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.
The principal goal of this Project was to obtain new mathematical results by exploiting ideas coming from two different areas. The first area, which is part of Theoretical Physics, is usually referred to as "Quantization". The subject of quantization has a long history and takes its origins in classic works on quantum theory, as developed by P. Dirac, J. Von Neumann, and others. In mathematics, deformation quantization theory is by now a well established theory. The second area that plays a major role in the Project is called "Noncommutative geometry". This is a relatively new subject, initiated by M. Kontsevich around 15 years ago. Noncommutative geometry has already influenced several other branches of mathematics and that has a number of deep applications. The main theme of the Project is applying the general machinery of quantization and noncommutative geometry to concrete problems in algebra and representation theory. The results of the Project have far reaching applications in, and establish new unexpected links between, many different areas of Mathematics. Furthermore, there are also applications in modern Mathematical and Theoretical Physics. The theory of Calabi-Yau algebras, for instance, has a direct connection with Mirror symmetry, an important part of Quantum Field Theory. Our mathematical results on quantization of del Pezzo surfaces have applications to models considered by physicists in String Theory. In addition, the Proposal opens up a wide variety of new directions for mathematical research.
