Algebraic K-Theory and Equivariant Homotopy Theory

代数 K 理论和等变同伦理论

基本信息

批准号：
1810575
负责人：
Teena Gerhardt
金额：
$ 21.72万
依托单位：
Michigan State University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2022-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1810575&HistoricalAwards=false
关键词：
Algebraic Theory Equivariant Homotopy

项目摘要

This project will study foundational objects in topology and algebra, centering on the study of algebraic K-theory. 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 40 years ago, computational progress has been slow. Even for many basic rings the K-theory groups still aren't known today. Despite the difficulties, interest in K-theory computations remains strong. Algebraic K-groups have significant applications to algebraic geometry, number theory, topology, and other mathematical areas. Many of these applications are quite surprising, and the role of algebraic K-theory across mathematical fields drives a great interest in the subject. In recent years, advances in the field of algebraic topology have made it possible to study questions in algebraic K-theory which were previously thought to be inaccessible. 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 study of algebraic K-theory and related invariants. In addition to the mathematics research goals, the project also includes work in undergraduate and graduate education, undergraduate research, conference organization, and efforts to increase the participation of women and underrepresented groups in mathematics. This project uses the tools of equivariant stable homotopy to study algebraic K-theory and related invariants. Algebraic K-theory is an invariant of a ring which is generally very difficult to compute. However, there is a homotopy theoretic approach to K-theory computations that has been quite fruitful. Despite the fact that algebraic K-theory is not itself an equivariant object, the tools of equivariant stable homotopy theory have proven very useful for K-theory computations. This project explores the intricate relationship between equivariant homotopy theory, algebraic K-theory, and related invariants such as topological Hochschild homology. The project will produce new K-theory computations as well as deepening our understanding of the invariants and tools used to make such computations. Further, this research will provide important foundations, structures, and examples for further work in equivariant stable homotopy theory. Specific research goals of the project are organized into three broader objectives: One, use recent results and new methods from equivariant stable homotopy theory to compute algebraic K-theory groups which were previously inaccessible. Two, develop the theory around related invariants such as topological Hochschild homology and topological coHochschild homology. Three, define and study equivariant algebraic structures that arise in K-theory computations.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.

该项目将研究拓扑和代数的基础对象，重点是代数K理论的研究。代数 K 理论是一个不变量，可应用于研究多个数学领域的基本对象。特别是，代数 K 理论可用于研究代数中称为环的基本对象的属性。尽管高等代数 K 理论已在 40 多年前定义，但计算进展缓慢。即使对于许多基本环，K 理论群至今仍不为人所知。尽管存在困难，人们对 K 理论计算的兴趣仍然很浓厚。代数 K 群在代数几何、数论、拓扑和其他数学领域有着重要的应用。其中许多应用都非常令人惊讶，代数 K 理论在数学领域中的作用引起了人们对该学科的极大兴趣。近年来，代数拓扑领域的进步使得研究代数K理论中以前被认为难以解决的问题成为可能。该项目的目标是使用代数拓扑工具不仅产生新的代数 K 理论计算，而且开发框架和理论以促进代数 K 理论和相关不变量的未来研究。除了数学研究目标外，该项目还包括本科生和研究生教育、本科生研究、会议组织以及增加女性和弱势群体在数学领域的参与的努力。该项目使用等变稳定同伦的工具来研究代数K理论和相关不变量。代数 K 理论是环的不变量，通常很难计算。然而，有一种用于 K 理论计算的同伦理论方法已经非常富有成效。尽管代数 K 理论本身并不是等变对象，但事实证明，等变稳定同伦理论的工具对于 K 理论计算非常有用。该项目探讨了等变同伦理论、代数 K 理论和拓扑 Hochschild 同调等相关不变量之间的复杂关系。该项目将产生新的 K 理论计算，并加深我们对用于进行此类计算的不变量和工具的理解。此外，这项研究将为等变稳定同伦理论的进一步研究提供重要的基础、结构和实例。该项目的具体研究目标分为三个更广泛的目标：第一，利用等变稳定同伦理论的最新成果和新方法来计算以前无法实现的代数K理论群。第二，围绕拓扑Hochschild同调和拓扑coHochschild同调等相关不变量发展理论。第三，定义和研究 K 理论计算中出现的等变代数结构。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

期刊论文数量（5）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Topological coHochschild homology and the homology of free loop spaces

拓扑coHochschild同调与自由环空间同调

DOI：
10.1007/s00209-021-02879-4
发表时间：
2022-05
期刊：
Mathematische Zeitschrift
影响因子：
0.8
作者：
Bohmann, Anna Marie;Gerhardt, Teena;Shipley, Brooke
通讯作者：
Shipley, Brooke

The Witt vectors for Green functors

格林函子的维特向量

DOI：
10.1016/j.jalgebra.2019.07.014
发表时间：
2019-11
期刊：
Journal of Algebra
影响因子：
0.9
作者：
Blumberg, Andrew J.;Gerhardt, Teena;Hill, Michael A.;Lawson, Tyler
通讯作者：
Lawson, Tyler

Topological Cyclic Homology Via the Norm

通过范数的拓扑循环同调

DOI：
发表时间：
2018-10
期刊：
Documenta mathematica
影响因子：
0.9
作者：
Angeltveit, Vigleik;Blumberg, Andrew J.;Gerhardt, Teena;Hill, Michael A.;Lawson, Tyler;Mandell, Michael A.
通讯作者：
Mandell, Michael A.

A Shadow Perspective on Equivariant Hochschild Homologies

等变 Hochschild 同调的影子视角