### Research interests

Algebraic geometry, motivic homotopy theory, motivic cohomology

Homotopy theory, higher category theory, algebraic K-theory

### Collaborators

Toni Annala, Aravind Asok, Tom Bachmann, Clark Barwick, Elden Elmanto, Saul Glasman, Ryomei Iwasa, Joachim Jelisiejew, Shane Kelly, Adeel A. Khan, Amalendu Krishna, Denis Nardin, Pavel Safronov, Sarah Scherotzke, Jay Shah, Nicolò Sibilla, Vladimir Sosnilo, Burt Totaro, Matthias Wendt, Maria Yakerson, Paul Arne Østvær

### Publications and preprints

**Remarks on the motivic sphere without A¹-invariance**(October 2024) [pdf, arXiv]**Atiyah duality for motivic spectra**, with T. Annala and R. Iwasa (March 2024) [pdf, arXiv]**Algebraic cobordism and a Conner–Floyd isomorphism for algebraic K-theory**, with T. Annala and R. Iwasa (to appear in JAMS) [pdf, arXiv]**Introduction to framed correspondences**, with N. Opdan (to appear in The Open Book Series) [pdf, arXiv]

Based on a series of lectures given in Oslo in August 2020.**Hermitian K-theory via oriented Gorenstein algebras**, with J. Jelisiejew, D. Nardin, and M. Yakerson (J. reine angew. Math. 793, 2022) [pdf, arXiv]- Appendix in
**Remarks on étale motivic stable homotopy theory**by T. Bachmann (March 2021) [pdf, arXiv] **Milnor excision for motivic spectra**, with E. Elmanto, R. Iwasa, and S. Kelly (J. reine angew. Math. 779, 2021) [pdf, arXiv]**The Hilbert scheme of infinite affine space and algebraic K-theory**, with J. Jelisiejew, D. Nardin, B. Totaro, and M. Yakerson (to appear in JEMS) [pdf, arXiv]**Cdh descent, cdarc descent, and Milnor excision**, with E. Elmanto, R. Iwasa, and S. Kelly (Math. Ann. 379, 2021) [pdf, arXiv]**On the infinite loop spaces of algebraic cobordism and the motivic sphere**, with T. Bachmann, E. Elmanto, A. A. Khan, V. Sosnilo, and M. Yakerson (Épijournal Géom. Algébrique 5, 2021) [pdf, arXiv]**Modules over algebraic cobordism**, with E. Elmanto, A. A. Khan, V. Sosnilo, and M. Yakerson (Forum Math. Pi 8, 2020) [pdf, arXiv]**Framed transfers and motivic fundamental classes**, with E. Elmanto, A. A. Khan, V. Sosnilo, and M. Yakerson (J. Topol. 13, 2020) [pdf, arXiv]**Affine representability results in A¹-homotopy theory III: finite fields and complements**, with A. Asok and M. Wendt (Algebr. Geom. 7, no. 5, 2020) [pdf, arXiv]**The localization theorem for framed motivic spaces**(Compos. Math. 157, 2021) [pdf, arXiv]**The categorified Grothendieck–Riemann–Roch theorem**, with P. Safronov, S. Scherotzke, and N. Sibilla (Compos. Math. 157, 2021) [pdf, arXiv]**Motivic infinite loop spaces**, with E. Elmanto, A. A. Khan, V. Sosnilo, and M. Yakerson (Cambridge J. Math. 9, 2021) [pdf, arXiv]**Norms in motivic homotopy theory**, with T. Bachmann (Astérisque 425, 2021) [pdf, arXiv]**Vanishing theorems for the negative K-theory of stacks**, with A. Krishna (Ann. K-Theory 4, 2019) [pdf, arXiv]**Generically split octonion algebras and A¹-homotopy theory**, with A. Asok and M. Wendt (Algebra Number Theory 13, no. 3, 2019) [pdf, arXiv]**Topoi of parametrized objects**(Theory Appl. Categ. 34, no. 9, 2019) [pdf, arXiv]**Categorifying rationalization**, with C. Barwick, S. Glasman, D. Nardin, and J. Shah (Forum Math. Sigma 7, 2019) [pdf, arXiv]**Cdh descent in equivariant homotopy K-theory**(Doc. Math. 25, 2020) [pdf, arXiv]**Higher traces, noncommutative motives, and the categorified Chern character**, with S. Scherotzke and N. Sibilla (Adv. Math. 309, 2017) [pdf, arXiv]**The six operations in equivariant motivic homotopy theory**(Adv. Math. 305, 2017) [pdf, arXiv]**Affine representability results in A¹-homotopy theory II: principal bundles and homogeneous spaces**, with A. Asok and M. Wendt (Geom. Topol. 22, no. 2, 2018) [pdf, arXiv]**Affine representability results in A¹-homotopy theory I: vector bundles**, with A. Asok and M. Wendt (Duke Math. J. 166, no. 10, 2017) [pdf, arXiv]**Higher Galois theory**(J. Pure Appl. Algebra 222, no. 7, 2018) [pdf, arXiv]

