Algebraic K-Theory and Equivariant Homotopy Theory

本课题将以代数k理论为中心，研究拓扑学和代数的基础对象。代数k理论是一个不变量，可以应用于研究数学多个领域的基本对象。特别是，代数k理论可以用来研究代数中称为环的基本对象的性质。虽然高等代数k理论在40多年前就被定义了，但计算进展缓慢。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，人们对k理论计算的兴趣依然浓厚。代数k群在代数几何、数论、拓扑学和其他数学领域有重要的应用。这些应用中的许多都是相当令人惊讶的，代数k理论在数学领域中的作用引起了人们对这一主题的极大兴趣。近年来，代数拓扑领域的进展使得研究代数k理论中以前被认为难以接近的问题成为可能。该项目的目标是利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。除了数学研究目标之外，该项目还包括本科和研究生教育、本科研究、会议组织以及努力增加妇女和代表性不足群体在数学领域的参与。本课题利用等变稳定同伦的工具研究代数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，对于k理论的计算，有一种同伦理论的方法已经相当富有成果。尽管代数k理论本身不是一个等变对象，但等变稳定同伦理论的工具已被证明对k理论计算非常有用。本项目探讨了等变同伦理论、代数k理论和相关不变量（如拓扑Hochschild同伦）之间的复杂关系。该项目将产生新的k理论计算，并加深我们对不变量和用于进行此类计算的工具的理解。本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：一，利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获得的代数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。

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.