On this page, you will find my publications, preprints and current research projects organized by research themes (the same paper may appear in different themes). If you want a chronological list, just click here.
On this page, you will find my publications, preprints and current research projects organized by research themes (the same paper may appear in different themes). If you want a chronological list, just click here.
Part of my research focuses on the theory of localizing invariants such as K-theory, THH and its relatives (TR, TC, TP) and more globally the study of stable oo-categories.
Publications:
A multiplicative comparison of MacLane homology and topological Hochschild homology, with Geoffroy Horel. arXiv link , paper on my site (last updated 05/02/21). Annals of K-theory, Vol. 6 (2021), No.3, 571-605
Preprints:
K(1)-local K-theory of Azumaya algebras. arXiv link.
Localizing invariants of Azumaya algebras. arXiv link.
Every motive is the motive of a stable oo-category, with Vova Sosnilo and Christoph Winges. arXiv link.
On endomorphisms of topological Hochschild homology. arXiv link
Dualizable presentable ∞-categories. Paper on my site, arXiV link
Every spectrum is the K-theory spectrum of a stable ∞-category, with Vova Sosnilo and Christoph Winges. arXiv link, paper on my site (submitted).
The additivity of traces in stable ∞-categories. arXiV link , paper on my site.
Projects:
Polynomial functoriality of TR and TC: 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.
Polynomial spectra and localizing invariants: 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".
Norms on equivariant algebraic K-theory: with Kaif Hilman, we construct a G-symmetric monoidal structure on the G-category of G-motives, for G a finite group, which in particular makes the equivariant K-theory of a G-symmetric monoidal G-category into a normed commutative ring G-spectrum
Separable algebras are a framework that classically encompasses both étale algebras and Azumaya algebras. In my work, I study the homotopical version of these objects, first introduced in homotopy theory by Balmer. I am also interested in generalizations of the Brauer group in homotopy theory.
Preprints:
K(1)-local K-theory of Azumaya algebras. arXiv link.
Localizing motives of Azumaya algebras. arXiv link.
Separable commutative algebras in equivariant homotopy theory, with Niko Naumann and Luca Pol. arXiv link (submitted)
Separability in homotopical algebra. arXiV link, paper on my site (Submitted - the version that is in my thesis).
Projects:
Separable algebras over the L_n-local sphere: with Anish Chedalavada, we compute the "étale site" of the L_n-local sphere, and give a sufficient condition to guarantee that commutative separable algebras have a finite tt-degree.
Br(KU): with Sven van Nigtevecht, we compute the Brauer group of KU (in progress).
Dualizable Brauer groups: with Ben Antieau, we extend on work of Stefanich and prove that over (classical) noetherian schemes, all invertible QCoh(X)-linear categories are compactly generated (Update: Stefanich has himself updated his paper and was in fact able to get rid of the noetherian assumption. This project has now become more of a work in progress where we see how general this Br^dual = Br phenomenon is; we have found a rigid counterexample).
Strict Brauer group of the sphere: with Robert Burklund and Shachar Carmeli, we compute the strict Brauer group of the sphere spectrum.
The category Pr^L of presentable oo-categories is a place where one can do surprising amounts of "algebra", and the study of presentable categories with extra structure often helps in the study of objects within those presentable categories. More generally, categorification is one of my favourite tricks in math.
Preprints:
A symmetric monoidal fracture square, with Niko Naumann and Luca Pol. arXiv link (submitted)
Locally rigid ∞-categories. Paper on my site, arXiV link
Dualizable presentable ∞-categories. Paper on my site, arXiV link
Characters and transfer maps via categorified traces, with Shachar Carmeli, Bastiaan Cnossen and Lior Yanovski. arXiV link, paper on my site (Accepted for publication in Forum of Mathematics, Sigma ).
Projects:
I am generally studying dualizable categories, doubly-presentable 2-categories, 2-stable 2-categories and am interesting in anything related to these topics.
