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Thursday, October 11, 2007

Kobi Kremnizer, MIT

**Quantum localization**

In this talk I will describe the quantum analog of the Beilinson-Bernstein localization theorem. The main object is the quantum flag variety. I will define it and D-modules over it. At roots of unity this non-commutative object localizes to the Springer resolution. I will explain how one can use this to prove the DeConcini-Kac-Procesi conjecture and Lusztig's conjectures. Then I will describe how this relates to derived loop spaces that appear in the work of Ben-Zvi and Nadler.

Thursday, October 18, 2007

Ciprian Manolescu, Columbia

**Symplectic Instanton Homology**

Thursday, October 18, 2007

Pramod Achar, LSU

**The category of staggered sheaves**

Let $G$ be an algebraic group, and let $X$ be a variety with a $G$-action. The category of ($G$-equivariant) perverse coherent sheaves on $X$ has shown up in various applications in geometric representation theory. However, this category has good properties only under strong conditions on the $G$-action, conditions that fail on many common spaces, including the flag variety and the Steinberg variety of a reductive group.

I will describe the construction of a $t$-structure on the derived category of equivariant coherent sheaves on $X$ that resembles the perverse coherent $t$-structure, but incorporates additional information from the $G$-action. The resulting abelian category, known as the category of "staggered sheaves," has many desirable properties - e.g., every object has finite length, and simple objects are given by an "IC" functor - under weaker conditions than what the perverse coherent category requires. I will also discuss some small examples, and speculate on potential uses of this category in representation theory.

Friday, October 19, 2007

Lev Rozansky, UNC at Chapel Hill

**Virtual links and algebraic categorification of quantum polynomial invariants**

Quantum polynomial invariants of links in $S^3$, such as the Jones, HOMFLY and Kauffman polynomials, were invented about 20-25 years ago, but their topological nature remains largely mysterious. Their rigorous definition is combinatorial, while their conceptual definition in terms of Chern-Simons-Witten path integral is mathematically non-rigorous.

Categorification program, whose pratcital implementation was pioneered by M. Khovanov, suggests to interpret the quantum polynomial invariants as graded Euler characteristics of special chain complexes of graded modules (or vector spaces), associated to links up to homotopy. For the Jones, HOMFLY and Kauffman polynomials, these complexes can be constructed combinatorially with the help of commutative algebra tools.

Virtual links were invented by L. Kauffman (and independently by M. Polyak and O. Viro) and they represent links in ‘thick’ surfaces. Some quantum polynomial invariants can be extended to virtual links, but the virtual crossings play a minor role in these extensions.

It turns out that virtual crossings play a central role in commutative algebra categorification of quantum polynomials. The ordinary crossings appear to be a certain ‘homological’ deformation of the virtual ones. I will explain this construction at the example of the categorification of the 2-variable HOMFLY polynomial.

The talk is based on my joint work with M. Khovanov (math.QA/0505056, math.QA/0701333).

Thursday, October 25, 2007

Boris Tsygan, Northwestern University

**Oscillatory Modules**

It was noticed long time ago that the Fukaya category of a symplectic manifold has some resemblance to the category of modules over a deformed algebra of functions on the manifold. In this talk we will define a category of modules with additional structure for which the analogy with the Fukaya category is somewhat closer. These modules, that we call oscilatory modules, are motivated both by the basics of noncommutative geometry and by the Hormander-Maslov theory of Lagrangian distributions. We will discuss the example of the two-torus.

Thursday, November 1, 2007

Alexei Oblomkov, Princeton

**Gromov-Witten/Donaldson-Thomas correspondence for toric threefolds**

There are two ways to obtain a curve $C$ in some ambient space $Y$. We can give the ideal $I$ of defining equation or we can choose an ‘`etalon" curve $C$ and define a map $f: C\to Y$. In the case when $dim Y=3$ the spaces of $f$’s and $I$'s (with fixed topological invariants) admit natural compatification.

For a collection of curves $K_1,\dots, K_m\subset Y$ and fixed topological invariants of $C$ we can ask the question: how many there are curves $C\subset Y$ such that $C$ meets all of $K_i$. The answer, depends on our choice of the compactification of the space of curves. If we use the first methods to describe the curves $C$ then the answer is called Donaldson-Thomas invariant, if we use the second method the answer is Gromov-Witten invariant.

Even though the DT and GW invariants are very far from being equal, Nekrasov, Maulik, Okounkov and Panharipande (MNOP) conjectured the correspondence between them. This correspondence is a particular case gauge/string theory duality. The conjecture is proved for the large class of threefolds (for toric threefolds) by Maulik, Okounkov, O. and Pandharipande.

The aim of the talk is to discuss the definition of DT, GW invariant and to sketch the proof in the toric case.

Thursday, November 8, 2007

David Treumann, Northwestern University

**Nearby cycles over general bases and stratified homotopy type**

Let $S$ be either a Riemann surface or a scheme of dimension one. To a morphism $f:X \to S$, a construction of Milnor and Grothendieck associates a sheaf on each fiber $f^{-1}(s)$ of $f$, called the sheaf of nearby cycles. If $S$ is replaced by a space or scheme of higher dimension (a general base" in the title), this construction sometimes goes wrong. Deligne proposed a framework -- the vanishing topos" of $f$ — for analyzing this behavior. His proposal has been carried out in the complex analytic setting by Le and Sabbah, and much more recently in the etale setting by Orgogozo. There does not yet exist a comparison theorem between the analytic and etale constructions of nearby cycles over general bases. In the talk I will survey the theory and discuss a work in progress toward obtaining such a comparison result.

Thursday, November 15, 2007

Jake Rasmussen, Princeton

**Odd Khovanov homology**

Khovanov homology is an invariant which assigns to a link in $S^3$ a family of bigraded homology groups whose graded Euler characteristic is the Jones polynomial. In this talk, I'll describe a variant of Khovanov's construction (joint work with Peter Ozsvath and Zoltan Szabo), which replaces the symmetric algebra appearing in Khovanov's definition with an exterior algebra. The resulting homology groups agree with Khovanov's over Z/2, but differ over Q.

Thursday, November 29, 2007

Sergiy Koshkin, Northwestern University

**Holomorphic fusion rules in Chern-Simons theory**

I will describe a class of complex threefold singularities that produce large N dual pairs. Each pair consists of a 3-manifold and a Calabi-Yau threefold with conjecturally related Chern-Simons and Gromov-Witten invariants respectively. The 3-manifolds that occur are of a special kind (graph manifolds), and their Chern-Simons invariants can be computed using only fusion rules, a part of the theory that does not involve Drinfeld's R-matrix. This is fortuitous because as I will explain, the fusion rules can be modified to produce holomorphic functions that reduce to the conventional invariants under a specialization of variables. These functions are expected to generate the Gromov-Witten invariants of the dual Calabi-Yau threefolds.

Thursday, December 6, 2007

Johannes Walcher, IAS

**Progress on Mirror Symmetry for Open Strings**

The classical mirror theorems relate the Gromov-Witten theory of a Calabi-Yau manifold at genus 0 to the variation of Hodge structure of an associated mirror manifold. I will review recent progress in extending these closed string results to the open string sector. Specifically, the open Gromov-Witten theory of a particular Lagrangian submanifold of the quintic threefold is related to the Abel-Jacobi map for a particular object in the derived category of coherent sheaves of the mirror quintic. I will explain the relevance for the homological mirror symmetry program, as well as the extension to higher genus Riemann surfaces.