CAREER: The Arithmetic of Fields and the Complexity of Algebraic Structures

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The PI will investigate a number of problems, using homotopy aspects of arithmetic geometry, that may be thought of as generalizations of the period-index problem for the Brauer group, the u-invariant problem for quadratic forms, and the Hasse principle for a quadratic form over a number field. The project will focus on the interaction of field arithmetic and the complexity of algebraic structures, such as quadratic forms, Brauer groups, linear algebraic groups and homogeneous varieties. The project represents a set of techniques and experiments designed to capitalize on the new topological perspective in the study of algebraic structures. It will aim to support and further the interactions of these areas with arithmetic algebraic geometry, algebraic topology and the algebraic geometry of stacks and moduli.The relevance of these topics, rooted in algebra within pure mathematics, is exhibited by their connections to a wide range of other subjects in recent years. Besides their many ties to other branches of mathematics, the algebraic structures at the core of this proposal have also found applications within diverse areas from theoretical physics to wireless communications. The research component of this project will seek to capitalize on and enrich our understanding of the many connections to other areas of study in order to gain more leverage in understanding these fundamental and important algebraic structures. The project's outreach components include funding conferences aimed at establishing a community inclusive of graduate students and young researchers, undergraduate colloquium aimed at increasing interest in mathematics, and facilitating a mentoring program for high school students in northeast Georgia interested in mathematics. It will also pursue strategies towards improving the mentoring and retention of graduate students in mathematics.