Amnon Yekutieli (Ben-Gurion University)

Algebraic Aspects of Deformation Quantization

Abstract
: Deformation quantization arose in mathematical physics, as a model for the transition from classical to quantum mechanics. Mathematically the question is this: given a commutative Poisson algebra C, can one construct a noncommutative associative algebra A, depending on a parameter h, such that C=A/(h), and the Poisson bracket on C can be recovered from the commutator of A.

The original problem considered the ring C of smooth functions on a differentiable manifold X. The complete solution (by Kontsevich in 1997) was amazing, due to its use of ideas from string theory and the combinatorics of graphs.

In this talk I will explain some of the background and Kontsevich's work. Then I will move to the algebraic aspects: instead of differentiable manifolds, the geometric objects considered will be algebraic varieties. I will explain my own work in this area, state the main results, and list some of the techniques used. Finally I will discuss recent developments, and links with other problems in algebraic geometry and mathematical physics.