Most of my research is centered around homotopy theory in a broad sense, or homotopical algebra. 

More precise current research themes are (in no particular order) : traces and dualizability; separable algebras and Brauer groups; ambidexterity; higher category theory and the theory of modes ; algebraic K-theory, noncommutative algebraic geometry and noncommutative motives. I'll explain what I mean by them in a bit more detail below.

There are also various objects/theories that I like to think about quite often, but about which I currently don't have projects or "a big picture". But I'm always happy to think and chat about them. Among them are equivariant homotopy theory, chromatic homotopy theory, tensor-triangular geometry, the homotopy theory of spaces. Of course, these interests and the themes mentioned above all interact and mix really well, and it's especially exciting when they interact. 

Traces and dualizability : The notion of the trace of an endomorphism of a dualizable object in a symmetric monoidal category has had many applications, to character theory and fixed point theory. It is remarkably general and useful at the same time. I am interested in character theories in various settings, but also more generally phenomena related to dualizability. 

One very interesting aspect of traces is how well they categorify: thus topological Hochschild homology (THH) is both the universal recipient for traces and a trace itself. I am very interested in THH for various reasons, among others its relation to K-theory, and its relation to dualizability. 

Separable algebras and Brauer groups : This research theme is part of the larger programme of exporting ideas, proofs and results from classical algebra to homotopical algebra. I am particularly interested in understanding how well we can export the notion of separability for algebras (some kind of strengthening and generalization of the notion of semi-simplicity) to homotopy coherent settings - this notion is particularly well-suited for such a study because much of the theory is controlled by what happens in the homotopy category. 

Separable algebras are in particular related to Brauer groups and Azumaya algebras, and I am also interested in studying how this notion generalizes to homotopical algebra. 

Ambidexterity : Poincaré duality relates the homology and the cohomology of a manifold. Hopkins and Lurie observed that other spaces exhibited a similar kind of duality in certain categories of localized spectra. Ambidexterity is the idea that homology (understood as a homotopy colimit) is often comparable to cohomology (understood as a homotopy limit) - from this idea arise among other things the Tate construction; but also the theory of higher semi-additivity, and higher commutative monoids. Understood this generally, this is a very diverse and very general idea. I am currently coming to grips with some of it, and am working on some projects relating it to my other research themes. 

Higher category theory and the theory of modes : I am also interested in pure (higher) category theory. One particularly exciting aspect is the theory of modes. Modes are certain presentable categories that govern what any presentable category can "look like" to some extent - understanding them is understanding the structure of presentable categories. I am often thinking about how one can try to understand and "classify" them (even if a complete classification is probably essentially hopeless), and how this can help our understanding of homotopical algebra, stability, and the macrocosm principle. 

Algebraic K-theory, noncommutative algebraic geometry and motives: Most of homological algebra and quite a bit of stable homotopy theory can be understood as being the study of stable (oo-)categories. These can be in turn understood as "noncommutative" versions of schemes, in the sense 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". 

This idea leads naturally to the theory of noncommutative algebraic geometry, and noncommutative motives. I'm very interested in the category of noncommutative motives, and variants thereof (linear variants; topological variants; A^1-invariant versions etc.). At the forefront of this category lies the study of algebraic K-theory and I am interested in studying it as well as its trace-theoretic cousins (THH, TC, TP etc.)