Maximilien Péroux

My general research is in algebraic topology. I am particulary interested in stable homotopy theory and its interplay with higher category theory. Homotopy theorists have generalized the notion of rings with multiplications that are associative and/or commutative up to higher homotopies. These objects are called A_∞ or E_∞-ring spectra. My research focuses on using higher categorical methods to study the homotopy theory of spectra with additional structures. Quasicategories (i.e. ∞-categories) provide a modern approach to homotopy theory and bypass obstacles from the rigid structure of a model category.

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There is a natural notion of coalgebras, which are spectra or modules X, in which, instead of an associative unital multiplication X ⊗X → X, we require a co-unital comultiplication X → X ⊗X, which is coassociative up to higher homotopies. These are called A∞-coalgebras. If we require cocommutativity to be up to higher homotopies, we call them E∞-coalgebras. There is also a notion of comodules over a coalgebra, dual to modules over an algebra. Topological Hochschild homology (THH) is an important invariant for algebras in spectra thanks to its connections with K-theory. Similarly, we can study the topological coHochschild homology as an invariant for coalgebras. One of my research goals is to showcase properties of this invariant.

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I also study modules and rings in spectra endowed with a group action. In equivariant stable homotopy theory, the notion of genuine G-spectrum allows the suspension by representation spheres SV which are one point compactifications of representations V of the group of equivariance G. It turns out that the equivariant setting is confronted with many challenges and my research aims to show how higher categories can solve these issues and study an equivariant version of THH.