results from the theory of automorphic forms. In recent joint work with Bourgain and Sarnak tools from additive combinatorics were used to prove that a wide class of "congruence graphs" are expanders; this expansion property plays crucial role in establishing novel sieving results pertaining to primes in orbits of linear groups.

November 26, 2009

No Colloquium - Thanksgiving Day Observed

December 3, 2009

What is $G_2$?

Professor Marty Weissman 

The "exceptional group" $G_2$ means different things to different people. Mostly ignoring its geometric origins, I will discuss $G_2$ from an algebraic standpoint, introducing Hurwitz algebras and octonions along the way. I will then interpret all of the basic properties of $G_2$ (root system, Dynkin diagram, etc..) using algebraic properties of octonions. Audience members will receive commemorative handouts :)