I will continue my talk from last week. I will quickly review material from last week and then discuss examples of equivariant cohomology of smooth toric vatieties and flag varieties as well as the localization theorem. Finally, will discuss an application of the Chinese remainder theorem in this setting.
Abstract:
We will discuss applications of the Atiyah, Guillemin-Sternberg theorem, such as classification of toric manifolds, the Schur-Horn theorem, and a brief introduction about viewing Bernstein–Kushnirenko theorem via symplectic geometry.
In this talk I will briefly introduce Cartan decompositions of semisimple Lie groups/algebras with some applications to quantum systems. I will also discuss how Riemannian and sub-Riemannian geometry can be used to study the cut locus of control systems arising from these decompo- sitions, after passing to a singular quotient space.
Abstract:
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.
TBA
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.
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