Topics in Algebraic Geometry, Non-commutative Geometry and Representation Theory

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Principal Investigator: Alexander Polishchuk Proposal Number: 0302215Institution: Boston UniversityTitle: Topics in algebraic geometry, noncommutative geometry and representation theoryAbstract.The investigator proposes a research program in algebraic geometry and representation theory. The proposed projects are: 1) the study of higher Massey products on derived categories of coherent sheaves on projective varieties; 2) the determination of the rings of algebraic cycles on Jacobians of curves modulo algebraic equivalence; 3) the study of intersection theory on the moduli spaces of higher spin curves in connection with the generalized Witten's conjecture; 4) the study of categories of holomorphic vector bundles on noncommutative tori; 5) the study of minimal representations of p-adic groups and their relation with the theory of automorphic forms.A significant part of this proposal is motivated by mathematical conjectures originating from physics. In particular, one of the five projects is motivated by a conjecture of Kontsevich, the "homological mirror conjecture", which is an attempt to give a mathematical foundation to mirror symmetry phenomena discovered in string theory. Another of the projects is devoted to the study of geometric objects relevant for a certain conjecture of Witten in connection with the theory of gravity. From a mathematical point of view, this proposal belongs to the fields of algebraic geometry and representation theory. Algebraic geometry is the part of mathematics studying geometric objects defined by polynomial equations. The richness of this area is due to the fact that it combines algebraic techniques with geometric intuition. Representation theory is essentially the study of symmetries given by linear transformations. Because of the ubiquity of symmetries of this type, representation theory is relevant for many other fields, including number theory and physics.