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.