Homotopy Theory and Its Applications

9802852米切尔 本项目主要研究同伦理论中三个不同但又密切相关的领域的问题：（1）Morava稳定子群的闭子群作用于Landweber精确谱的同伦不动点谱，其系数环是Lubin-Tate模提升空间上的函数（2）乘法稳定同伦理论、算子同伦理论、单算子上代数的Andre-Quillen同调（Goerss）;（3）代数整数环的etale同伦型，以期在Iwasawa理论和代数K-理论（Mitchell）中得到应用。 该项目更广泛的目标是进一步理解同伦理论，这是拓扑学的分支，涉及高维曲面和其他几何对象在变形下保持不变的性质。 同伦理论与数论和代数几何等看似无关的领域之间存在着一些令人惊讶且相当神秘的联系，研究人员希望阐明的正是这些联系。 ***

9802852 Mitchell This project addresses problems in three distinct but closely related areas of homotopy theory: (1) homotopy fixed point spectra arising from the action of closed subgroups of the Morava stabilizer group on the Landweber exact spectrum whose coefficient ring is the functions on the Lubin-Tate moduli space of lifts (Devinatz); (2) multiplicative stable homotopy theory, the homotopy theory of operads, Andre-Quillen homology of algebras over simplicial operads (Goerss); and (3) etale homotopy type of rings of algebraic integers, with a view to applications in Iwasawa theory and algebraic K-theory (Mitchell). The broader goal of the project is to further our understanding of homotopy theory, the branch of topology concerned with those properties of higher-dimensional surfaces and other geometric objects that remain invariant under deformation. There are some surprising and rather mysterious connections between homotopy theory and seemingly unrelated fields such as number theory and algebraic geometry, and it is these connections that the investigators hope to elucidate. ***