Alternatively have a look at the program.

## Bispaces and bibundles

The theory of non-abelian gerbes or bibundle gerbes requires the notion of a bibundle. This in turn requires the notion of a bispace which is a set which has commuting left and right G transitive G actions. We consider the structure of G bibundles and their classifying theory. In particular we give examples and also explain why examples are hard to find. This is joint work with David Roberts and Danny Stevenson.

## Families of Dirac operators and affine quantum groups

Families of Dirac type operators constructed from the supersymmetric Wess-Zumino-Witten model are a useful tool in Fredholm operator realization of twisted K-theory classes on compact Lie groups. They transform in a covariant manner with respect to the action of a central extension of a loop group, the level of the representation giving directly the Dixmier-Douady class of the twisting gerbe. I want to describe a deformation of this system in the language of quantum affine algebras.

## "Looped" principal connections and applications to loop space geometry

This talk reports on joint work with Mauro Spera. Given a principal bundle with connection one can define the induced "looped connection" on the looped principal bundle. We explain this procedure and apply it to the construction of Spin-c connections on the loop space of a finite-dimensional string manifold, and to show that the loop space of an arbitrary finite-dimensional manifold possesses a good cover in the sense of Leray.

## From infinite-dimensional Teichmueller theory to conformal field theory and back

The mathematical definition (in the original sense of G. Segal) and construction of Conformal Field Theory (CFT) requires deep developments in algebra, analysis and geometry. The algebraic aspects involving vertex operator algebras have been well developed over the past twenty-five years, and the construction of CFT is nearing completion. However, many problems in analysis and geometry must be urgently addressed. These problems involve the infinite-dimensional moduli and Teichmuller spaces of Riemann surfaces with parametrized boundaries.

## Generalized Witten genus and vanishing theorems

We report our joint work with Qingtao Chen and Fei Han, where we construct a mod 2 analogue of the Witten genus for $8k+2$ dimensional spin manifolds, as well as modular characteristic numbers for a class of spin^c manifolds which we call string^c manifolds. When these spin^c manifolds are actually spin, one recovers the original Witten genus on string manifolds. These genera vanish on string and string^c complete intersections respectively in complex projective spaces.

## Lie conformal algebra complex and integrable hierarchies

Lie conformal algebras encode the singular operator product expansion of chiral

fields in conformal field theory. As for Lie algebras, there exists a cohomology

theory which parametrizes first order deformations and abelian extensions.

A recent application of Lie conformal algebras is to the classification and

construction of integrable hierarchies. An important component in this theory is

the variational complex, defined as a reduction of the de Rham complex on phase

space.

## Cantor systems and number fields

We discuss C*-algebras associated with dynamical systems on a Cantor space. Of particular interest are systems arising from global fields.

## Twisted spectral triples and conformal noncommutative geometry

## Transgression of Bundle Gerbes to Loop Spaces and its Inverse

Brylinski and McLaughlin have introduced a "transgression" functor that takes abelian gerbes with connection over a smooth manifold M to certain principal bundles over the free loop space LM of M. In my talk I will present a characterization of the image of this functor. Then I describe an inverse functor, called "regression", so that an equivalence of geometric categories over M and LM is obtained. It will become clear how the concept of a bundle gerbe fits nicely into this context.

## Twisted longitudinal index theorem for foliations

I report on the joint work with Paulo Carrillo-Rouse. For a foliation with a twisting on its leaf space, we establish the equivalence between the twisted topological index and the twisted analytic index, both taking values in the K-theory of the twisted C*-algebra of the honolomy groupoid. We also develop a notion of geometric cycles and the geometric K-homology for a foliation with a twisting. As an application of our twisted longitudinal index theorem, we

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