Talk

In set theory, the Gelfand spectrum of a C*-algebra is only defined in the commutative case. Topos theory overcomes this restriction, especially through the oncept of a locale as the constructive analogue of a topological space. This approach unravels the logical structure of noncommutative spaces, which turns out to be intuitionistic.

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.

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

Abstract:

Harmonic functions are real-analytic and so automatically extend from being functions of real variables to being functions of complex variables. But how far do they extend? This question may be answered by twistor theory, the Penrose transform, and associated geometry. I shall base the constructions on a formula of Bateman from 1904. This is joint work with Feng Xu.

About the speaker:

I will present an outline of the Carey-Phillips-Rennie-Sukochev approach to the (Odd) Local Index Theorem. Some recent simplifications will be mentioned; as well as numerous amusing stories about Alan Carey.

Based on joint work with Elena Pierpaoli and Kevin Teh, I will show how the nonperturbative form of the spectral action can be used to build cosmological model that exhibit a coupling between topology and possible inflation scenarios.

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

Dualities have not only greatly increased our theoretical understanding of String Theory and Quantum Field Theory, but in recent years have found important applications in diverse areas as well, such as integrable systems, condensed matter physics, heavy ion collisions, hydrodynamics and particle physics. In this talk I will give an overview of the various dualities  (T-duality, S-duality, Mirror symmetry, AdS/CFT, Geometric Langlands, ... ), in particular their topological aspects, and our contributions to this research area.

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.