RUI: Problems in Varieties of Commuting Matrices, Jet Schemes, and Division Algebras

The University Corporation, Northridge, Northridge CA

Investigators

Abstract

This project deals with two broad classes of problems. The first concerns varieties of commuting matrices and jet schemes, and arises from algebro-geometric interpretations of an open problem in commuting matrices. The other concerns division algebras, and arises from the classical question of crossed products, and as well, from recently discovered applications of division algebras to coding theory. In the first class of problems, the PI, Sethuraman, wishes to study various varieties defined by the condition that certain matrices commute, such as varieties of commuting pairs, of commuting triples, and of commuting pairs in the centralizers of fixed matrices. The idea is to gain insight into three generated commutative matrix algebras. Jet schemes of commuting pairs of matrices arise naturally in these considerations, as do jet schemes of determinantal varieties. In the second class of problems, Sethuraman wishes to study certain generically defined symbol division algebras of prime exponent and index various powers of the prime to determine if these are crossed products. The generic nature of the symbol algebra makes this a significant object to study. In a different direction, Sethuraman wishes to continue to study applications of division algebras to wireless communication, including determining large subgroups of units in orders in division algebras, and the construction of orthogonal lattices inside number fields for appropriate trace forms. The problems considered in this proposal arise from algebra and algebraic geometry. There are two kinds of problems studied here: one where the motivation comes from within mathematics, and the other where, very significantly, the motivation comes from wireless communication. The PI has had a successful collaboration with engineers where he was able to utilize certain abstractly defined mathematical objects (``division algebras'') to solve a very practical telecommunications problem. The synergy appeared most unexpectedly, and has led to a very fruitful body of results in the field. In addition to this work with engineers, it is to be noted that the PI teaches in a primarily undergraduate institution, with a large immigrant population, where he has guided several undergraduate and masters level students on research projects in mathematics. Some of these students will go on to be school teachers. He proposes to continue working with students on research projects.