Interactions between noncommutative algebra, algebraic geometry and representation theory

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The proposed projects in this proposal are in three directions. The first project studies noncommutative analogs of algebraic surfaces. Following previous work on classification of maximal orders on projective, nonsingular surfaces, this project studies remaining classes of maximal orders. Another goal for this project is to give explicit constructions of orders on surfaces and then use them to study their module and derived categories. The last goal in this direction is to initiate study of maximal orders on surfaces in characteristic p. The second project continues previous work on self dual sheaves compatible with the intersection chain sheaves on reductive Borel-Serre compactifications of locally symmetric spaces. The goal is to see if such sheaves exist for compactifications of certain unitary Shimura varieties and to see to what extent these sheaves can be used in analysis of the Lefschetz numbers of Hecke correspondences. The last project investigates structure of double Hurwitz numbers of genus zero curves by studying the expected polynomiality phenomenon.In the last few decades, there have been very fruitful interactionsbetween various areas of mathematics and theoretical physics. Oneimportant thread in these connections has been algebraic geometry, a very old subject that dates back at least to ancient Greece. Algebraic geometry is a subject in which solutions of many variable polynomial equations are studied as geometric objects. The proposed projects use tools from algebraic geometry to solve problems in noncommutative algebra. In noncommutative algebra, the main objects of study are polynomials in which the xy is not the same as yx. Such noncommutative algebras are of interest to physicists as well. The second project of this proposal is at the intersection of three subjects: number theory (where number systems are studied), representation theory (where symmetries are studied), and topology. In topology those properties of spaces are studied which do not change under elastic deformations such as twisting and stretching. The spaces to be studied in this project encode number theoretic information. These spaces have some natural symmetries. The goal is to investigate fixed points of these symmetries. The last project is at the intersection of combinatorics and algebraic geometry. The goal is to understand the number of ways in which a sphere can be covered by similar geometric objects called algebraic curves with specified restrictions.