Interactions between Algebra, Algebraic Geometry and Topology

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In this proposal, the investigator and collaborators study problems inalgebra, algebraic geometry and topology. Three of these projectsstudy families of noncommutative algebras arising naturally in variousbranches of algebra. The first project is to study finite dimensionalirreducible representations of Clifford algebras of forms of higherdegree. The approach in studying these algebras is to work on areformulation of the questions in terms of moduli spaces of certainreflexive sheaves on hypersurfaces. Another goal is to investigatearithmetic applications of these results. The second project (jointwith D. Chan) is continuation of previous work on sheaves of maximalorders on complex, projective, algebraic surfaces. The main goal is toclassify various classes of such sheaves of maximal orders. This workfollows recently developed ideas in noncommutative algebraicgeometry. The third project (joint with A. Ram) is to investigatewhether the Springer correspondence can be set up for rationalCherednik algebras along the lines of single graded Heckealgebras. Another goal is to investigate whether these algebras can berelated to N. Wallach's approach to the Springer correspondence. Thelast project (joint with M. Banagl) is to investigate whether thereexist self-dual sheaves compatible with the intersection chain sheaveson reductive Borel-Serre compactifications of locally symmetric spacesassociated to semisimple algebraic groups defined over rationalnumbers. In the last few decades, the interaction between various branches ofmathematics and theoretical physics has proved to be especiallyenriching for all the fields. One important thread in theseconnections has been algebraic geometry, a very old subject that datesback 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 inmathematics, but also in computer science, coding theory, robotics andstring theory in physics to name a few areas impacted by it. The firsttwo projects of this proposal use tools from this subject to study"algebras" that are of interest to a wide range of mathematicians andphysicists. It is useful to describe the whole family of such classesof algebras as a geometric object, and a goal of this project is toobtain such descriptions. The third aspect of this research is partof a subject called representation theory. The main objects of studyhere are "representations", processes which encode information aboutsymmetry in nature. The last part of this proposal stands at theintersection of three subjects: number theory (where number systemsare studied), representation theory, and topology. In topology spacesare studied to determine which properties do not change under elasticdeformations such as twisting and stretching. The spaces to bestudied in this research encode number theoretic information. The goalis to investigate whether there are invariants of these spaces whichthemselves satisfy some symmetry.