Orateur
M.
Rudy Rodsphon
Description
One way to describe succinctly local index theory on closed spin manifolds could be the following slogan of Quillen : Dirac operators are a "quantization" of connections, and index theory is a "quantization" of the Chern character. As is well-known, this leads to proofs of the index theorem using heat kernel methods. For non necessarily spin manifolds, pseudodifferential operators and their symbolic calculus play a crucial role in the original proofs of the index theorem. However, symbols may also be viewed as a deformation quantization of functions on the cotangent bundle, which has led to other fruitful approaches to index theory through a "quantization" process. Methods used in these two different quantization pictures do not seem to be quite related a priori. Based on ideas of Perrot, the upshot of the talk will be that it is possible to implement an algebraic version of the heat kernel method in the deformation quantization picture, which has many avantages over the original one. In particular, we recover Nest-Tsygan's algebraic index theorem for a certain class of symplectic manifolds in a very natural way.
