Classifying spaces, proper actions and stable homotopy theory

这个项目的主要目标是计算来自稳定同伦理论的离散群的不变量。第二个目标是探索这些计算与数论的联系。该项目涉及来自稳定同伦理论的无限离散群的不变量，如Morava K理论和Morava E理论。这些不变量推广了经典的欧拉特征线和拓扑K-理论。对于有限群，它们是众所周知的，并由Hopkins-Kuhn-Ravenel研究。对于K-理论，这些无限群的不变量是由Lück和Coathors计算的。我们计划推广这两种计算，对于无限群的Morava K-理论，我们期望得到一个公式的Morava K-理论欧拉特征使用中心化。对于摩拉瓦E-理论，我们计划推广的霍普金斯-库恩-拉文埃尔字符理论从有限到无限群。最后，我们将把这些结果应用到具体的群体中。由此得出的具体公式应该是数论性质的。其中一些将包含戴德金和黎曼zeta函数的值。

Primary goal of this project is to compute invariants of discrete groups coming from stable homotopy theory. Secondary goal is to explore connections of these computations with number theory. The project deals with invariants of infinite discrete groups coming from stable homotopy theory, such as Morava K-theories and Morava E-theories. These invariants generalise the classical Euler characteristic and topological K-theory. For finite groups they are well-known and studied by Hopkins-Kuhn-Ravenel. For K-theory these invariants for infinite groups were computed by Lück and coathors. We plan to generalise both these computations.For Morava K-theories of infinite groups we expect to obtain a formula for the Morava K-theoretic Euler characteristic using centralisers. For Morava E-theory we plan to generalise the Hopkins-Kuhn-Ravenel character theory from finite to infinite groups. Finally, we will apply these results to concrete groups. Concrete formulas coming out of these should be of number theoretic nature. Some of them will contain values of Dedekind and Riemann zeta functions.