Alternatively have a look at the program.

## Moduli varieties of real and quaternionic bundles over a curve

We examine the moduli problem for real and quaternionic vector bundles over a curve, and we give a gauge-theoretic construction of would-be moduli varieties for such bundles. These moduli varieties are irreducible subsets of real points inside a complex projective variety. We relate our point of view to previous work by Biswas, Huisman and Hurtubise, and we use this to study Gal(C/R)-actions on moduli varieties of semistable holomorphic bundles over a complex curve with a given real structure.

## Brieskorn varietes and fake lens spaces

Brieskorn varietes admit $S^1$-actions whose isotropy groups are finite. Choose a Brieskorn variety diffeomorphic to a sphere and an odd prime p which is relatively prime to the order of all isotropy groups. Then the orbit space of the induced $Z/p$-action is a homotopy lens space -- a closed manifold homotopy equivalent to a lens space. A homotopy lens space is fake if it is not homeomorphic to a classical lens space. Using Reidemeister torsion, we show some of these homotopy lens spaces are fake lens spaces.

## Poincaré duality complexes

Poincaré duality complexes are homotopy generalisations of manifolds. This talk will provide an introduction and an overview of results and open questions concerning Poincaré duality complexes in low dimensions.

## Euler Characteristics of Categories and Homotopy Colimits

In this talk I will present a topological approach to Euler characteristics of categories in terms of finiteness obstructions. This approach is compatible with almost anything one would want, for example products, coproducts, covering maps, isofibrations, and homotopy colimits. Classical constructions are special cases, for example, under appropriate hypotheses the functorial $L^2$-Euler characteristic of the proper orbit category for a group $G$ is the equivariant Euler characteristic of the classifying space for proper $G$-actions.

## Lusternik-Schnirelmann Theory; Old and New

The Lusternik-Schnirelmann theory has been originated in 1930th, and last decade it meets a renaissance. The contemporary level of the theory relates topology, analysis, geometry, complexity theory and other interesting subjects. In the talk I want to recall the main points of the development of the theory, as well as to show recent results and applications.

## Grothendieck's Problem for 3-Manifold Groups

The following problem was posed by Grothendieck: Let $G$ be a finitely presented residually finite group and $u : H \to G$ the inclusion homomorphism of a finitely presented subgroup. Suppose that the extension $\hat u$ to the profinite completion is an isomorphism. Is $u$ an isomorphism? This talk will discuss this problem in the context of the fundamental groups of compact 3-manifolds.

## On Gromov's macroscopic dimension

Gromov introduced the notion of macroscopic dimension to study closed manifolds with positive scalar curvature (PSC). In the talk we will discuss two his conjectures on the subject: I. If a closed $n$-manifold $M$ admits a PSC metric, then the macroscopic dimension of its universal cover is less than $n-1$. II. If the universal cover of a closed $n$-manifold $M$ has the macroscopic dimension less than $n$, then the image of the fundamental class under a map classifying the universal cover is trivial, $f_*([M])=0$, in the rational homology of the classifying space $H_*(B\pi)$.

## Chiral differential operators and topology

In this talk, I will first describe a reformulation of a construction by Gorbounov, Malikov and Schechtman in terms of global geometric data. This will then be applied to define vertex algebraic versions of Dolbeault complexes and obtain a geometric description of the Witten and elliptic genera of complex manifolds.

## M-theory and generalized cohomology

Anomalies and partition functions in physics have played a major role in uncovering the mathematics behind physical theories, such as quantum field theory. We will consider the case of string theory and M-theory and show how their anomalies and partition functions point to generalized cohomology theories such as Morava K-theory and variants of elliptic cohomology. Viewed as obstructions, anomalies also give rise to structures such as string structures and even lead to generalizations.

## On homomorphisms of groups into the Mapping class groups

This is a joint work with J. Behrstock and C. Drutu. We show that if a group G has infinitely many pairwise non-conjugate homomorphisms into a mapping class group of a surface, then G has a finite index subgroup (the index depends on the surface only) that acts non-trivially on a real tree. If G is finitely presented, then the real tree can be replaced by a simplicial tree.

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