PhD defence by Maxime Ramzi

Title: Separability in homotopy theory and topological Hochschild homology

Abstract: The first part of the thesis is concerned with the notion of separable algebras in homotopy theory, specifically in the context of monoidal stable ∞-categories. Separable algebras are a common generalization of étale algebras in the commutative setting and Azumaya algebras in the noncommutative setting, and in Part I, I study their foundational properties: I prove rigidity results with respect to the homotopy category which generalize previously known rigidity results, and I try to bring the well-developed theory from classical algebra to homotopical algebra.
        The second part is devoted to the study of topological Hochschild homology (THH) and related invariants. In the first chapter of Part II, I explain how to rephrase a theorem of Dundas and McCarthy relating THH and algebraic K-theory in terms of Kaledin and Nikolaus’ trace theories, and how to use this formalism to extend the theorem to nonconnective ring spectra and their bimodules, as well as to more general invariants than algebraic K-theory. In the second chapter of the second part, I explain how to use this result to compute invariants of THH itself, such as its endomorphism ring spectrum, and variants thereof.

Thesis for download

Advisors: Jesper Grodal, University of Copenhagen
Markus Land, L.M.U. Munich

Assessment committe:
Nathalie Wahl (chair), University of Copenhagen
Benjamin Antieau, Northwestern University
Alexander Efimov, Hebrew University of Jerusalem

This work was supported by the Danish National Research Foundation
through the Copenhagen Centre for Geometry & Topology
(CPH-GEOTOP-DNRF151). Part of the work was also supported by
the Swedish Research Council (grant no. 2016-06596) during the
research program “Higher algebraic structures in algebra, topology
and geometry” held at Institut Mittag–Leffler, Sweden in the spring
of 2022.