Alternatively have a look at the program.

## Nested operads

I will present nested operads, which generalize operads of various kinds (May, cyclic, modular, PROPs, Batanin's theory of globular operads, etc.), by allowing operations to be organized in hierarchies. This provides, for example, a unifying framework for different models of higher categorical structures, or higher dimensional fundamental groupoids. I will describe connections to existing theories of higher dimensional operads.

## The Anick conjecture on the minimal Hilbert series of quadratic algebras

We present several results on the Anick conjecture which asserts that the lower bound for the Hilbert series, known as the Golod-Shafarevich estimate is attained on generic quadratic algebra. The technique (due to Anick), allowing to write down precisely the formula for the Hilbert series will be demonstrated. We will discuss also related questions of Koszulity and being noncommutative complete intersection (NCCI).

## Variational Lie algebroids and homological evolutionary vector fields

We define Lie algebroids over infinite jet spaces for vector bundles over smooth manifolds. We also establish their equivalent representation through homological evolutionary vector fields on certain infinite jet super-bundles. Our new definition is a nontrivial generalization of the classical one for usual manifolds, which is no longer applicable, and more fully grasps the geometry of strings in space-time. (The talk is based on the joint work in progress with J.van de Leur:math.DG/1006.4227v2.)

## Extended Picard complexes and homogeneous spaces

Let k be an algebraically closed field of characteristic 0. Let G be a connected linear k-group acting on a smooth irreducible k-variety X. In works of Popov, Kraft, Knop, Vust, and others the equivariant Picard group Pic_G(X) of G-line bundles on X was investigated. We introduce the extended equivariant Picard complex UPic_G(X) in degrees 0 and 1, whose first cohomology is Pic_G(X). In other words, we compute Pic_G(X) in terms of divisors and rational functions on X and X\times G.

## An abstract Torelli theorem

We consider the abstract pair 'group J with a distinguished subset C' where J is the K-points of a Jacobian of a smooth curve C, K algebraically closed. We prove that this pair determines the field, the curve and the Jacobian uniquely up to an isomorphism of fields and a bijective morphism of the varieties, which is a bijective isogeny on Jacobians. In characteristic 0 the bijective morphism is an isomorphism. Over finite fields the theorem proves a conjecture from the recent paper by Bogomolov, Korotaev and Tschinkel.

## Rationality of instanton moduli

We consider the moduli space $I_n$ of rank-2 mathematical instanton vector bundles with second Chern class $n\ge1$ on the projective space $P^3$. The irreducibility, respectively, the rationality of $I_n$ was known for $n\le5$, respectively, for $n=1,2,3$ and 5. It was recently proved by A.Tikhomirov that $I_n$ is irreducible for all odd values of $n$. Now the question of rationality of $I_n$ for arbitrary $n$ is in order. In this talk we discuss the recent result of D.Markushevich and A.Tikhomirov answering the question of irreducibility of $I_n$.

## Nahm's equations and modular forms

We consider certain $q$-series depending on parameters $(A,B,C)$, where $A$ is a positive definite $r \times r$ matrix, $B$ is an $r$-vector and $C$ is a scalar, and ask when these $q$-series are modular forms. Werner Nahm has formulated a partial answer to this question: he conjectured a criterion for which $A$'s can occur, in terms of torsion in the Bloch group. The conjecture was proved by Don Zagier for $r=1$. Recently me an Sander Zwegers found several counterexamples for $r\ge 2$, so the correct formulation of Nahm's conjecture became an interesting open question.

## Hall algebras of weighted projective lines and quantum algebras

My talk is based on two joint papers with Olivier Schiffmann arXiv:0903.4828 and arXiv:1003.4412. The goal of our project is to apply the structure results on the derived category of coherent sheaves on a weighted projective line to a study of quantized enveloping algebras. In particular, we obtain a new realization of quantum affine algebras of simply-laced types, based on the orbifold quotients of P^1.

## Koszul categories of $sl(\infty)$-, $o(\infty)$-, $sp(\infty)$-modules

Various categories of integrable modules over the classical infinite- dimensional Lie algebras $sl(\infty), o(\infty), sp(\infty)$ have been studied in recent years. In this talk we construct a category based on the mixed tensor algebra (the tensor algebra of the natural and conatural modules). The indecomposable injectives in the category turn out to be simply indecomposable direct summands in the tensor algebra. In addition, we show that this category is antiequivalent to the category of locally unitary finite-dimensional modules over a Koszul algebra.

## Poisson traces, D-modules, and symplectic resolutions

The space of Poisson traces on a Poisson algebra is the dual to the zeroth Poisson (or Lie) homology, i.e., the functionals which annihilate Poisson brackets. I will recall a systematic way of understanding these in terms of a canonical D-module on the spectrum of the Poisson algebra, whose space of solutions is the space of Poisson traces. When the variety has finitely many symplectic leaves, this D-module is holonomic, and as a corollary, the space of Poisson traces is finite-dimensional.

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