Hochschild-Type Invariants of Ring Spectra

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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.