In this project, I prove that the category of dualizable presentable modules over a base presentably symmetric monoidal category V, and internal left adjoints between them, is itself a presentable category. I take this as an opportunity to survey the formal foundations of the theory of dualizable categories and relevant notions (compact maps, continuous K-theory), as well as that of rigid symmetric monoidal categories. 

In this project, joint with Thomas Nikolaus, we expand on the last section of the paper K-theory and polynomial functors (Barwick, Glasman, Mathew, Nikolaus) to give a complete proof that TR admits a refinement to a functor on Cat^poly, the category of stable categories and polynomial functors. Our proof also works for TC. 

In this project, I study the categorical version of the Dundas-McCarthy theorem, stating that the first Goodwillie derivative of K-theory is THH. Inspired by Raskin's sketch in On the Dundas-Goodwillie-McCarthy theorem, and Nikolaus's announced proof in his work on trace theories, I prove, using trace theory methods, that the derivative of K-theory is indeed THH. I prove this by studying trace theories - the main results are as follows: first, that trace theories in the sense of Kaledin-Nikolaus agree with shadows in the sense of Ponto-Shulman, a result easily motivated at an uncoherent level, of which I give a homotopy-coherent proof; and second, that the category of fiberwise cocontinuous shadows on (Pr^L)_{st, omega} with values in spectra (equivalently, of trace theories) is equivalent to the category of spectra with S^1-action, the equivalence being implemented by evaluation at Sp. Among other things, this resolves a conjecture of Hess-Rasekh about the existence of THH as a fully coherent shadow; and it extends the Dundas-McCarthy theorem to arbitrary finitary localizing invariants. 

In this paper, I propose a definition of what the "universal higher groupoid cardinality" is, in principle generalizing Baez-Dolan groupoid cardinality and providing a higher analogue thereof. However, I also prove that this definition turns out to trivialize and boils down to the discrete, usual groupoid cardinality. I deduce this from general results in higher semiadditivity theory following Harpaz, together with the following technical result: the free rational commutative monoid on a point is discrete. 

In this project, joint with Thomas Blom and based on ideas of Thomas Nikolaus, we study polynomial functoriality for localizing invariants. Specifically, we prove the following conjecture by Nikolaus: for any functor E : Cat^poly --> Spaces whose restriction to Cat^ex is a localizing invariant admits a canonical refinement to a functor Cat^poly --> Sp^poly to the category of "spectra and polynomial maps".

In this project, joint with Niko Naumann and Luca Pol, we compute the category of commutative separable algebras in genuine G-spectra for G a finite p-group : we prove that they are all "standard", i.e. of the form D(X_+) for some finite G-set X. We also prove that there exist some nonstandard ones for some non-p-groups, such as Sigma_3. 

In this project, joint with Vova Sosnilo and Christoph Winges, which is a kind of sequel to our previous paper on categorification of spectra, we show that every localizing motive arises as the motive of a stable category. More is true: we prove that the universal localizing invariant Cat^perf --> Mot_loc is a Dwyer-Kan localization. 

In this project which is still in its preliminary stages, I compute the endomorphism ring spectrum of the functor THH, defined on different variants of the category Cat_st (such as Alg_O(Cat_st) for any operad O).