Homotopy theory and arithmetic: a dialogue

数论中的问题很多，有些可以追溯到古代。 虽然看似简单的状态，他们可以非常难以解决。 相比之下，在拓扑学（一门探索几何学中对扰动不敏感的部分的学科）中，陈述问题可能很困难，但有许多既定的技术来解决它们。 这是本建议的目的是使用拓扑技术来解决长期开放的问题在数论，并使用结果在数论中建立新的和有趣的项目在拓扑结构。 中心的主题是，拓扑结构提供了一个令人难以置信的灵活的工具做计算的对象的重要性，在数论，数论提供了非常有趣的例子，对象有用的topologist.Specifically，这个建议的目的是建立数论approaches的马勒和其他人在设置的功能领域使用计算的同源性的赫维茨空间的分支覆盖。 我们将进一步发展的Lefschetz纤维使用这种机制的不变量。 在相反的方向上，我们将使用etale上同调的算术来重新想象p-紧群理论作为算术几何的一种形式。 这些技术应该，例如，导致这些对象的表征理论的进一步发展。 最后，我们将通过色同伦理论程序，使用算术和几何技术进行稳定同伦理论的计算。

Questions in number theory abound, some going back to the ancients. While deceptively simple to state, they can be exceedingly difficult to solve. In contrast, in topology (a subject that explores the part of geometry that is insensitive to perturbation) it can be difficult to state problems, but there are numerous established techniques for solving them. It is the purpose of this proposal to use techniques in topology to solve long-standing open problems in number theory, and to use results in number theory to establish new and interesting projects in topology. The central themes are that topology provides an incredibly flexible tool for doing computations of objects of importance in number theory, and that number theory provides very interesting examples of objects useful to topologists.Specifically, this proposal aims to establish number-theoretic conjectures of Malle and others in the setting of function fields using computations of the homology of Hurwitz spaces of branched covers. We will further develop invariants of Lefschetz fibrations using this machinery. In the opposite direction, we will use the arithmetic of etale cohomology to re-imagine the theory of p-compact groups as a form of arithmetic geometry. These techniques should, for instance, lead to a further development of the representation theory of these objects. Lastly, we will pursue computations in stable homotopy theory using arithmetic and geometric techniques via the chromatic homotopy theory program.