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

代数和拓扑学这两个数学领域是紧密交织在一起的。事实上，代数中的工具可以用来研究拓扑中的对象，反之亦然。这种深度相互作用的一个例证是代数k理论。代数k论是环的不变量，环是代数中的基本对象。由于代数k理论在代数几何、数论和拓扑学领域的重要应用，它引起了人们极大的兴趣。虽然代数k理论很难计算，并且仍然存在许多悬而未决的问题，但有一种使用拓扑工具的强大方法。近年来，代数拓扑学取得了令人兴奋的进展，这使得研究代数k理论中以前被认为难以接近的问题成为可能。这个项目的目标是产生新的代数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量，称为拓扑Hochschild同调。本项目的另一个目标是进一步发展拓扑Hochschild同调变体的框架和理论，并研究其在其他数学领域的应用。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、会议组织以及支持妇女和其他代表性不足的群体参与数学的工作。本课题利用等变稳定同伦的工具研究代数k理论和拓扑Hochschild同伦。代数k理论是环的不变量，通常很难计算。迹方法是研究代数k理论的一种富有成效的方法，它通过拓扑Hochschild同调和拓扑循环同调等更易于计算的理论来逼近代数k理论。迹法方法依赖于等变稳定同伦理论中的工具。本课题探讨了等变同伦理论、代数k理论和拓扑Hochschild同伦之间的复杂关系。该项目的具体研究目标被组织为三个广泛的目标：一，利用迹方法和等变稳定同伦理论的最新发展来计算以前无法获得的代数k理论群。二是利用等变同伦理论研究了代数和拓扑的Hochschild同调，如扭曲拓扑Hochschild同调和实拓扑Hochschild同调。三、研究拓扑Hochschild同调理论在几何和低维拓扑问题中的应用。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。

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.