Transchromatic homotopy theory

跨色同伦理论

• 批准号: 1406408
• 负责人: Nathaniel Stapleton
• 金额: $ 13.5万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406408&HistoricalAwards=false
• 关键词: Transchromatic; homotopy; theory

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.

拓扑空间是数学中一些最基本的对象。代数拓扑结合代数对象到拓扑空间，以便分开。 这样做的一些最强大的代数工具是“共同体理论”。一系列称为“摩拉瓦理论”的共同论理论具有与代数几何形状的深刻而有些理解的联系，并且与几何形状具有重要的猜想关系。前两个Morava E理论是古典的。但是，其余的非常神秘。该项目的目的是利用与代数几何形状的连接，以互相发挥不同的摩拉维亚理论，以揭示猜想的几何特性。 PI计划研究色层之间的关系，以揭示色素均匀同拷贝理论中潜伏在幕后的几何形状。他将通过追求三个相互关联的程序来做到这一点。 PI将使用P分解组的代数几何形状，广义特征理论和现场理论（从Stolz-Teichner的意义上）来攻击问题。在与Schommer-Pries的联合合作中，PI希望为诸如Morava E理论之类的更高色度共同体学理论提供领域的理论结构。同样特别感兴趣的是经典定理的“变性”证明，这些证明会导致更普遍的结果。例如，在与Tomer Schlank的合作中，PI在对称群体的Morava E理论上提供了新的证据，自然会导致定理将有限的Abelian群体与对称群体的有限的Abelian群体的产品进行概括。证明中的主要工具之一是从电子理论到p-adic K理论的角色图，它允许人们将电子理论中的某些问题减少到表示理论。该地图值得进一步研究。