Localizing invariants, motives and stable infinity-categories
This is currently my most active research theme. I study algebraic K-theory and related invariants, such as topological Hochschild homology (THH) and its variants (TP, TC, TR, TC^-), as well as stable \(\infty\)-categories more generally. Another relevant umbrella term here is "noncommutative geometry".
My most recent result in this area concerns the phenomenon of chromatic Redshift, a subtle interaction between algebraic K-theory and chromatic homotopy theory. I proved that a certain extension of this result to rigid symmetric monoidal stable \(\infty\)-categories actually fails in my paper on Chromatic Noshift.
I was particularly excited about this result because I had been thinking about the question for a while, from various angles (at some points trying to prove the opposite), and the solution ultimately crucially used some of the work I had done with my collaborators, Vova Sosnilo and Christoph Winges, where we clarified some things about the category of localizing motives, proving it is a Dwyer--Kan localization of the category of stable categories. This in turn built on and expanded upon our earlier work.
The Noshift result also built on some work that also fits in the "Algebra of presentable categories" themes, namely my description of free rigid commutative algebras. This work was based on many many discussions with Jan Steinebrunner which helped me build some intuition for these free gadgets. Together with Shaul Barkan, he's writing a book to explain more generally how free constructions work. We intend to come back to this and do a "pushout" of our methods to obtain more general results.
With Kaif Hilman, we also used the RSW work to construct equivariant norms on algebraic K-theory, a question which had caused some trouble in the past in the nonconnective setting. The results in there will be used in future work with Thomas Nikolaus and Thomas Blom in the context of polynomial functoriality of localizing invariants.
I have generally been trying to explore the structure and properties of the category of localizing motives. I settled the question of its compact generation, after Efimov had done it in the relative case (over \(\mathbb S[x]\)); and I also studied to what extent its invertibles recover Brauer groups in the two papers Localizing motives of Azumaya algebras and \(K(1)\)-local K-theory of Azumaya algebras.
I always like to think about THH, and in On endomorphisms of THH, I calculated its endomorphisms as a functor. This was inspired by earlier work of Nathalie Wahl and Angela Klamt who computed similar things.
Efimov's definition of "continuous K-theory" and more generally continuous localizing invariants spanned a lot of interest for dualizable presentable categories. In Dualizable presentable \(\infty\)-categories and its multiplicative companion Locally rigid \(\infty\)-categories, I laid down a lot of foundational material on this notion. My motivation was actually originally different, as I had been brought to thinking about dualizable categories after a question of Lior Yanovski's on the size of the "dualizable Brauer group" of a presentably symmetric monoidal \(\infty\)-category.
The additivity of traces in stable \(\infty\)-categories: I explain how the localizing property of THH can be used to recover the so-called ``additivity of traces''. This paper was subsumed by many other subsequent papers that did more things and in a better way.