Research

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. Some nontrivial examples of the BOS twisted spectral sequence (with Igor Kriz). New York J. Math. 22 (2016) [Arxiv:1604] [NYJM].
  2. 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].
  3. On modules over motivic ring spectra (with Håkon Andreas Kolderup) Ann. K-Theory. 5 (2020) [Arxiv:1708.05651][AKT].
  4. 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].
  5. Perfection in motivic homotopy theory. (with Adeel Khan, 10 pages). Proc. London. Math. Soc. 120 (2019) [Arxiv:1812][PLMS].
  6. Modules over algebraic cobordism (with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson, 40 pages) Forum Math. \Pi (2020) [Arxiv:1908] [ForumPi].
  7. 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.].
  8. THH and TC are (very) far from being homotopy functors Journal of Pure and Applied Algebra, 225 (2021) [Arxiv:2007] [JPAA].
  9. Descent for semiorthogonal decompositions (with Benjamin Antieau) Adv. Math, 380 (2021) [Arxiv:1912] [AiM].
  10. Scheiderer motives and equivariant higher topos theory (with Jay Shah) Adv. Math. 382 (2021) [Arxiv:1912] [AiM].
  11. 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].
  12. On nilpotent extensions of \infty-categories and the cyclotomic trace (with Vladimir SosniloInt. Math. Res. Not., 14 (2021) [Arxiv:2010] [IMRN] (slides).
  13. Voevodsky’s slice conjectures via Hilbert schemes (with Tom Bachmann) [Arxiv:1912] Algebr. Geom. 8 [Alg. Geom.].
  14. 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].
  15. Motivic infinite loop spaces (with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson[Arxiv:1711] Cambridge J. Math. 9 [CJM] (slides).
  16. Algebraic cobordism and étale cohomology (with Marc Levine, Markus Spitzweck and Paul Arne Østvær) [Arxiv:1711] to appear in Geometry & Topology [G&T].

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. Stable motivic invariants are eventually étale local  (with Tom Bachmann and Paul Arne Østvær, 32 pages) [Arxiv:2003] (submitted, comments welcome, last updated March 2020).
  4. A descent view on Mitchell’s theorem (with Denis Nardin and Lucy Yang, 5 pages) [Arxiv:2008] (submitted, comments welcome, last updated August 2020).
  5. On distributivity in higher algebra I: the universal property of bispans (with Rune Haugseng, 77 pages) [Arxiv:2010] (submitted, comments welcome, last updated January 2021).
  6. Motivic colimits and motivic extended powers (with Tom Bachmann and Jeremiah Heller, 32 pages) [Arxiv:2104] (preliminary version, comments welcome, last updated April 2021).
  7. \mathbb{A}^1-connected components of classifying spaces and purity for torsors (with Matthias Wendt and Girish Kulkarni, 22 pages) [Arxiv:2104] (submitted, comments welcome, last updated April 2021).
  8. Twisted K-theory in motivic homotopy theory (with Denis Nardin and Maria Yakerson, 20 pages) [Arxiv:2110] (preliminary version, comments welcome, last updated October 2021).

Mostly expository:

  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).

In Preparation:

    1. Mod 2 Power operations on normed motivic spectra (with Tom Bachmann and Jeremiah Heller).
    2. K-theory of universal homeomorphisms (with Akhil Mathew and Jakub Witaszek). (slides) (video).
    3. On distributivity for higher algebra II (with Rune Haugseng).
    4. The Real integral Hodge conjecture for Chow-Witt groups (with Jay Shah).
    5. Chow-Witt zero cycles of real schemes (with Aravind Asok)

Last updated: July 9 2021.