Groups, Manifolds, and Stable Homotopy Theory

空间，如平面或球体，很容易可视化，但对于高维对象，如具有许多参数的大型数据集的形状，这是不可能的。代数拓扑将代数不变量与空间联系起来，将理解空间的问题（直到连续变形）转移到更易于处理的代数世界。这个项目的核心问题是，当考虑到有问题的空间及其对称性的附加数据时，如何理解这些不变量。例如，不只是考虑一个球体，我们可以尝试建立额外的对称数据，将每个点映射到它的对映点；或者不考虑平面，我们可以考虑平面和平面的所有刚性运动，这些运动固定了一个以原点为中心的正方形。在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结构和分类结构。这个项目的一部分是完成一些必要的基础工作，使这些结构能够对对称的额外数据进行编码，另一部分是使用新开发的工具来严格定义和研究这些理论，同时保持对对称性的跟踪。这个提议还有一个教育方面的内容——PI计划模仿芝加哥大学的项目，在约翰霍普金斯大学启动一个定向阅读项目。其目标是通过将本科生与研究生配对阅读课程，为本科生提供指导，并在此过程中鼓励感兴趣的学生攻读数学研究生课程。在PI先前关于等变代数k理论的工作的基础上，PI将开发与g流形的等变伪同位素和h协体相关的a理论的等变版本。该项目符合项目组的长期研究计划，旨在建立一系列同伦理论结构，将紧g流形的几何行为与其潜在的等变同伦类型的几何行为联系起来。在目前的工作中，项目组已经定义了一个等变a理论谱，下一步是证明它符合一个等变稳定参数化h协定理。在与合作者的一个相关项目中，PI计划研究瓦尔德豪森g范畴的可加性定理的等变版本。对于可加性定理，k理论是非等价的是普适性的，我们可以等价地期待一个类似的描述。等变K理论和等变a理论进展缓慢，部分原因是没有足够的基础。等变无限环空间理论是将具有g作用的合适范畴转化为g谱的机制，近年来取得了很大的发展，并且由于需要为等变代数K和a理论奠定坚实的基础，PI与合作者一起完成了一系列关于等变无限环空间理论的项目。这项工作不仅产生了新的结果（其中一些已经预计了大约35年），而且还巩固了文献碎片化和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。在一个不同的方向上，PI和合作者已经定义并正在研究一个光谱字符映射，它概括了来自表示理论的经典字符映射。最后，PI计划研究动机伽罗瓦扩展。

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.