Deformation quantization, Lie groupoids and index theorems

Lecturer: dr. Hessel Posthuma

Abstract: In these lectures I will give an introduction to the theory of Lie groupoids and Lie algebroids, and explain their relevance in various areas of geometry, topology and mathematical physics. In particular, I will focus on the connections to Poisson geometry, one of which is given by the Lie-Poisson structure on the dual of a Lie algebroid. Deformation quantization of this Poisson structure, and the associated algebraic index theorems, lead to general index theorems in the framework of Lie groupoids and Lie algebroids. Finally I will explain how these index theorems generalize various well-known special cases, such as the classical Atiyah-Singer index theorem.