Time: Thursday October 13, 2.30-3.30, Carney 309

Speaker: Izzet Coskun (University of Illinois, Chicago)

Title:
Pictures and
homogeneous spaces

Abstract:

Many important problems in representation theory have analogues in geometry. For example, decomposing tensor products of representations of GL(n) into irreducible representations is very closely tied to the geometry of the Grassmannian. Similarly, studying the restriction of a representation of GL(n) to subgroups such as SO(n) or SP(n) has geometric analogues in terms of the geometry of flag varieties. In this talk, I will show you how drawing a few pictures can make studying such lofty problems a lot of fun. I will specifically concentrate on Littlewood-Richardson rules and geometric branching rules. I intend to make the talk accessible to anyone who is willing to be seduced by pictures.

Abstract:

Many important problems in representation theory have analogues in geometry. For example, decomposing tensor products of representations of GL(n) into irreducible representations is very closely tied to the geometry of the Grassmannian. Similarly, studying the restriction of a representation of GL(n) to subgroups such as SO(n) or SP(n) has geometric analogues in terms of the geometry of flag varieties. In this talk, I will show you how drawing a few pictures can make studying such lofty problems a lot of fun. I will specifically concentrate on Littlewood-Richardson rules and geometric branching rules. I intend to make the talk accessible to anyone who is willing to be seduced by pictures.

Time: Thursday
October
27,
4.00-5.00,
Fulton
Hall
230

Speaker: Richard Askey (University of Wisconsin)

Speaker: Richard Askey (University of Wisconsin)

Title:
The binomial
theorem, beta and gamma functions, and some extensions of each.

Abstract:

It is well known that the number of permutations of the set 1,2,...,n
is n!. An extension of this where one counts inversions was posed as a
problem by M. Stern in 1839. These will be the starting place to build
up the binomial theorem, the extension of n! which we now write as the
gamma function, the beta integral of John Wallis, Euler's integral
representation of the gamma function as an integral, and the connection
between these three things. This connection will be looked at in two
different settings, the classical one which most of you know reasonable
well, and what will be called q-extensions of these classical results
into a world which has finally started to come into its own.Abstract:

Time: Wednesday,
November
9,
from
4-5pm,
Carney
309

Abstract:

After reviewing results about the existence of tight contact structures on closed 3-manifolds, we show how to use Heegaard Floer theory (in particular, the contact Ozsvath-Szabo invariant) to verify tightness of certain contact structures on 3-maniolds given by surgery along specific knots in S^3.

Time: Thursday November 17, 2.30-3.30, Carney 309

Speaker: Joseph Harris (Harvard University)

Speaker:
András
Stipsicz (Rényi Institute of Mathematics)

Title:
3-dimensional contact topologyAbstract:

After reviewing results about the existence of tight contact structures on closed 3-manifolds, we show how to use Heegaard Floer theory (in particular, the contact Ozsvath-Szabo invariant) to verify tightness of certain contact structures on 3-maniolds given by surgery along specific knots in S^3.

Time: Thursday November 17, 2.30-3.30, Carney 309

Speaker: Joseph Harris (Harvard University)

Title:
The
Interpoloation
Problem

Abstract:

Time: Wednesday December 7, 4.15-5.15, Carney 309

Abstract: In his article "3-Dimensional manifolds, Kleinian groups, and hyperbolic geometry", William Thurston posed 24 problems related to the topology and geometry of Kleinian groups and hyperbolic 3-manifolds. We'll discuss four of the remaining open problems from this list, 15-18, having to do principally with finite-sheeted covers of hyperbolic 3-manifolds. We'll discuss how recent work of Kahn-Markovic and Wise implies that these problems are essentially equivalent, and the prospects for answering these questions combining their results.

Abstract:

Time: Wednesday December 7, 4.15-5.15, Carney 309

Speaker:
Ian Agol (University
of California, Berkeley)

Title: Virtual properties of
3-manifoldsAbstract: In his article "3-Dimensional manifolds, Kleinian groups, and hyperbolic geometry", William Thurston posed 24 problems related to the topology and geometry of Kleinian groups and hyperbolic 3-manifolds. We'll discuss four of the remaining open problems from this list, 15-18, having to do principally with finite-sheeted covers of hyperbolic 3-manifolds. We'll discuss how recent work of Kahn-Markovic and Wise implies that these problems are essentially equivalent, and the prospects for answering these questions combining their results.