Non commutative geometry, microlocal analysis, and symplectic geometry

Northwestern University, Evanston IL

Investigators

Abstract

The aim of this project is to extend the study of non-commutative differential geometry of deformation quantization algebras, and to apply it to symplectic geometry. A deformation quantization is a new multiplication law on an algebra of function on a manifold which depends on a formal parameter. When the value of the parameter is zero, then this product becomes the usual product of functions. All such deformation quantizations were classified by Kontsevich. The simplest examples of them arise from algebras of differential operators on manifolds. In our previous work, we computed for all deformed algebras basic invariants of non-commutative differential geometry (Hochschild and cyclic homology, etc.). We applied these results to prove generalized Atiyah-Singer index theorems. Our main tool was what we call non-commutative differential calculus, which is an extension of classical algebraic constructions with forms and multi-vectors to non-commutative setting. The new project is aimed at developing both non-commutative geometry and algebra of deformation quantization, in particular a theory of modules over deformation quantization rings, and at applying them to symplectic geometry, in particular to the Fukaya theory of Lagrangian intersections and to mirror symmetry. The main aim of this project is to develop what we call non-commutative differential calculus. By this we mean an extension of the clasical multi-variable calculus to the case when the variables no longer commute, i.e. when the value of the product is no longer independent of the order of factors. Such situations arise very naturally in mathematics and physics; in quantum mechanics, the non-commutativity expresses mathematically the uncertainty principle of Heisenberg. We intend to apply the non-commutative calculus to so called deformation quantization, a geometric setup very much motivated by quantum mechanics. Our previous work in this direction yielded new proofs and generalizations of classical theorems about solutions of partial differential equations; our new project aims at applications to geometric questions of mathematical physics, such as string theory, mirror symmetry, and Lagrangian intersections.