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

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.    

For more information about this event, please contact Department of Mathematics at 313-577-2479 or math@wayne.edu.