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 (oo-)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 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".
Since localizing invariants in the sense of Blumberg-Gepner-Tabuada give rise to interesting cohomology theories on actual schemes (and in other contexts, such as analytic geometry), this research theme also leads to interactions and collaborations with algebraic and arithmetic geometers.
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 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.
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.