I work in homotopy theory. More precisely my research generally concerns the following themes: algebraic K-theory and related invariants, homotopical algebra, categorical algebra, ambidexterity and algebraic topology. I also like to keep an eye on developments in tt-geometry, chromatic homotopy theory, representation theory, motivic homotopy theory, Goodwillie calculus and arithmetic geometry.

Algebraic K-theory and related invariants: These days especially, a lot of my research is focused on algebraic K-theory and the related theory of noncommutative motives. I am interested in foundational questions that help us actually work with motives, and understand the global structure of these objects; though I also sometimes focus on more concrete calculational questions.

I generally focus more on the categorical aspect of these topics, and so a large part of my study of K-theory is actually a study of stable (\(\infty\)-)categories, or equivalently noncommutative algebraic geometry à la Kontsevich. The naming comes from the idea that any scheme \(X\) has an associated stable category of perfect modules \(\mathrm{Perf}(X)\), and that many tools and ideas for studying schemes can be extended to stable categories through this lens, pretending that any stable category \(C\) is that of perfect modules over some "noncommutative scheme". One of my recent interests is seeing how true this really is, by studying various forms of the "geometricity conjecture", which basically says that all very nice "noncommutative schemes" (technically, smooth and proper, and over an algebraically closed field) are close to honest schemes.

Apart from K-theory, one of my favourite invariants is topological Hochschild homology, THH, and I often try to say new things about it.

Homotopical algebra: This is sometimes called "higher" algebra, but I find the term "homotopical" more descriptive. For me, this is essentially the study of the algebra of the category of spectra, and more generally of ((symmetric) monoidal) stable \(\infty\)-categories, where concepts of classical algebra (coming from e.g. representation theory or commutative algebra) are interpreted and studied in this "new" type of algebra. As a main example, I studied the notion of separability in homotopical algebra, on the way to understanding the notion of étale morphisms, a centerpiece of classical algebra.

I am also interested specifically in new kinds of geometries and in Brauer groups, which is related to, and feeds into, the theme below.

Categorial algebra: One particularly exciting aspect of pure (higher) category theory is the algebra of presentable categories which truly behaves like some kind of higher algebra. One strength of category theory is its self-applicability, so that studying algebra in higher categories also gives insights on the algebra of higher categories. This area is rapidly developing, with an impetus coming from Efimov's definition of continuous K-theory which led to a need for an understanding of this categorical algebra, as well as from Scholze's work with, among others, Aoki and Stefanich. Scholze and Stefanich for example recently introduced the notion of a Gestalt, some kind of higher categorical gadget whose definition comes from (quote) "an extreme Tannakian perspective".

Ambidexterity: Following Hopkins and Lurie's work on the \(K(n)\)-local category, there has been a certain industry devoted to studying the ambidexterity phenomenon, in which cohomology and homology are canonically identified. These ideas have participated in exciting developments in chromatic homotopy theory (such as the solution of (part of) the redshift conjectures, or the telescope conjecture), and are also related to fundamental questions in the homotopy theory of spaces that I really want to see answered.

Algebraic topology: The original motivation for much of homotopy theory is algebraic topology, a large subset of which is the homotopy theory of spaces. Though my research is typically more focused on stable homotopy theory and algebraic K-theory, I maintain an ongoing interest for these fundamental questions, and it sometimes happens that some of these "fancy" tools can have some applications in algebraic topology. I have, for example, a particular obsession for the problem of composability of Becker--Gottlieb transfers; and I also hope that some of the relevant methods are helpful in some Farrel-Jones type questions.

"Homotopical algebra" studies the algebra one gets when one thinks of numbers this way. Just like in classical algebra, where there are many different number systems with vastly different properties (the natural numbers, or rational numbers, or real numbers, or complex numbers, where the equation x^2+1 = 0 has a solution!), there are even more different "homotopical number systems". Historically, these arose as ways of studying shapes, but they became a subject of study in their own right.

I am interested in understanding these "homotopical number systems" : what they look like, how they behave and relate to one another, how they relate to classical number systems and finally what light they shed on topology, the study of shapes. So I like to explore these new worlds, filled with peculiar objects and understand examples and general behaviours.