CAREER: Applications of equivariant homotopy theory to manifolds

许多数学领域中的一个强大思想是通过将多边形等几何对象与代数不变量（例如在连续变形下不改变的数字）联系起来来研究它。一个早期的例子是欧拉，他指出，对于多边形曲面，顶点减去边加上面的数量总是2，这使他能够证明只有5个柏拉图立体。在20世纪20年代，Emmy Noether将焦点从数值不变量转移到它们背后的代数结构，例如具有代数运算的集合。在同伦理论中，重点进一步扩展到无限维空间的更高级数学概念，其连通分量的数量恢复了底层的数值不变量。这个CAREER奖资助的项目旨在将几何对象的经典代数不变量提升到更高的同伦世界，并从这个更丰富的角度研究它们。该项目的教育部分包括不仅通过会议在学术界扩大对数学的接触，而且还超越了通常的学术环境，以监禁个人：PI将在学士学位课程中引入和教授基于探究学习的大学数学选修课。南森林监狱的一个项目该项目的主要研究目标是利用等变稳定同伦理论的新成果来研究具有群作用的流形及其等变同胚。稳定同伦理论，研究谱，在流形分类的长期计划中发挥了至关重要的作用;然而，群G的G-流形的分类远不如经典的非等变故事。部分建立在以前的工作PI对等变代数K-理论和她的合作基础工作等变无限循环空间理论，机械，把合适的G-范畴到G-谱，PI和马尔凯维奇是建立了一个等变推广瓦尔德豪森的代数K-理论的空间。受古德威利的启发，他们猜想，对于紧致光滑G流形M，等变A理论分裂出一个稳定的等变h-协边空间。这个项目涉及到这些努力的一个重要扩展，重点是证明这个猜想，并利用它来进一步了解等变h-协边和G-流形的单同态。第二个研究目标是应用稳定同伦技术来研究经典不变量，即，将它们提升到光谱。PI将继续进行两个这种风格的项目。首先，PI将与Malkiewich和Moi一起，从他们定义的表示理论中研究经典字符映射的派生版本。第二个是PI与Rovi共同领导的拓扑学女性研讨会项目，旨在将Zakharevich和坎贝尔最近关于剪刀同余K理论谱的工作应用和扩展到流形。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估而被认为值得支持。

A powerful idea in many areas of mathematics has been that of studying a geometric object such as a polygon by associating it with an algebraic invariant, for example a number that does not change under continuous deformation. An early instance of this is Euler who noted that for a polygonal surface the number of vertices minus edges plus faces is always two, which allowed him to prove that there are only five platonic solids. In the 1920s, Emmy Noether shifted the focus from numerical invariants to the algebraic structures underlying them, such as sets with algebraic operations. In Homotopy Theory, the focus is extended further to more advanced mathematical notions of infinite-dimensional spaces whose number of connected components recover the underlying numerical invariant. The projects funded by this CAREER award aim to lift classical algebraic invariants of geometric objects to the higher homotopical world and study them from this richer perspective. The educational part of the project consists in expanding access to mathematics not only within the academic world via conferences, but also beyond the usual academic settings to incarcerated individuals: the PI will introduce and teach an inquiry-based learning college level mathematics elective in the B.A. program offered at South Woods Prison. The primary research goal of the project is to use new results in equivariant stable homotopy theory to study manifolds with group actions and their equivariant diffeomorphisms. Stable homotopy theory, the study of spectra, has played a crucial role in the longstanding program to classify manifolds; however, the classification of G-manifolds for a group G is far less understood than the classical nonequivariant story. Building in part on prior work of the PI on equivariant algebraic K-theory and her collaborative foundational work on equivariant infinite loop space theory, the machinery that turns suitable G-categories into G-spectra, the PI and Malkiewich were have built an equivariant generalization of Waldhausen's algebraic K-theory of spaces. Inspired by Goodwillie’s vision, they conjecture that, for a compact smooth G-manifold M, equivariant A-theory splits off a stable space of equivariant h-cobordisms. This project involves a significant extension of these efforts focused on proving this conjecture and exploiting it to further our understanding of equivariant h-cobordisms and diffeomorphisms of G-manifolds. The second research goal is to apply stable homotopy techniques to study classical invariants by deriving these, i.e., lifting them to spectra. The PI will pursue two projects of this flavor. First, together with Malkiewich and Moi, the PI will study a derived version of the classical character map from representation theory that they have defined. The second, which is a project that the PI is co-leading with Rovi for the Women in Topology Workshop, is to apply and extend recent work of Zakharevich and Campbell on scissors congruence K-theory spectra to manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.