Algebraic Structures and the Arithmetic of Fields

某些类型的高度对称代数结构，如二次型和除法代数，在数学及其应用中无处不在，在多天线无线通信、空间旋转的有效表示、理论物理的规范对称和数论中的伽罗瓦表示等不同领域发挥着重要作用。虽然我们对这些结构的理解已经相当丰富，但仍存在许多基本问题。该项目将采用和开发快速发展的算术几何领域的新工具，以加深我们对这些代数结构的理解。首席研究员将参与培训学生和组织该领域的会议。本研究项目将研究线性代数群的环量与域算法之间的相互作用，目的是开发Brauer群的周期指数型问题的改进版本以及其他代数结构类的相关概念。这样做还将涉及扩展“领域修补”和相关方法的范围和适用性，在研究者与合作者先前工作的基础上，为更广泛的对象类别和更一般的领域产生新的局部-全局原则。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Certain types of highly symmetric algebraic structures, such as quadratic forms and division algebras, have been ubiquitous in mathematics and its applications, playing important roles in such diverse fields as multiple-antenna wireless communications, efficient representations of spatial rotations, gauge symmetries of theoretical physics, and Galois representations in number theory. While our understanding of these structures has become quite rich, many fundamental questions still remain. This project will employ and develop new tools from the rapidly growing field of arithmetic geometry to deepen our understanding of these algebraic structures. The principal investigator will be involved in the training of students and the organization of conferences in the field.This research project will study the interplay between torsors for linear algebraic groups and field arithmetic, with the aim of developing refined versions of period-index type problems for the Brauer group and related notions for other classes of algebraic structures. Doing this will also involve extending both the scope and applicability of "field patching" and related methods, building on prior work of the investigator with collaborators to produce new local-global principles for wider classes of objects as well as for more general fields.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.