Mathematics Topology Seminar - Jay Shah

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Date: February 25, 2020
Time: 2:00 p.m. - 3:00 p.m.
Location: Faculty/Administration #1140 | Map
656 W. Kirby
Detroit, MI 48202
Category: Seminar

Speaker: Jay Shah, University of Notre Dame

Title: Two theories of real cyclotomic spectra

Abstract: The topological Hochschild homology $THH(R)$ constitutes a powerful and well-studied invariant of an associative ring $R$. As originally shown by Bokstedt, Hsiang and Madsen, $THH(R)$ admits the elaborate structure of a cyclotomic spectrum, whose formulation depends upon equivariant stable homotopy theory. More recently, inspired by considerations in p-adic Hodge theory, Nikolaus and Scholze demonstrated (under a bounded-below assumption) that the data of a cyclotomic spectrum is entirely captured by a system of circle-equivariant Frobenius maps, one for each prime p. They also give a formula for the topological cyclic homology $TC(R)$ directly from these maps.

The purpose of this talk is to extend the work of Nikolaus and Scholze in order to accommodate the study of real topological Hochschild homology $THR$, which is a $C_2$-equivariant refinement of $THH$ defined for an associative ring with an anti-involution, or more generally an $E_sigma$-algebra in $C_2$-spectra. The key idea is to make use of the $C_2$-parametrized Tate construction. This is joint work with J.D. Quigley and is based on the arXiv preprint 1909.03920.    


Department of Mathematics