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Thematic Program on Quantization, Spring 2011

Poster

Quantization is an important topic in mathematics and physics. From the physics point of view, quantization seeks to associate a quantum analogue to each classical mechanical system. This process is what one uses to determine the correct quantum description of a physical problem. For example, the quantum model of the hydrogen atom is obtained by quantizing the system of a particle moving in a $1/r$ potential. In mathematics, quantization includes Geometric Quantization, which seeks as natural as possible a procedure for associating a Hilbert space to each symplectic manifold satisfying an integrality condition. The mathematical theory also includes Deformation Quantization, which seeks to deform the commutative algebra of functions on a Poisson manifold into a noncommutative algebra in which the leading-order noncommutativity is given by the Poisson bracket on the functions. Further, one may consider quantum analogues of various classical objects. One often finds--for example in representation theory and combinatorics--that the quantum analogue facilitates proof of a conjecture about the classical object. The summer schools and conference will be concerned with these topics, among others.

Quantization has been quite important in mathematics. This can be seen through the development of Geometric Quantization by Kirillov, Kostant, and Souriau in the 1960's, and the proof of various versions of the "quantization commutes with reduction conjecture," beginning with the ground-breaking work of Guillemin and Sternberg in the 1980's and continued by Meinrenken and others in the 1990's, and recent study of the $L^2$-analogue. Much work has been done on the semiclassical asymptotics of geometric quantization, such as the proof by Bordemann, Meinrenken, and Schlichenmaier of a general asymptotic formula for Berezin--Toeplitz quantization on compact Kahler manifolds. Semiclassical analysis has resulted in applications in number theory, as in the work of Borthwick and Uribe on relative Poincare series. Kontsevich's proof of his formality conjecture showed that every Poisson manifold has a star product, and this was one of the main results that earned him the Fields Medal. This line of reasoning led to the development of infinity algebras in mathematical physics, which has recently been used by Costello to give a rigorous geometric construction of the elliptic genus. Deformation quantization is also used by Etingof and Ginzburg to give a better geometric understanding of the rational Cherednik algebra, and more generally, to give a method for geometric understanding of representations of associative algebras. Quantum groups and Hecke algebra give examples of quantum analogues of classical objects, and an important direction in mathematics since 1990 has been to construct topological invariants using quantum groups. Quantum cohomology provides a further example of this phenomenon with deep geometric and physical roots.

We will produce a volume with proceedings from the different thematic programs.