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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.

本课题以代数K理论为中心，研究拓扑学和代数学的基础对象。代数K-理论是一个不变量，它可以应用于研究数学的几个领域的基本对象。特别是，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理论在40多年前就被定义了，但计算的进展一直很缓慢。即使对于许多基本环，K-理论群今天仍然不为人所知。尽管困难重重，但对K理论计算的兴趣仍然很强。代数K-群在代数几何、数论、拓扑学和其他数学领域有着重要的应用。许多这些应用是相当令人惊讶的，代数K理论在数学领域的作用推动了人们对这一主题的极大兴趣。近年来，在代数拓扑领域的进展，使人们有可能研究问题的代数K-理论，以前被认为是无法访问的。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，而且还开发框架和理论，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，该项目还包括本科和研究生教育、本科研究、会议组织以及增加女性和代表性不足群体参与数学的努力。本计画以等变稳定同伦为工具，研究代数K-理论及相关不变量。代数K-理论是环的不变量，通常很难计算。然而，有一个同伦理论的方法来K理论计算已经相当富有成效。尽管代数K-理论本身并不是一个等变对象，但等变稳定同伦理论的工具已经被证明对K-理论计算非常有用。这个项目探讨了等变同伦理论，代数K-理论和相关的不变量，如拓扑Hochschild同调之间的复杂关系。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，这项研究将提供重要的基础，结构和例子，为进一步的工作等变稳定同伦理论。该项目的具体研究目标分为三个更广泛的目标：一，使用等变稳定同伦理论的最新结果和新方法来计算以前无法访问的代数K理论群。二是围绕拓扑Hochschild同调和拓扑coHochschild同调等相关不变量展开理论。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，并被认为值得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来支持。

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 Cyclic Homology Via the Norm

通过范数的拓扑循环同调

DOI：
发表时间：
2018
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.

The Witt vectors for Green functors

格林函子的维特向量