# Research

Research:

Papers on framed motives: these represent contributions to the “framed motivic program” as envisioned by Voevodsky and initiated by Ananyevskiy, Garkusha, Neshitov and Panin.

1. Motivic infinite loop spaces (with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson[Arxiv:1711] (submitted, last updated: April 2020).
2. Framed transfers and motivic fundamental classes (with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson, 39 pages)  Journal of Topology. 13 (2020), 460-500. [Arxiv:1809] [JTop].
3. Modules over algebraic cobordism (with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson, 40 pages). [Arxiv:1908] (submitted, comments welcome, last updated September 2019).
4.  On the infinite loop spaces of algebraic cobordism and the motivic sphere (with Tom Bachmann, Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson, 11 pages). [Arxiv:1911] (preliminary version, comments welcome, last updated March 2020)

Papers on excision and cdh descent: these are papers in and around the idea of excision (in the sense of Milnor) and its relationship with Voevodsky’s cdh topology and motivic homotopy theory.

1. Milnor excision for motivic spectra (with Marc Hoyois, Ryomei Iwasa and Shane Kelly, 9 pages) [Arxiv:2004] (submitted, comments welcome, last updated, April 2020)
2. Cdh descent, cdarc descent and Milnor excision (with Marc Hoyois, Ryomei Iwasa and Shane Kelly, 27 pages) [Arxiv:2002] (submitted, comments welcome, last updated April 2020)

Papers on motives in other contexts: these are papers on analogs of motivic homotopy theory in various contexts; so far, they concern motives in different Grothendieck topologies.

1. 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)
2. Scheiderer motives and equivariant higher topos theory (with Jay Shah, 78 pages). [Arxiv:1912] (submitted, comments welcome, last updated December 2019)
3. 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).
4. Algebraic cobordism and étale cohomology (with Marc Levine, Markus Spitzweck and Paul Arne Østvær). [Arxiv:1711] (submitted, comments welcome, last updated Feb 2019).

Papers on derived categories and derived algebraic geometry: these represent papers using or on derived techniques in algebraic geometry.

1. Descent for semiorthogonal decompositions (with Benjamin Antieau, 30 pages) [Arxiv:1912] (submitted, comments welcome, last updated December 2019)

Others:

1. Voevodsky’s slice conjectures via Hilbert schemes (with Tom Bachmann, 8 pages) [(submitted, comments welcome, last updated December 2019)
2. Perfection in motivic homotopy theory. (with Adeel Khan, 10 pages). Proc. London. Math. Soc. 120 (2019), 28-38 [Arxiv:1812][PLMS].
3. Thesis: Motivic contractibility of the space of rational maps (preliminary version, comments welcome, last updated April 2018) (To be split into two papers).
4. On modules over motivic ring spectra (with Håkon Andreas Kolderup) [Arxiv:1708.05651] to appear in Ann. K-Theory.

From a previous life:

1. Some nontrivial examples of the BOS twisted spectral sequence (with Igor Kriz). New York J. Math. 22 (2016) 363–378. [Arxiv:1604] [NYJM].

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) (Oberwolfach Report, last updated May 2018).
3. A primer to unstable motivic homotopy theory (with Benjamin Antieau). Surveys on Recent Developments in Algebraic Geometry, Proc. Sympos. Pure Math. 95 (2017), 305-370. [Arxiv:1605].

In Preparation:

1. Twisted homotopy $K$theory and twisted cycle class maps.
2. Étale framed motives.
3. Mod 2 Power operations on normed motivic spectra (with Tom Bachmann and Jeremiah Heller).
4. K-theory of universal homeomorphisms (with Akhil Mathew and Jakub Witaszek). (slides).
5. The universal property of bispans I & II (with Rune Haugseng).
6. Something about continuous invariants of analytic spaces (with Martin Speirs and Yifei Zhao).