Algebraic K-theory and Equivariant Homotopy Theory

这个项目的主题是利用等变稳定同伦理论的工具来研究代数K-理论，特别是奇异环和滤环的K-理论。虽然代数K-理论的定义在本质上不是等变的，但等变稳定同伦理论的工具已被证明对K-理论的计算是有用的。特别是，一种卓有成效的方法利用了拓扑Hochschild同调(THH)的等变结构来计算代数K-理论。在某些奇异环的情况下，该方法将K-理论的计算归结为按圆的实表示环分级的等变稳定同伦群的计算。要计算K-理论，需要确定产生哪些等变同伦群，计算这些群，然后将它们组合起来恢复K-理论。这些步骤中的每一个都是困难的，只有在少数情况下才能被理解。这个项目试图为各种具体的K理论计算解决这些问题，以及定义一个抽象的代数对象，包含在这些计算中出现的结构。该项目的其他具体目标包括为滤环的K理论开发一种方法，并回答几个关于THH结构的问题。代数K-理论是一种不变量，它可以应用于从数学的多个领域研究基本对象。特别地，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-理论早在30多年前就被定义了，但计算进展缓慢。事实上，即使对于一些非常基本的环，代数K-理论仍然是未知的。然而，K-理论计算在许多数学领域都有重要的应用：代数数论、流形分类、动机同伦理论、L函数的特殊值等，这些重要计算的一种方法在于代数拓扑学领域，更具体地说，是研究等变同伦理论。这个项目的目标是使用这些工具不仅产生新的代数K-理论计算，而且开发框架和理论，以促进未来的计算。

The theme of this project is to use the tools of equivariant stable homotopy theory to study algebraic K-theory, particularly the K-theory of singular and filtered rings. Although the definition of algebraic K-theory is not inherently equivariant, the tools of equivariant stable homotopy theory have proven useful for K-theory computations. In particular, one fruitful approach exploits the equivariant structure of topological Hochschild homology (THH) to compute algebraic K-theory. In the case of certain singular rings, this approach reduces the computation of K-theory to the computation of equivariant stable homotopy groups of THH, graded by the real representation ring of the circle. To compute K-theory one needs to determine which equivariant homotopy groups arise, compute these groups, and then assemble them to recover K-theory. Each of these steps is difficult and understood only in a small number of cases. This project seeks to address these issues for various specific K-theory computations, as well as defining an abstract algebraic object embodying structures that arise in these computations. Other specific goals of the project include developing an approach for the K-theory of filtered rings, and answering several questions about the structure of THH. 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 30 years ago, computational progress has been slow. Indeed, even for some very basic rings, the algebraic K-theory is still not known. K-theory computations, however, have important applications to many areas of mathematics: algebraic number theory, classification of manifolds, motivic homotopy theory, special values of L-functions, etc. An approach to these important computations lies in the field of algebraic topology, and more specifically, in the study of equivariant homotopy theory. The goal of this project is to use these tools to not only produce new algebraic K-theory computations, but also to develop the framework and theory to facilitate future computations.