Abstracts for Geometry and Quantum Field Theory

Suppose $G$ is a compact Lie group, $LG$ its (free) loop group and $\Omega G$ its based loop group. Let $P \to M$ be a principal bundle with structure group one of these loop groups. In general, differential form representatives of characteristic classes for principal bundles can be easily obtained using the Chern-Weil homomorphism, however for infinite-dimensional bundles such as $P$ this runs into analytical problems and classes are more difficult to construct.

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.

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.

From known results on classical pseudodifferential operators, we derive a new approach of the problem of existence and uniqueness of traces on the algebra of log-polyhomogeneous pseudodifferential operators. In particular, we show that if the manifold is odd-dimensionnal, the canonical trace is the unique trace on odd-class log-polyhomogeneous operators.