Hochschild-Type Invariants of Ring Spectra

系统，其中一个可以添加和乘以下的基本规则的算术被称为环。 整数是最基本的例子，但其他具有非常不同性质的环出现在所有数学领域，无论是纯数学还是应用数学。 50多年来，环研究的一个主要目标是理解它们的代数K-理论。 这是一种通过拓扑学来思考环的方法，拓扑学是对空间形状的定性研究。 该奖项支持的研究使用跟踪方法将代数K理论与更容易理解的事物联系起来，如Hochschild同调及其变体。 此外，如果用拓扑环（称为环谱）来代替环，那么Hochschild型不变量就成为原环的代数K理论的更好近似，而且这个领域最近有了很大的发展。主要研究者已经使用这些方法计算了代数K理论，并正在研究其他方法。 理解环谱的Hochschild型不变量也为通过与传统离散环的类比对其进行分类提供了很好的工具。 该项目为研究生提供研究培训机会，并将鼓励和指导研究生和本科生一级的年轻女数学家。迄今为止，主要研究员的工作主要是计算拓扑Hochschild同源性，Hochschild同源性的环谱版本。 事实证明，这是一个特殊的情况下，与一个圆，一个更一般的想法张紧任何拓扑空间与环谱，并查看圆计算在这个更广泛的背景下已经使一个更好地理解拓扑Hochschild同调计算在某些情况下。 就像环与圆的张量积有助于理解代数K-理论一样，环与高维环面的张量积有助于理解迭代代数K-理论，因此人们对理解环（或环谱）与空间之间的相互作用感兴趣，这使得一些这样的张量积计算比其他的要容易得多。 为了获得代数K理论计算，这些项目将使用Nikolaus和Scholze的新公式解决拓扑循环同源计算。该奖项反映了NSF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。

Systems where one can add and multiply following the basic rules of arithmetic are called rings. The integers are the most basic example, but other rings with very diverse properties occur in all areas of mathematics, pure and applied. For more than fifty years, a major goal in the study of rings has been to understand their algebraic K-theory. This is a way of thinking about rings via topology, the qualitative study of the shape of spaces. The research supported by this award uses the trace method to connect algebraic K-theory to things which are easier to understand, like Hochschild homology and its variants. It turns out, moreover, that if one replaces the rings by topological versions of rings that are called ring spectra, the Hochschild-type invariants become even better approximations of algebraic K-theory of the original rings, and this field has seen a lot of recent development. The principal investigator has made calculations of algebraic K-theory using these methods, and is working on others. Understanding the Hochschild-type invariants of ring spectra also gives good tools for classifying them by analogies with traditional discrete rings. This project provides research training opportunities for graduate students and will encourage and mentor young women mathematicians at the graduate and undergraduate levels.Most of the principal investigator's work to date has been in calculations of topological Hochschild homology, the ring spectrum version of Hochschild homology. It turns out that this is a special case, associated to a circle, of a more general idea of tensoring any topological space with a ring spectrum, and viewing the circle calculation in this wider context has already enabled a better understanding of topological Hochschild homology calculations in some cases. Just as the tensor product of a ring with a circle is instrumental in understanding algebraic K-theory, the tensor product of a ring with a higher dimensional torus is instrumental in understanding iterated algebraic K-theory, and so one is interested in understanding the interaction between rings (or ring spectra) and spaces that makes some such tensor product calculations much easier than others. To get algebraic K-theory calculations, these projects will address topological cyclic homology calculations using Nikolaus and Scholze's new formulation.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.