Algebraic Structures and the Arithmetic of Fields

某些类型的高度对称的代数结构，如二次型和除法代数，在数学及其应用中普遍存在，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性和数论中的伽罗瓦表示等领域发挥着重要作用。虽然我们对这些结构的了解已经变得相当丰富，但许多根本问题仍然存在。这个项目将使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。这项研究项目将研究线性代数群的扭量与场算术之间的相互作用，目的是为Brauer群开发周期指数型问题的精化版本，并为其他类型的代数结构开发相关概念。要做到这一点，还将涉及扩展“现场修补”和相关方法的范围和适用性，以研究者与合作者先前的工作为基础，为更广泛的对象类别以及更一般的领域制定新的局部-全球原则。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.