Intersection theory à la Fulton, in the motivic homotopy category.
This applies to higher Chow groups but also to exotic theories like Chow–Witt groups and hermitian K-theory.
As an application of the full generality, we prove a motivic Gauss–Bonnet formula.
Joint with F. Déglise and F. Jin.
Preprint, 47 pages, last updated June 2018.
Develops a motivic version of the theory of (grouplike) E∞-spaces.
Applications include a recognition principle for motivic infinite loop spaces, a motivic Barratt–Priddy–Quillen theorem, and a representability result for the infinite loop space of the motivic sphere spectrum.
Joint work with Elden Elmanto, Marc Hoyois, Vladimir Sosnilo, and Maria Yakerson.
Preprint (submitted), 77 pages, last updated June 2018.
Homotopy invariant K-theory in spectral algebraic geometry
A revised version of an older preprint called Brave new motivic homotopy theory II.
Compares two versions of A1-homotopy invariance in the setting of spectral algebraic geometry, and introduces a generalization of Weibel’s homotopy invariant K-theory.
The latter is applied to show that non-discrete commutative ring spectra are rarely K-regular.
Joint work with D.-C. Cisinski, in preparation.
A revised version of an older preprint called Brave new motivic homotopy theory I.
The main result is an analogue of Kashiwara's lemma for A1-homotopy invariant Nisnevich sheaves over spectral algebraic spaces.
This version features a greatly condensed exposition.
Preprint (submitted), 27 pages, last updated July 2018.
The foundational results of motivic homotopy theory, generalized to the setting of derived algebraic geometry.
Notably, the formalism of Grothendieck’s six operations is extended to this setting.
Ph.D. thesis, 122 pages, last updated July 2016.
A proof of Orlov’s result that the derived category of a smooth projective variety determines its rational Chow motive up to Tate twists, that passes through the noncommutative world.
Master thesis, 7 pages, last updated January 2014.