Interactions between noncommutative algebra, algebraic geometry and representation theory

点击翻译按钮获取中文摘要

The proposed research is at the intersection of noncommutative algebra, algebraic geometry and representation theory. The projects in the first direction concern noncommutative analogs of algebraic surfaces called maximal orders on surfaces. These are coherent sheaves of noncommutative algebras on surfaces whose generic stalk is a central simple algebra. The proposed projects related to maximal orders include studying coarse moduli spaces of terminal orders, orders on 3-folds, module and derived categories of certain terminal orders and Brauer groups of some K3 surfaces in characteristic p. The second direction of proposed projects concerns Clifford algebras of higher degree forms and their representations. The projects in this direction include using arithmetically Cohen-Macaulay bundles to construct moduli spaces of irreducible representations of Clifford algebras of ternary cubic forms, and using Clifford algebras of binary forms to construct maximal orders on surfaces. The third direction of proposed projects concerns computations of Lefschetz numbers for Hecke correspondences of certain Shimura varieties. Over the past few decades there have been increasing and fruitful interactions between various areas of mathematics and theoretical physics. 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 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. A great deal of progress has been achieved in the past 15 years in understanding such algebras and the projects in this proposal will advance this knowledge. The proposed projects in another direction lie at the intersection of number theory (where number systems are studied), representation theory (where symmetries are studied), and topology. In topology, 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 and possess various kinds of symmetries.