Quantization, Noncommutative Geometry, and Applications

University Of Chicago, Chicago IL

Investigators

Abstract

The Proposal explores new structures on classical geometric objects, like manifolds, vector bundles, etc., that can be interpreted, most naturally, from the point of view of noncommutative, rather than commutative, geometry. This involves developing a new general theory of deformation quantization of vector bundles on an algebraic variety. The theory provides an explanation for recently discovered invariants associated with singularities of intersections of lagrangian submanifolds. Other applications include explicit constructions of deformation quantizations of an important class of 2-dimentional algebraic surfaces known as del Pezzo sufaces. The subject of quantization has a long history and takes its origins in classic works on quantum mechanics by Dirac, Heisenberg, Pauli, and others. In mathematics, the idea of quantization involves replacing familiar geometric objects by appropriate noncommutative analogues thought of as some deformations of the corresponding geometric objects. The resulting theory is often referred to as noncommutative geometry. The present Project is concerned with a more specific direction known as noncommutative algebraic geometry. This is a relatively recent area, 10-15 years old, at the crossroads between algebra, geometry and theoretical physics. The developments in noncommutative algebraic geometry were strongly influenced by, and have important applications to, string theory, a part of theoretical physics describing fundamental laws of elementary particles at very high energy.