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）转移到更容易处理的代数世界。在这个项目的核心是理解这些不变量的问题时，空间的问题是考虑其对称性的额外数据。例如，我们可以不考虑球体，而尝试建立将每个点映射到其对跖点的对称性的额外数据;或者不考虑平面，我们可以考虑平面以及平面的所有刚性运动，这些运动使一个正方形以其原点为中心。在这个项目中考虑的特定不变量是代数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和分类结构。这个项目的一部分是关于完成一些必要的基础工作，使这些结构编码对称性的额外数据，它的一部分是关于使用新开发的工具来严格定义和研究这些理论，同时保持对称性的跟踪。这个提议还有一个教育部分--PI计划在约翰霍普金斯大学启动一个定向阅读项目，以芝加哥大学为蓝本。目标是通过将本科生与研究生配对进行阅读课程，为本科生提供指导，并在此过程中鼓励感兴趣的学生继续从事数学研究生工作。PI将在等变代数K理论的基础上，开发与G流形的等变伪同位素和h-协边相关的A理论的等变版本。该项目符合项目团队的长期研究计划，旨在建立一系列同伦理论构造，将紧致G流形的几何行为与其潜在的等变同伦类型联系起来。在目前的工作中，项目团队已经定义了一个等变A-理论谱，下一步是证明它符合等变稳定参数化h-协边定理。在与合作者的相关项目中，PI计划研究Waldhausen G-范畴的可加性定理的等变版本。非等变地，K-理论在可加性定理方面是普适的，并且等变地，人们可以希望得到类似的表征。等变K和A理论进展缓慢，部分原因是没有足够的基础。等变无限循环空间理论，将合适的G作用范畴转化为G谱的机器，在过去的几年里有了很大的发展，并且，由于需要为等变代数K和A理论打下坚实的基础，PI与合作者一起完成了一系列关于等变无限循环空间理论的项目。这项工作不仅产生了新的成果（其中一些成果已经预期了大约35年），而且还巩固了文献零散和需要澄清的重要领域的工作。它们应该为上文提议的项目以及一些相关项目提供一些工具。在另一个不同的方向，PI和合作者已经定义并正在研究一个谱特征映射，它从表示论中推广了经典的特征映射。最后，PI计划研究motivic Galois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.