Algebraic K-Theory, Topological Hochschild Homology, and Equivariant Homotopy Theory

代数 K 理论、拓扑 Hochschild 同调和等变同伦理论

基本信息

批准号：
2104233
负责人：
Teena Gerhardt
金额：
$ 23.28万
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：
2021
资助国家：
美国
起止时间：
2021-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104233&HistoricalAwards=false
关键词：
Algebraic Theory Topological Hochschild Homology

项目摘要

The mathematical fields of algebra and topology are deeply intertwined. Indeed, tools from algebra can be used to study objects in topology, and vice versa. One illustration of this deep interaction is through algebraic K-theory. Algebraic K-theory is an invariant of rings, fundamental objects in algebra. There is great interest in algebraic K-theory due to its significant applications in the fields of algebraic geometry, number theory, and topology. While algebraic K-theory is difficult to compute, and many open questions remain, there is a powerful approach using tools from topology. In recent years, exciting advances in 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 produce new algebraic K-theory computations. A key step in computing algebraic K-theory is studying a related invariant called topological Hochschild homology. Another goal of this project is to further develop the framework and theory around variants of topological Hochschild homology, and study applications to several other areas of mathematics. In addition to the mathematics research goals, the project also includes work in undergraduate and graduate education, undergraduate research, conference organization, and efforts to support the participation of women and other underrepresented groups in mathematics. This project uses the tools of equivariant stable homotopy to study algebraic K-theory and topological Hochschild homology. Algebraic K-theory is an invariant of a ring which is generally very difficult to compute. A fruitful approach to the study of algebraic K-theory is the trace method approach, which approximates algebraic K-theory by theories that are more computable, such as topological Hochschild homology and topological cyclic homology. The trace method approach relies on tools from equivariant stable homotopy theory. This project explores the intricate relationship between equivariant homotopy theory, algebraic K-theory, and topological Hochschild homology. Specific research goals of the project are organized into three broad objectives: One, use recent developments in trace methods and equivariant stable homotopy theory to compute algebraic K-theory groups which were previously inaccessible. Two, use equivariant homotopy theory to study algebraic and topological Hochschild homologies such as twisted topological Hochschild homology and Real topological Hochschild homology. Three, study applications of topological Hochschild homology theories to questions in geometry and low-dimensional topology.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.

代数和拓扑的数学领域与之交织在一起。实际上，代数的工具可用于研究拓扑中的对象，反之亦然。这种深度互动的一个例证是通过代数K理论。代数K理论是戒指的不变，代数的基本对象。由于其在代数几何，数字理论和拓扑结构中的重要应用，因此对代数K理论引起了极大的兴趣。虽然代数K理论很难计算，并且仍然存在许多开放问题，但使用拓扑工具有一种强大的方法。近年来，代数拓扑的令人兴奋的进步使得在代数K理论中研究问题以前被认为是无法访问的。该项目的一个目标是生成新的代数K理论计算。计算代数K理论的关键步骤是研究一个相关的不变式，称为拓扑Hochschild同源性。该项目的另一个目标是进一步开发围绕拓扑Hochschild同源性变体的框架和理论，并研究其他几个数学领域的应用。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，会议组织的工作，以及支持妇女和其他代表性不足的数学群体的参与。该项目使用稳定同型的工具来研究代数K理论和拓扑Hochschild同源性。代数K理论是一个不变的环，通常很难计算。对代数K理论研究的一种富有成果的方法是痕量方法方法，该方法通过更可计算的理论（例如拓扑性Hochschild同源性和拓扑周期性同源性）近似代数K理论。痕量方法方法依赖于稳定同型理论的工具。该项目探讨了均质同义理论，代数K理论和拓扑Hochschild同源性之间的复杂关系。该项目的具体研究目标分为三个广泛的目标：一个，使用痕量方法中的最新发展和模棱两可的稳定同型理论来计算以前无法访问的代数K理论组。第二，使用均等均匀理论来研究代数和拓扑霍基柴尔德同源物，例如扭曲的拓扑霍基柴尔德同源性和真实的拓扑霍基柴尔德同源性。第三，拓扑Hochschild同源性理论的研究应用于几何学和低维拓扑的问题。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来支持。