Groups, Manifolds, and Stable Homotopy Theory

空间，如平面或球体，很容易可视化，但对于像具有许多参数的大型数据集的形状这样的高维对象来说，这是不可能的。代数拓扑学将代数不变量与空间联系起来，将理解空间(直到连续变形)的问题转移到更容易处理的代数世界。这个项目的核心是，当所讨论的空间与其对称性的附加数据一起考虑时，理解其中一些不变量的问题。这个项目中考虑的特殊不变量是代数K理论和A理论：与数论和几何中的问题有非常重要联系的深层不变量。然而，它们的多功能性需要复杂的拓扑和范畴结构。这个项目的一部分涉及完成一些必要的基础工作，允许这些构造对对称的额外数据进行编码，另一部分涉及使用新开发的工具来严格定义和研究这些理论，同时跟踪对称。这项提议还有一个教育方面的因素--PI计划在约翰·霍普金斯大学启动一个定向阅读项目，效仿芝加哥大学的项目。这个项目符合项目组的一个长期研究计划，该计划旨在建立一系列同伦理论构造链，将紧致G-流形的几何行为与其基础同变同伦类型的几何行为联系起来。在目前的工作中，项目组定义了一个等变A-理论谱，下一步是证明它符合一个等变稳定的参数化h-余边化定理。在与合作者的一个相关项目中，PI计划研究Waldhausen G-范畴的等变版本的可加性定理。不同的是，K-理论在可加性定理方面是普适的，等同的，人们可以希望得到类似的刻画。等变K和A理论进展缓慢，部分原因是没有足够的基础。这项工作不仅产生了新的成果(其中一些成果已经被期待了大约35年)，而且还巩固了文献支离破碎、需要澄清的重要领域的工作。他们应该为上面提出的项目提供一些工具，以及一些相关的工具。在不同的方向上，PI和合作者已经定义并正在研究一种从表示理论中推广经典特征映射的谱特征映射。最后，PI计划研究Motivic Galois扩张。

Spaces, like a plane or a sphere, are easy to visualize, but for high-dimensional objects like the shape of a large data set with many parameters, this is impossible. Algebraic topology associates algebraic invariants to spaces, moving the problem of understanding spaces (up to continuous deformation), to the more tractable world of algebra. At the heart of this project is the problem of understanding some of these invariants when the space in question is considered with the additional data of its symmetries. For example, instead of just considering a sphere, we can try to build in the additional data of the symmetry which maps each point to its antipodal point; or instead of considering a plane, we can consider a plane together with all the rigid motions of the plane that fix a square centered at its origin. The particular invariants considered in this project are algebraic K-theory and A-theory: deep invariants that have very significant connections to problems in number theory and geometry. Their versatility, however, requires sophisticated topological and categorical constructions. Part of this project is concerned with completing some necessary foundational work that allows these constructions to encode the extra data of symmetries, and part of it is concerned with using newly developed tools to rigorously define and study these theories, while keeping track of symmetries. There is also an educational component to the proposal-the PI plans to start a Directed Reading Program at the Johns Hopkins University, modeled on the one at the University of Chicago. The goal is to provide mentorship to undergraduates by pairing them with graduate students for reading courses, and in the process encourage interested students to pursue graduate work in mathematics.Building on the PI's previous work on equivariant algebraic K-theory, the PI will develop an equivariant version of A-theory that is related to equivariant pseudo-isotopies and h-cobordisms of G-manifolds. Th project fits into a long-term research program of the project team aimed at establishing a chain of homotopy-theoretic constructions that relate the geometric behavior of compact G-manifolds to that of their underlying equivariant homotopy types. In current work, the project team has defined an equivariant A-theory spectrum, and the next step is to show that it fits into an equivariant stable parametrized h-cobordism theorem. In a related project with collaborators, the PI plans to study an equivariant version of the additivity theorem for Waldhausen G-categories. Nonequivariantly, K-theory is universal with respect to the additivity theorem, and equivariantly one can hope for a similar characterization. Progress has been slow in equivariant K and A-theory partly because adequate foundations were not in place. Equivariant infinite loop space theory, the machinery which turns suitable categories with G-action into G-spectra, has seen a lot of development in the last few years, and, motivated by the need for a solid foundation for equivariant algebraic K and A-theory, the PI, together with collaborators, is completing a series of projects on equivariant infinite loop space theory. This work not only produces new results (some of which have been anticipated for about 35 years), but they also consolidate work in important areas where the literature is fragmented and in need of clarification. They should provide some of the tools for the projects proposed above, as well as some related ones. In a different direction, the PI and collaborators have defined and are studying a spectral character map that generalizes the classical character map from representation theory. Lastly, the PI plans to study motivic Galois extensions.