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Fall 2008 | Winter 2009 | Spring 2009

Tuesday, April 7, 2009

Valerio Toledano Laredo, Northeastern University

**Hall Algebras (pre-talk)**

I will review the construction of the Hall algebra H(A) of an abelian category. In examples coming from quivers, H(A) is a quantum group, where the role of the deformation parameter is played by the cardinality q of the field over which A is defined.

I will also explain how one may set q=1 and obtain the enveloping algebras of semisimple Lie algebras.

**Stability conditions and Stokes factors**

The space Stab(A) of all such conditions has the structure of a complex manifold and has attracted a lot of attention recently. D. Joyce defined invariants which count, in a suitable sense, semistable objects in A and showed how to organise these in a holomorphic generating function on Stab(A) which satisfies an intriguing non-linear PDE.

I will explain how the transformation rules of Joyce's invariants and his PDE can be understood as Stokes phenomena. The latter describe the monodromy of solutions of ODE's on the complex plane around irregular singularities, that is poles of order greater or equal to 2.

Tuesday, April 14, 2009

Ezra Getzler

**Topological field theory in two dimensions and Teichmuller space**

I'll explain my recent work, in both the closed and open/closed cases, on filtering the modular operad which governs topological field theory in two dimensions. Most of the talk will be spent reclling the definition of this operad, and defining the filtration.

Tuesday, April 21, 2009

Tony Pantev, University of Pennsylvania

Geometric Langlands and non-abelian Hodge theory (and pre-talk)

Tuesday, May 12, 2009

Susan Tolman, University of Illinois

**Hamiltonian circle actions with minimal fixed sets (and pre-talk)**

The purpose of this talk is to show that there are very few "extremely simple" symplectic manifolds with Hamiltonian actions. More precisely, consider a Hamiltonian circle action on a compact symplectic manifold (M,ω). It is easy to check that the sum of dim(F) + 2 over all fixed components F is greater than or equal to dim(M) + 2. We show that, in certain cases, equality implies that the manifold "looks like" one of a handful of standard examples. This can be viewed as a symplectic analog of the Petrie conjecture.

Monday, June 1, 2009

Young Heon Kim, UBC and IAS

**Optimal Transportation (pre-talk)**

Optimal transportation theory was first initiated by Monge in 19th century and it studies the phenomena arising when mass is allocated in a cheapest way. We will introduce some basic concepts and results in this rapidly growing research area that sees a lot of connections and applications, for example, in geometric inequalities, partial differential equations and Ricci curvature.

**Geometry and regularity of optimal transportation**

In optimal transport theory, one wants to understand the phenomena arising when mass is transported in a cheapest way. This variational problem is governed by the structure of the transportation cost function defined on the product of the source and target domains. For regularity of optimal transportation maps, a crucial condition for the cost was found recently by Ma, Trudinger and Wang, which later was confirmed to be necessary by Loeper. This so-called MTW condition turned out to be a curvature condition of a certain pseudo-metric defined by the cost function as observed by McCann and myself. I will explain certain results in this direction.