The principal investigator's proposed research lies on the boundary between non-commutative algebra and algebraic geometry. In particular, the principal investigator continues his research into non-commutative algebraic geometry, investigating particular non-commutative spaces and using the intuition obtained from examination of those particular examples to simultaneously develop the foundations of the subject. Among the particular examples examined are quadric hypersurfaces and, more generally, complete intersections of quadrics, in non-commutative analogues of projective n-space. Another family of particular examples is the rational Cherednik algebras of Etingof and Ginzburg viewed as (possible) non-commutative resolutions of important and naturally occurring deformations of quotient singularities. Recent work of M. Van den Bergh has shown many situations in which certain non-commutative rings provide "resolutions" of the singular variety corresponding to their centers: for example, the derived categories of the non-commutative rings are equivalent to the derived categories of a crepant resolution of the center. The principal investigator's research develops and extends this idea to other varieties, particularly the varieties that arise in several papers by string theorists.
Non-commutative algebra arises from the need to solve equations where the solutions are matrices, or more generally linear operators, rather than simply numbers. Such equations arise throughout science. Motivated in part by the wonderfully fruitful interaction between algebra and geometry that lies at the heart of algebraic geometry, there has been great interest in developing a geometry that is an appropriate counterpart to non-commutative algebra. This theme, in both an algebraic and analytic version, has been developed with some vigor and success for more than a decade now. There is growing evidence, provided by developments in string theory, that the universe in which we live is in part governed by a mildly non-commutative geometry. The proposed research can be thought of as an exploration of toy models of this universe, and also as developing basic tools and concepts for understanding non-commutative spaces.
