CAREER: Equivariant Homotopy and Algebraic K-Theory

该项目的主要研究目标是利用等变稳定同伦理论的工具来研究代数K-理论及其相关不变量。虽然代数K-理论的定义在本质上不是等变的，但等变稳定同伦理论的工具已被证明对K-理论的计算是有用的。特别是，一种卓有成效的方法利用了拓扑Hochschild同调(THH)的等变结构来计算代数K-理论。在某些情况下，K-理论的计算可以归结为THH的等变稳定同伦群的计算，按圆的实表示环分级。确定需要计算的群，计算它们，并组装群以恢复代数K-理论，这些都是该方法的重要组成部分。这个项目的目标包括为各种具体的K理论计算完成这些步骤，以及定义包含在这些计算中出现的等变结构的抽象代数对象。这个研究项目的其他目标包括描述更高拓扑的Hochschild同调的结构，以及开发和探索THH的新的等变模型的应用。代数K-理论是一个不变量，可以用于从多个数学领域研究基本对象。特别地，代数K-理论可以用来研究代数中基本对象的性质，称为环。虽然高等代数K-理论早在30多年前就被定义了，但计算进展缓慢。事实上，即使对于一些非常基本的环，代数K-理论仍然是未知的。然而，K理论计算在数学的许多领域都有重要的应用。代数K-理论是代数拓扑、代数几何和数论的交集，在动机同伦理论、流形分类、L函数的特殊值等方面都有应用。本项目的一个目标是利用代数拓扑学中的工具，不仅产生新的代数K-理论计算，而且发展框架和理论，以便于将来的计算。该项目还包括几个教育和辅导方案，以招募和留住妇女和其他在数学领域任职人数不足的群体为中心。针对本科生的计划包括开发女性数学课程，创建科学职业讲座系列，以及针对职业生涯早期本科生的本科生研究机会。对于研究生，职业指导研讨会将既面向密歇根州立大学的学生，也将更广泛地面向国际代数拓扑界的学生和博士后，通过即将到来的为期一学期的项目的静修。还包括为来自代表性不足群体的K-12学生提供机会，以及为科学、数学和工程领域的女性教职员工提供的计划。此外，格哈特提出了一个研究项目，解决了为什么许多成功的女性数学家选择离开学术数学的问题。

The primary research goal of the proposed project is to use the tools of equivariant stable homotopy theory to study algebraic K-theory and related invariants. Although the definition of algebraic K-theory is not inherently equivariant, the tools of equivariant stable homotopy theory have proven useful for K-theory computations. In particular, one fruitful approach exploits the equivariant structure of topological Hochschild homology (THH) to compute algebraic K-theory. In some cases K-theory computations can be reduced to the computation of equivariant stable homotopy groups of THH, graded by the real representation ring of the circle. Determining which groups need to be computed, computing them, and assembling the groups to recover algebraic K-theory are all important components of this approach. Goals of this project include completing these steps for various specific K-theory computations, as well as defining abstract algebraic objects embodying equivariant structures arising in such computations. Other goals of this research program include describing the structure of higher topological Hochschild homology, and developing and exploring applications of a new equivariant model for THH.Algebraic K-theory is an invariant which can be applied to study basic objects from several fields of mathematics. In particular, algebraic K-theory can be used to study properties of fundamental objects in algebra, called rings. Although higher algebraic K-theory was defined more than 30 years ago, computational progress has been slow. Indeed, even for some very basic rings, the algebraic K-theory is still not known. K-theory computations, however, have important applications to many areas of mathematics. Algebraic K-theory lies in the intersection of algebraic topology, algebraic geometry, and number theory, with applications to motivic homotopy theory, classification of manifolds, special values of L-functions, etc. A goal of this project is to use tools from algebraic topology to not only produce new algebraic K-theory computations, but also to develop the framework and theory to facilitate future computations. This project also includes several educational and mentoring programs centered around the recruitment and retention of women and other underrepresented groups in mathematics. Programs aimed at undergraduate students include the development of a Women in Mathematics course, the creation of a Careers in Science lecture series, and undergraduate research opportunities aimed at early-career undergraduates. For graduate students, career mentoring seminars will be developed both for students at Michigan State University, and more broadly for students and post-docs in the international Algebraic Topology community through a retreat at an upcoming semester-long program. Also included are opportunities for K-12 students from underrepresented groups, as well as a program for female faculty members in science, mathematics, and engineering. Additionally, Gerhardt proposes a research project addressing the question of why many successful female mathematicians choose to leave academic math.