NSCPHD1128 Deformation Quantization and Algebraic Index Theorems

Deformation quantization has its origin in the problems related to quantum mechanics, to be more precise the question of constructing a quantum mechanical system associated to a given symplectic structure and Hamiltonian flow on the phase space. While there is no unambiguous prescription for the analytic constructions (sometimes called geometric quantization), the formal deformation quantization has by now become a well understood theory. It has its applications in both analysis, being directly related to semi-classical analysis of both elliptic and hyperbolic differential equations, and in Poisson geometry which underlines the theory. Methods from deformation quantization lead to the algebraic index theorems, which, in particular, give various generalizations of the classical Index theorem of Atiyah and Singer in the context of smooth structures and to proofs of the Riemann-Roch type theorems for D-modules on complex analytic manifolds.

The present research activity in the subject centers around the questions related to quantization of stratified Poisson manifolds and actions of (quantum) groups, where most of the natural questions related to deformation quantization like: existence, classification, index theorems and computations related to algebraic K-theory are open.

The goal of the masterclass is to present the students with the basics of modern methods in deformation quantization, as

• Formality theorems and their applications to deformation quantization;

• Computations involving Fedosov quantization;

• Examples of algebraic index theorems;

• Quantization of symplectic manifolds with Hamiltonian action of a compact Lie group;

• Deformation quantization of manifolds with singularities like orbifolds and stratified symplectic manifolds.