CAREER: Applications of equivariant homotopy theory to manifolds

在许多数学领域中，一个强有力的思想是通过将几何对象（如多边形）与代数不变量（例如在连续变形下不变化的数字）联系起来来研究几何对象（如多边形）。一个早期的例子是欧拉，他注意到对于一个多边形曲面，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，埃米·诺特将重点从数值不变量转移到它们背后的代数结构，比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的数学概念，其连通分量的数量恢复了潜在的数值不变量。由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调世界，并从更丰富的角度研究它们。该项目的教育部分不仅包括通过学术会议在学术世界中扩大数学的接触，而且还包括在通常的学术环境之外为被监禁的人提供数学：PI将在南伍兹监狱提供的学士学位课程中引入并教授基于探究的大学水平数学选修课。本课题的主要研究目标是利用等变稳定同伦理论的新结果来研究具有群作用的流形及其等变微分同胚。稳定同伦理论，光谱的研究，在流形分类的长期计划中起着至关重要的作用；然而，G群的G流形的分类远不如经典的非等变故事那样容易理解。基于PI先前在等变代数k理论方面的工作和她在等变无限环空间理论方面的合作基础工作，将合适的g范畴转化为g谱的机制，PI和Malkiewich已经建立了Waldhausen的空间代数k理论的等变推广。受Goodwillie设想的启发，他们推测，对于紧致光滑g流形M，等变a理论分离出一个由等变h协阵组成的稳定空间。这个项目是对这些努力的一个重要扩展，重点是证明这个猜想，并利用它来进一步理解g流形的等变h协象和微分同态。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，即提升到谱来研究经典不变量。PI将进行两个这样的项目。首先，PI将与Malkiewich和Moi一起，从他们定义的表征理论中研究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“拓扑女性研讨会”的项目，是将zakharrevich和Campbell关于剪刀同余k理论谱的最新研究应用并扩展到流形。该奖项反映了美国国家科学基金会的法定使命，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，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.