The main theme of the present proposal is applying the general machinery of non-commutative geometry to concrete problems in algebra and geometry. The proposal establishes several new unexpected links between seemingly unrelated branches of mathematics, such as noncommutative algebra, algebraic geometry, representation theory, and physics. Specifically, given a finite dimensional symplectic vector space and a finite group of symplectic automorphisms of that vector space, the authors study symplectic resolutions and symplectic deformations (both commutative and noncommutative) of the corresponding orbifold. This gives a new approach, through Hochschild cohomology, to various "stringy" topological invariants of the orbifold. Thus, one of the main ideas of the proposal is to approach the algebraic geometry of an orbifold via the techniques of noncommutative geometry. The "symplectic reflection algebras" provide a basic tool for implementing this idea. A systematic use of symplectic reflection algebras leads to various new results in conventional "commutative" algebraic geometry, which have been inaccessible by purely "commutative" methods.
The proposal opens up a wide variety of independent directions for research. One of the basic goals of the proposal is in applying the methods and results developed in one branch of mathematics to a totally different area of mathematics. For example the proposal involves systematic applications of various algebraic techniques to problems in geometry and representation theory. Thus, the proposal enhances the idea of "unity" of mathematics. This proposal also has far reaching applications not only in various areas of mathematics, but also has applications in modern mathematical and theoretical physics, especially in quantum physics. Mathematical science as a whole makes a profound contribution to society. The current proposal goes a long way toward promoting interactions between different disciplines and enhancing scientific understanding.
