Most of my research is centered around homotopy theory in a broad sense, or homotopical algebra.
Most of my research is centered around homotopy theory in a broad sense, or homotopical algebra.
More precise current research themes are: algebraic K-theory and noncommutative motives; separable algebras and Brauer groups; ambidexterity; higher category theory and the theory of modes; the homotopy theory of spaces. 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 and am always happy to think and chat about. Among them are equivariant homotopy theory, chromatic homotopy theory, and tensor-triangular geometry. Of course, these interests and the themes mentioned above all interact and mix really well, and it's especially exciting when they interact.
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.).
One of my main research goals is to try and understand these variants of the category of motives and find organizing principles.
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 and categorical algebra. I am also interested in more classical Brauer groups and questions about these, like the Artin-Tate conjecture or the Period-Index problem.
Higher category theory and the theory of modes : I am also interested in pure (higher) category theory. One particularly exciting aspect is the algebra of presentable categories which truly behaves like some kind of higher algebra.
Among other things, I'm interested in 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.
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. Parts of it also relate to extremely fundamental questions in homotopy theory which I want to see answered.
The homotopy theory of spaces: The original motivation for much, if not all of homotopy theory is specifically the homotopy theory of spaces, nowadays also known as oo-groupoids. This is the part of homotopy theory that is the closest to algebraic topology.
When one is in luck, some of the fancy tools developed for other purposes in homotopy theory can come back and strike in that area, and I'm particularly interested in applications of K-theoretic methods in related problems.