Maxime Ramzi PhD defence
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.