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

PI将研究一些问题，使用算术几何的同伦方面，可以被认为是Brauer群的周期指数问题，二次型的u-不变问题和数域上二次型的Hasse原理的推广。该项目将侧重于领域算术的相互作用和代数结构的复杂性，如二次型，布劳尔群，线性代数群和齐次簇。该项目代表了一套技术和实验，旨在利用新的拓扑角度研究代数结构。它的目的是支持和促进这些领域的相互作用与算术代数几何，代数拓扑和代数几何的堆栈和moduli.The相关性，这些主题，植根于纯数学代数，是展示了他们的连接到广泛的其他科目在最近几年。除了与其他数学分支的许多联系之外，这个提议的核心代数结构也在从理论物理到无线通信的不同领域中找到了应用。 该项目的研究部分将寻求利用和丰富我们对与其他研究领域的许多联系的理解，以便在理解这些基本和重要的代数结构方面获得更多的杠杆作用。 该项目的推广部分包括资助会议，旨在建立一个社区，包括研究生和年轻的研究人员，本科生座谈会，旨在提高对数学的兴趣，并促进辅导计划，高中学生在东北部格鲁吉亚感兴趣的数学。它还将寻求改善数学研究生的指导和保留的战略。

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.