Seminar

Macdonald polynomials and level two Demazure modules for affine sl(n+1)

An important result due to Sanderson and Ion says that characters of level one Demazure modules are specialized Macdonald polynomials. In this talk, I will introduce a new class of symmetric polynomials indexed by a pair of dominant weights of $sl_{n+1}$ which is expressed as linear combination of specialized symmetric Macdonald polynomials with coefficients defined recursively. These polynomials arose in my own work while investigating the characters of higher level Demazure modules.

Classification of the irreducible representations of braid groups

Abstract: In this talk I'm going to discuss the classification of the
irreducible representations of the Artin braid group $B_n$ on $n$ strings. All
irreducible representations of $B_n$ of dimension  less or equal to $n-1$
were classified by Ed Formanek in 1996; the irreducible representations of
$B_n$ of dimension  $n$ for $n\geq 9$ were classified by the speaker in 1999,
and for $n \leq 8$ they were classified by Formanek, Lee, Vazirani and the
speaker in 2003.

The Margulis Superrigidity Theorem

A lattice is a special kind of discrete subgroup of a topological group. The Margulis superrigidity theorem says, roughly, that if the group satisfies certain conditions then the structure of the lat-tice has a surprising amount of influence on the structure of the group. For this and related work, Grigory Margulis won the Fields Medal in 1978. I’ll try to present some of these ideas in a way un-derstandable to grad students of all backgrounds

Equivariant cohomology, momentum graphs and Chinese remainder theorem

Abstract: I will start with defining the notion of equivariant cohomology for a group action on a topological space. It is a ring that encodes information both about the topology of the space as well as the action of the group. Often equivariant cohomology is easier to compute and one can recover the usual cohomology of a space from its equivariant cohomology.