CAREER: Applications of equivariant homotopy theory to manifolds

在许多数学领域中，一个强大的想法是通过将几何对象(如多边形)与代数不变量(例如，在连续变形下不变的数字)相关联来研究该几何对象。一个早期的例子是欧拉，他指出，对于多边形曲面，顶点减去边和面的数量总是两个，这使他能够证明只有五个柏拉图实体。在20世纪20年代，艾米·诺特将重点从数值不变量转移到它们背后的代数结构上，例如带有代数运算的集合。在同伦理论中，重点进一步扩展到无限维空间的更高级的数学概念，其连通分支的数量恢复了潜在的数值不变量。该职业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来研究它们。该项目的教育部分不仅包括通过会议在学术界扩大接触数学的机会，而且还包括在通常的学术环境之外扩大对被监禁个人的接触：PI将在南伍兹监狱提供的学士学位课程中引入并教授一门基于探究学习的大学水平的数学选修课。该项目的主要研究目标是利用等变稳定同伦理论中的新结果来研究具有群作用的流形及其等变微分同态。稳定同伦理论，即谱的研究，在流形分类的长期程序中起着至关重要的作用；然而，对群G的G-流形的分类远不如经典的非等变故事所了解的那么多。在PI关于等变代数K-理论的部分工作和她在等变无限循环空间理论方面的合作基础工作的基础上，PI和Malkiewich建立了Waldhausen的代数K-空间理论的等变推广。受古德威利的启发，他们猜想，对于紧致光滑的G-流形M，等变A-理论从一个稳定的等变h-余弦线空间中分离出来。这个项目涉及到这些工作的重要扩展，集中在证明这一猜想并利用它来加深我们对G-流形的等变h-余边同构和微分同态的理解。第二个研究目标是应用稳定同伦技术通过推导经典不变量，即将它们提升到谱来研究经典不变量。PI将进行两个这种风格的项目。首先，PI将与Malkiewich和Moi一起研究他们定义的表示理论中的经典字符映射的派生版本。第二个项目是PI与Rovi共同领导的女性拓扑学研讨会，是将Zakharevich和Campbell最近在剪刀同余K理论谱方面的工作应用和推广到多种领域。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，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.