Research

At present, I am interested in motivic cohomology and its interaction with various sub-fields of algebra and algebraic geometry such as p-adic Hodge theory, birational geometry, classification of vector bundles, Brauer groups and Azumaya algebras, stacks and chromatic homotopy theory etc.

Keywords: (algebraic) K-theory and cobordism, motivic cohomology and algebraic cycles, (topological) cyclic homology in arithmetic geometry, cdh and valuation-theoretic methods in algebraic geometry, \mathbb{R}-algebraic geometry, Hilbert schemes and variants, abstract homotopy theory.

 Published/to appear:

  1. A primer to unstable motivic homotopy theory (with Benjamin Antieau). Surveys on Recent Developments in Algebraic Geometry, Proc. Sympos. Pure Math. 95 (2017) [Arxiv:1605].
  2. On modules over motivic ring spectra (with Håkon Andreas Kolderup) Ann. K-Theory. 5 (2020) [Arxiv:1708.05651][AKT].
  3. Framed transfers and motivic fundamental classes (with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson, 39 pages)  Journal of Topology. 13 (2020) [Arxiv:1809] [JTop].
  4. Perfection in motivic homotopy theory. (with Adeel Khan, 10 pages). Proc. London. Math. Soc. 120 (2019) [Arxiv:1812][PLMS].
  5. Modules over algebraic cobordism (with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson, 40 pages) Forum Math. \Pi 8 (2020) [Arxiv:1908] [ForumPi].
  6. Cdh descent, cdarc descent and Milnor excision (with Marc Hoyois, Ryomei Iwasa and Shane Kelly, 27 pages) Math. Annalen. 379 (2021) [Arxiv:2002] [Math. Ann.].
  7. THH and TC are (very) far from being homotopy functors Journal of Pure and Applied Algebra, 225 (2021) [Arxiv:2007] [JPAA].
  8. Descent for semiorthogonal decompositions (with Benjamin Antieau) Adv. Math, 380 (2021) [Arxiv:1912] [AiM].
  9. Scheiderer motives and equivariant higher topos theory (with Jay Shah) Adv. Math. 382 (2021) [Arxiv:1912] [AiM].
  10. On the infinite loop spaces of algebraic cobordism and the motivic sphere (with Tom Bachmann, Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson) Épijournal Géom. Algébrique, Vol. 5. (2021) [Arxiv:1911] [EPIGA].
  11. On nilpotent extensions of \infty-categories and the cyclotomic trace (with Vladimir Sosnilo) Int. Math. Res. Not., 14 (2021) [Arxiv:2010] [IMRN] (slides).
  12. Voevodsky’s slice conjectures via Hilbert schemes (with Tom Bachmann) [Arxiv:1912] Algebr. Geom. 8 [Alg. Geom.].
  13. Milnor excision for motivic spectra (with Marc Hoyois, Ryomei Iwasa and Shane Kelly, 9 pages) [Arxiv:2004]  J. reine angew. Math. (Crelle’s journal) 779 [Crelle].
  14. Motivic infinite loop spaces (with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson)  [Arxiv:1711] Cambridge J. Math. 9 [CJM] (slides).
  15. Algebraic cobordism and étale cohomology (with Marc Levine, Markus Spitzweck and Paul Arne Østvær) [Arxiv:1711] Geometry & Topology 26 [G&T].
  16. \mathbb{A}^1-connected components of classifying spaces and purity for torsors (with Matthias Wendt and Girish Kulkarni, 22 pages) [Arxiv:2104] Doc. Math. 27 [DM].
  17. On distributivity in higher algebra I: the universal property of bispans (with Rune Haugseng, 77 pages) [Arxiv:2010] Compos. Math. 159 [CM].
  18. On étale motivic spectra and Voevodsky’s covergence conjecture (with Tom Bachmann and Paul Arne Østvær, 32 pages) [Arxiv:2003] J. Eur Math. Soc. [JEMS].
  19. A descent view on Mitchell’s theorem (with Denis Nardin and Lucy Yang, 5 pages) [Arxiv:2008] (to appear in Israel J. Math.).

Preprints:

  1. Thesis: Motivic contractibility of the space of rational maps (preliminary version, comments welcome, last updated April 2018) (To be split into two papers).
  2. Relative étale realizations of motivic spaces and Dwyer-Friedlander K-theory of noncommutative schemes. (with David Carchedi, 81 pages) [Arxiv:1810] (preliminary version, comments welcome, last updated October 2018).
  3. Motivic colimits and motivic extended powers (with Tom Bachmann and Jeremiah Heller, 32 pages) [Arxiv:2104] (submitted, comments welcome, last updated April 2021).
  4. Twisted K-theory in motivic homotopy theory (with Denis Nardin and Maria Yakerson, 20 pages) [Arxiv:2110] (submitted, comments welcome, last updated October 2021).
  5. Normed motivic spectra and power operations (with Tom Bachmann and Jeremiah Heller, 38 pages) [Arxiv:2210] (submitted, comments welcome, last updated October 2022).
  6. Splitting results for normed motivic spectra (with Tom Bachmann and Jeremiah Heller, 12 pages) [Arxiv:2210] (preliminary version, comments welcome, last updated October 2022).
  7. Motivic cohomology of equicharacteristic schemes (with Matthew Morrow, 74 pages) [Arxiv:2309] (preliminary version, comments welcome, last updated September 2023) (MFO report) (IHES lecture series) (NY number theory) (Matthew’s talk in Tokyo).
  8. The motivic Adams conjecture (with Alexey Ananyevskiy, Oliver Röndigs and Maria Yakerson, 24 pages) [Arxiv:2310] (submitted, comments welcome, last updated October 2023).
  9. The Quillen-Lichtenbaum dimension of complex varieties (with Nick Addington) [Arxiv:2312] (submitted, comments welcome, last updated April 2024).
  10. Equvariant algebraic K-theory and Artin L-functions (with Ningchuan Zhang) [Arxiv:2405] (preliminary version, comments, welcome, last updated May 2024).

Notes:

  1. Notes on motivic infinite loop space theory (with Tom Bachmann). [Arxiv:1912].
  2. Topological periodic cyclic homology of smooth \mathbb{F}_p-algebras (after Bhatt-Morrow-Scholze) (MFO report, last updated May 2018).
  3. On the motivic cohomology of schemes (MFO report, last updated June 2022).

In Preparation:

    1. K-theory of universal homeomorphisms (with Akhil Mathew and Jakub Witaszek). (slides) (video).
    2. On distributivity for higher algebra II (with Rune Haugseng).
    3. The Real integral Hodge conjecture for Chow-Witt groups (with Jay Shah).
    4. Chow-Witt zero cycles of real schemes (with Aravind Asok).
    5. Cdh motivic cohomology of schemes in mixed characteristics (with Tom Bachmann and Matthew Morrow) (Matthew’s talk at the Fields) (Tom’s talk in Mainz).
    6. Quadratic forms and purity for torsors (with Matthias Wendt).
    7. Atiyah-Segal completion theorems for localizing invariants (with Dimitri Kubrak and Vladimir Sosnilo).