Transchromatic homotopy theory

拓扑空间是数学中最基本的对象之一。代数拓扑学将代数对象与拓扑空间联系起来，以便将它们区分开来。实现这一点的一些最强大的代数工具是“上同调理论”。一系列被称为Morava E-理论的上同调理论与代数几何有着深刻而又有些了解的联系，并与几何有着重要的猜想关系。前两个莫拉瓦E理论是经典的；然而，其余的都是相当神秘的。这个项目的目标是利用与代数几何的联系来相互播放不同的Morava E-理论，以揭示猜想几何的性质。PI计划研究色层之间的关系，以揭示隐藏在色同伦理论背后的几何图形。他将通过追求三个相互关联的项目来做到这一点。PI将使用p-可分群的代数几何、广义特征标理论和场论(在Stolz-Teichner意义下)来解决这个问题。在与Schommer-Pries的合作中，PI希望给出高色上同调理论的场论构造，例如Morava E-理论。同样令人特别感兴趣的还有经典定理的“变色”证明，这些证明导致了更一般的结果。例如，在与Tomer Schlank的合作中，PI给出了关于对称群的Morava E-理论的Strickland定理的一个新的证明，这自然导致了该定理的推广到具有对称群的有限交换群的圈积。证明中的主要工具之一是从E-理论到p-进K-理论的特征标映射，它允许人们将E-理论中的某些问题归结为表示理论。这张地图值得进一步研究。

Topological spaces are some of the most fundamental objects in mathematics. Algebraic topology associates algebraic objects to topological spaces in order to tell them apart. Some of the most powerful algebraic tools for doing this are "cohomology theories". A certain sequence of cohomology theories called the Morava E-theories have a deep and somewhat understood connection to algebraic geometry and an important conjectural relationship with geometry. The first two Morava E-theories are classical; however, the rest are quite mysterious. The goal of this project is to use the connection to algebraic geometry to play different Morava E-theories off each other in order to expose properties of the conjectural geometry. The PI plans to study the relationship between the chromatic layers in order to expose the geometry that lurks behind the scenes in chromatic homotopy theory. He will do this by pursuing three interrelated programs. The PI will use the algebraic geometry of p-divisible groups, generalized character theory, and field theories (in the sense of Stolz-Teichner) to attack the problem. In joint work with Schommer-Pries, the PI hopes to give a field theoretic construction of higher chromatic cohomology theories such as Morava E-theory. Also of particular interest are "transchromatic" proofs of classical theorems that lead to more general results. For instance, in work with Tomer Schlank, the PI has given a new proof of Strickland's theorem on the Morava E-theory of symmetric groups that naturally leads to a generalization of the theorem to wreath products of finite abelian groups with symmetric groups. One of the main tools in the proof is a character map from E-theory to p-adic K-theory that allows one to reduce certain problems in E-theory to representation theory. This map deserves further study.