Adeel A. Khan

Lecture course (joint with F. Binda), SS 2018, course page.

Lecture course, WS 2017/18, course page

Exercise sessions for lecture course, WS 2016/17 – SS 2017, exercise sheets

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.

A complete description of the functor of points of a blow-up in regularly immersed centre, using the language of derived algebraic geometry. Preprint, 12 pages, last updated March 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.

A revised version of an older preprint called Brave new motivic homotopy theory II. Compares A¹-homotopy theory over a spectral scheme with classical A¹-homotopy theory over the underlying classical scheme. We also deduce some consequences for the homotopy invariant K-theory of commutative ring spectra. 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.

My Ph.D. thesis (2016). The foundational results of motivic homotopy theory are generalized to the setting of derived algebraic geometry. Notably, the formalism of Grothendieck’s six operations is extended to this setting. 122 pages, minor update in September 2018.

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.

Mini-course, link, Essen, May 2016