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，这些问题可以被认为是Brauer群的周期指数问题的推广，二次型的u不变问题，以及数字域上二次型的Hasse原理。该项目将侧重于场算法与代数结构的复杂性的相互作用，如二次型、Brauer群、线性代数群和齐次变异。该项目代表了一套技术和实验，旨在利用代数结构研究中的新拓扑视角。它将旨在支持和进一步这些领域与算术代数几何，代数拓扑和堆栈和模的代数几何的相互作用。这些主题的相关性，植根于纯数学中的代数，近年来通过它们与广泛的其他学科的联系表现出来。除了与数学的其他分支有许多联系外，这个提议的核心代数结构也在从理论物理到无线通信的各个领域得到了应用。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。该项目的外联部分包括资助旨在建立一个包括研究生和年轻研究人员在内的社区的会议，旨在提高对数学兴趣的本科生讨论会，以及为佐治亚州东北部对数学感兴趣的高中生提供指导计划。它还将寻求改进指导和留住数学研究生的战略。

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.