In this proposal, the investigator and collaborators study problems in algebra, algebraic geometry and topology. Three of these projects study families of noncommutative algebras arising naturally in various branches of algebra. The first project is to study finite dimensional irreducible representations of Clifford algebras of forms of higher degree. The approach in studying these algebras is to work on a reformulation of the questions in terms of moduli spaces of certain reflexive sheaves on hypersurfaces. Another goal is to investigate arithmetic applications of these results. The second project (joint with D. Chan) is continuation of previous work on sheaves of maximal orders on complex, projective, algebraic surfaces. The main goal is to classify various classes of such sheaves of maximal orders. This work follows recently developed ideas in noncommutative algebraic geometry. The third project (joint with A. Ram) is to investigate whether the Springer correspondence can be set up for rational Cherednik algebras along the lines of single graded Hecke algebras. Another goal is to investigate whether these algebras can be related to N. Wallach's approach to the Springer correspondence. The last project (joint with M. Banagl) is to investigate whether there exist self-dual sheaves compatible with the intersection chain sheaves on reductive Borel-Serre compactifications of locally symmetric spaces associated to semisimple algebraic groups defined over rational numbers.
In the last few decades, the interaction between various branches of mathematics and theoretical physics has proved to be especially enriching for all the fields. One important thread in these connections has been algebraic geometry, a very old subject that dates back at least to ancient Greece. Algebraic geometry is the area of mathematics that studies solutions to multi-variable polynomial equations as geometric objects. It has found applications not only in mathematics, but also in computer science, coding theory, robotics and string theory in physics to name a few areas impacted by it. The first two projects of this proposal use tools from this subject to study "algebras" that are of interest to a wide range of mathematicians and physicists. It is useful to describe the whole family of such classes of algebras as a geometric object, and a goal of this project is to obtain such descriptions. The third aspect of this research is part of a subject called representation theory. The main objects of study here are "representations", processes which encode information about symmetry in nature. The last part of this proposal stands at the intersection of three subjects: number theory (where number systems are studied), representation theory, and topology. In topology spaces are studied to determine which properties do not change under elastic deformations such as twisting and stretching. The spaces to be studied in this research encode number theoretic information. The goal is to investigate whether there are invariants of these spaces which themselves satisfy some symmetry.
