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Graduate and Postdoc Summer School at the Center for Mathematics at Notre Dame

May 31 - June 4, 2011

This summer school is designed for graduate students and postdoctoral students.  We anticipate that each lecturer will give 3 or 4 lectures on a topic of current mathematical interest.

Featured speakers will be:

Yuri Berest (Cornell University)
"Rational Cherednik Algebras, Calogero-Moser Spaces and Applications"
In these lectures, I will give a brief introduction to representation theory of rational Cherednik algebras associated to a finite (complex) reflection group and discuss some applications in algebra, geometry and mathematical physics.

Vasily Dolgushev (Temple)
"Graph Complexes, GRT, and Willwacher's Construction"
In preprint arXiv:1009.1654, Thomas Willwacher established very interesting links between three objects which play an important role in deformation quantization. These objects are Kontsevich's graph complex, the Grothendieck-Teichmueller Lie algebra, and the deformation complex of the operad GER governing homotopy Gerstenhaber algebras. In my lectures, I will describe these links. A large part of my lectures will be devoted to various prerequisites. I will recall (co)operads as well as (co)bar construction for (co)operads. I will talk about convolution Lie algebra, about deformation complex of an operad, and about twisting procedure for operads. I will also talk about various ``incarnations'' of the operad GER. Then I will introduce graph operads and a few versions of graph complexes. I will show how Kontsevich's graph complex allows us to compute cohomology of the deformation complex for the operad GER. We will also prove that the zeroth cohomology of Kontsevich's graph complex is isomorphic to the Grothendieck-Teichmueller Lie algebra. Part of the results, I will be talking about, were also obtained by Benoit Fresse.

Rinat Kedem (UIUC)
"Cluster algebras, integrable models and total positivity"
The basic definitions, theorems and conjectures about cluster algebras of geometric type; A few examples of finite and infinite types; Applications to integrable spin chains and the character theory of Lie algebras and quantum affine algebras; Methods of solution using statistical mechanics and combinatorics; Relation to total positivity and the Toda system; Results on the quantization of these systems.

Eugene Leman (Univ. of Illinois at Urbana-Champaign)
"Introduction to geometric quantization, old and new"
We present an overview of geometric quantization on manifolds. We describe what's involved in extending it to orbifolds.

Eckhard Meinrenken (Toronto)
"Group-valued moment maps and Verlinde formulas"
In a 1998 paper with Alekseev and Malkin, we introduced the notion of a quasi-Hamiltonian G-space, with moment map taking values in G. (For ordinary Hamiltonian spaces, the moment map takes values in the dual of the Lie algebra.) Recently, a theory of quantization of such moment maps was developed. In these lectures, I will give a description of this quantization procedure, with applications to the quantization of moduli spaces of flat G-bundles.