Galois theory for ∞-topoi and its connection to the étale homotopy type of Artin–Mazur–Friedlander.**The homotopy fixed points of the circle action on Hochschild homology**(April 2018, not intended for publication) [pdf, arXiv]**A¹-contractibility of Koras–Russell threefolds**, with A. Krishna and P. A. Østvær (Algebr. Geom. 3, no. 4, 2016) [pdf, arXiv]**A quadratic refinement of the Grothendieck–Lefschetz–Verdier trace formula**(Algebr. Geom. Topol. 14, no. 6, 2014) [pdf, arXiv]

A Lefschetz fixed-point theorem in stable motivic homotopy theory.**The motivic Steenrod algebra in positive characteristic**, with S. Kelly and P. A. Østvær (J. Eur. Math. Soc. 19, no. 12, 2017) [pdf, arXiv]**From algebraic cobordism to motivic cohomology**(J. reine angew. Math. 702, 2015) [pdf, arXiv]

A proof of the Hopkins–Morel isomorphism in motivic homotopy theory.**The étale symmetric Künneth theorem**(Math. Z. 304, no. 1, 2023) [pdf, arXiv]

### Reports

**Milnor excision for motivic spectra**(Oberwolfach Reports 24/2022) [pdf]**Algebraic cobordism, Hilbert schemes, and derived algebraic geometry**(2019 IAS Summer Collaborators report) [pdf]**A¹-homotopical classification of principal G-bundles**(Oberwolfach Reports 5/2016) [pdf]

### Miscellaneous notes

**Sequential vs. symmetric spectra**(after J. Lurie) [pdf]**Homotopy theory and the Bloch–Kato conjecture**[slides]

An introduction to homotopy theory and its role in the proof of the Bloch–Kato conjecture.**Finite Brown representability**[pdf]

A modern exposition of Adams’ variant of Brown representability.**Coherent six operations**[pdf]

A symmetric monoidal (∞,2)-category encoding the six operations of Grothendieck with Poincaré duality.**K-theory of dualizable categories**(after A. Efimov) [pdf]**The Steinberg relation**[pdf]

Proof of the Steinberg relation in unstable motivic homotopy theory.**On Quillen’s plus construction**[pdf]

A discussion of acyclic maps in ∞-topoi.**A trivial remark on the Nisnevich topology**[pdf]

Equivalence of two definitions of the Nisnevich topology.

### Older mathematical writings

- Notes on the birational classification of surfaces (Northwestern, 2nd year) [pdf]
- Notes on the first reconstruction theorem in Gromov–Witten theory (Northwestern, 2nd year) [pdf]
- Notes on the Nisnevich topology and Thom spaces in motivic homotopy theory (Northwestern, 1st year) [pdf]
**Chern character and derived algebraic geometry**(Master thesis, EPFL) [pdf, slides, poster]**Sur la cohomologie des schémas**(EPFL, 8th semester) [pdf]

Basic scheme theory in the language of locally ringed spaces. Definition and properties of sheaf cohomology and of Čech cohomology. Computation of the cohomology of affine schemes and of projective spaces.**The Syntax of First-Order Logic**(EPFL, 6th and 7th semesters) [pdf]

A constructive treatment of various topics in first-order logic: Herbrand’s theorem and the epsilon theorems; Craig’s interpolation lemma; Gödel’s first incompleteness theorem (two proofs); a detailed account of Gödel’s second incompleteness theorem; basic Zermelo–Fraenkel set theory; the consistency of the axiom of choice and of the generalized continuum hypothesis.**Décompositions paradoxales**(EPFL, 5th semester) [pdf]

This is a survey of the notion of*equidecomposability*in a set being acted upon by a group. Applications include the generalized Banach–Tarski paradox, the Sierpiński–Mazurkiewicz paradox, and the von Neumann paradox in the plane. The equivalence between the amenability and the nonparadoxality of discrete groups is proved (Tarski’s theorem).**Sur la constante de Khinchin**, with Stéphane Flotron and Ludovic Pirl (EPFL, 4th semester) [pdf]

Khinchin’s theorem states that there exists a real number*K*, Khinchin’s constant, such that the sequence of partial geometric means of the elements of the continuous fraction of almost any real number converges to*K*. This text contains an elementary discussion of continuous fractions and two different proofs of this result: Khinchin’s original proof and a more conceptual proof using ergodic theory.