Shadows, trace theories and categorified THH: part of this project is already recorded in my thesis, but essentially I compare shadows in the sense of Ponto (and later Ponto-Shulman, Hess-Rasekh) and trace theories in the sense of Nikolaus (and Kaledin), and explain how to view "the universal shadow" as a categorified version of THH.
Much of homotopy theory originates from algebraic topology, the study of topological spaces (and of homotopy types) through algebraic means. Homotopy theory is now a separate field, but it does sometimes go back to its roots, and I sometimes do too.
Publications:
On the multiplicativity of the Euler characteristic, with John Klein and Cary Malkiewich. arXiV link, paper on my site . Proceedings of the American Mathematical Society. 151, 11, p. 4997-5006 10 p.
Preprints:
Characters and transfer maps via categorified traces, with Shachar Carmeli, Bastiaan Cnossen and Lior Yanovski. arXiV link, paper on my site (Accepted for publication in Forum of Mathematics, Sigma ).
Projects:
Assembly maps, Becker-Gottlieb transfers and the topological Dwyer-Weiss-Williams index theorem: with Marco Volpe and Basti Wolf, we extend work of Bartels-Efimov-Nikolaus on assembly maps in A-theory to prove composability of Becker-Gottlieb transfers in a wide range of situations, and also to give a new proof of the Dwyer-Weiss-Williams index theorem. Along the way, we study parametrized motives.
I sometimes on pure/abstract category theory.
Publications:
Universality of Barwick's unfurling construction, with Bastiaan Cnossen and Tobias Lenz. arXiv link, International Mathematical Research Notices, Volume 2025, Issue 18, September 2025
An elementary proof of the naturality of the Yoneda embedding. arXiV link, paper on my site. Proceedings of the American Mathematical Society. 151, 10, p. 4163-4171
Preprints:
Fully faithful functors and pushouts of oo-categories, with Peter Haine and Jan Steinebrunner. arXiv link
A monoidal Grothendieck construction for ∞-categories. arXiV link, paper on my site (Accepted for publication in the Nagoya Mathematical Journal)
Projects:
Localizations of categories of (commutative) algebras: with Stefan Schwede, Vova Sosnilo and Christoph Winges, we prove that for nice (symmetric) monoidal categories equipped with a cofibration structure, localizing (commutative) algebras at weak equivalences is monadic over commutative algebras in the localization. This has applications in vastly different examples.
The topic of ambidexterity, beyond the rational case, is born out of Kuhn's Tate vanishing theorem and Hopkins and Lurie's generalization of this to higher finite groups. This is a topic close to my heart, though I haven't worked much on it precisely.
Preprints:
Characters and transfer maps via categorified traces, with Shachar Carmeli, Bastiaan Cnossen and Lior Yanovski. arXiV link, paper on my site (Accepted for publication in Forum of Mathematics, Sigma ).
Projects:
Higher groupoid cardinality is groupoid cardinality : In this project, 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.
Morava-Euler characteristics (this is an "open" project, in that I often think about it but do not have results): in a beautiful paper, Lior Yanovski introduced his Morava-Euler characteristic, a function defined on (equivalence classes of) spaces belonging to a certain subclass containing both p-finite spaces and finite spaces, extending both Baez and Dolan's groupoid cardinality and the Euler characteristic, and sharing formal properties with the latter. I would like to figure out what the largest class of spaces on which this works is. This is related to calculations in higher semi-additive K-theory, following Ben Moshe-Schlank, as well as to foundational questions regarding Carmeli-Schlank-Yanovski's category Tsadi (specifically, regarding its "connective" variant, as well as regarding its "infinite height" part), and finally to elliptic cohomology (as in Yanovski's original paper).
Publications:
A note on quadratic forms, with Fabian Hebestreit and Achim Krause. arXiV link, paper on my site. Bulletin of the London Mathematical Society, vol. 56, no. 5 (2024), pp. 1803-1818
On the Degree of irreducible polynomials, RMS, Revue de la Filière Mathématiques, April 2018