Research

At present, I am interested in motivic cohomology and its interaction with various sub-fields of algebra and algebraic geometry. 

For papers posted on the Arxiv see here. Here are some selected preprints/publications by topics.

Motivic cohomology

  1. Motivic cohomology of equicharacteristic schemes (with Matthew Morrow) [Arxiv:2309] (MFO report) (IHES lecture series) (NY number theory) (Matthew’s talk in Tokyo) (Matthew’s lecture series).
  2. \mathbb{A}^1-invariant motivic cohomology of schemes (with Tom Bachmann and Matthew Morrow) [Arxiv:2508] (school at Osnabrück) (talk at Abel symposium) (Matthew’s talk at the Fields) (Matthew’s talk at the IAS).
  3. On the filtered algebraic K-theory of stacks I: characteristic zero (with Dmitry Kubrak and Vladimir Sosnilo) [Arxiv:2503].

Applied motivic cohomology/K-theory

  1. Equivariant algebraic K-theory and Artin L-functions (with Ningchuan Zhang) [Arxiv:2405].
  2. The Quillen-Lichtenbaum dimension of complex varieties (with Nick Addington) [Arxiv:2312].
  3. Motivic Steenrod operations at the characteristic via infinite ramifications (with Toni Annala) [Arxiv:2506].

Motivic homotopy theory

  1. On étale motivic spectra and Voevodsky’s covergence conjecture (with Tom Bachmann and Paul Arne Østvær) [Arxiv:2003] [J. Eur Math. Soc].
  2. Normed motivic spectra and power operations (with Tom Bachmann and Jeremiah Heller) [Arxiv:2210] [Ann. Sci. École Norm. Sup.].
  3. Voevodsky’s slice conjectures via Hilbert schemes (with Tom Bachmann) [Arxiv:1912] [Alg. Geom.].
  4. Milnor excision for motivic spectra (with Marc Hoyois, Ryomei Iwasa and Shane Kelly, 9 pages) [Arxiv:2004] [J. reine angew. Math.].
  5. Perfection in motivic homotopy theory. (with Adeel Khan) [Arxiv:1812][Proc. Lond. Math. Soc.]

Motivic infinite loop spaces

  1. Motivic infinite loop spaces (with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson)[Arxiv:1711] [Camb. J. Math.] (slides).
  2. Modules over algebraic cobordism (with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson) [Arxiv:1908] [Forum Math. Pi].
  3. Framed transfers and motivic fundamental classes (with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson) [Arxiv:1809] [J. Topol.].

Exotic topologies

  1. Cdh descent, cdarc descent and Milnor excision (with Marc Hoyois, Ryomei Iwasa and Shane Kelly, 27 pages) [Arxiv:2002] [Math. Ann.].
  2. Scheiderer motives and equivariant higher topos theory (with Jay Shah) [Arxiv:1912] [Adv. Math.]

Noncommutative motives

  1. On nilpotent extensions of \infty-categories and the cyclotomic trace (with Vladimir Sosnilo) [Arxiv:2010] [Int. Math. Res. Not.]

Higher algebra

  1. On distributivity in higher algebra I: the universal property of bispans (with Rune Haugseng) [Arxiv:2010] [Compos. Math.].

Notes:

  1. Notes on motivic infinite loop space theory (with Tom Bachmann) [Arxiv:1912] 2019.
  2. Topological periodic cyclic homology of smooth \mathbb{F}_p-algebras (after Bhatt-Morrow-Scholze) (MFO report) 2018.
  3. On the motivic cohomology of schemes (MFO report) 2022.
  4. Zero cycles on singular varieties (MFO report) 2025.

Notice: To the greatest extent I am legally able to, I refuse permission to use any of my work for AI training purposes. (stolen from RGH)

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