Topological methods in arithmetic and algebra

这个项目将拓扑学、算术和代数等不同领域的几条线索编织在一起。 算术中的问题--例如确定一个数字系统的对称性--可能很难直接解决，但有时更容易在概率或统计意义上解决。 这个项目的一个主要目标是使用拓扑学中的工具（特别是平面上点的配置空间的几何学）来解决算术中这些统计问题的几何类似问题。 沿着的方式，主要研究者将开发工具来研究这些问题的代数对象的结构，这些工具将强烈通知这些配置空间的几何设置。主要研究者将特别旨在建立和完善Malle，Delaunay，和其他人在功能字段设置的结构。 该方法涉及到一个新的工具，用于计算的辫子群的同调使用的同调代数的辫子Hopf代数。作为这项工作的一部分，首席研究员将开发一个结构理论的辫子霍普夫代数，并使用它来更好地理解他们的同调代数。 最后，首席研究员将颠倒所涉及的主题的角色，并使用算术工具来解决色同伦理论中的问题，特别是围绕色分裂和红移的问题。

This project weaves together several threads in the disparate areas of topology, arithmetic, and algebra. Questions in arithmetic - such as determining the symmetries of a number system - can be difficult to resolve directly, but are sometimes easier to address in a probabilistic or statistical sense. One main goal of this project is to use tools in topology (and specifically the geometry of configuration spaces of points in the plane) to resolve geometric analogues of these statistical questions in arithmetic. Along the way, the Principal Investigator will develop tools to study the structure of algebraic objects governing these questions; these tools will be strongly informed by the geometric setting of these configuration spaces.The Principal Investigator will particularly aim to establish and refine conjectures of Malle, Delaunay, and others in the function field setting. The methods involve a new tool for the computation of the homology of braid groups using the homological algebra of braided Hopf algebras. As part of this work, the Principal Investigator will develop a structure theory for braided Hopf algebras, and use this to better understand their homological algebra. Finally, the Principal Investigator will reverse the roles of the subjects involved, and use arithmetic tools to address questions in chromatic homotopy theory, particularly surrounding the chromatic splitting and redshift conjectures.