Adeel A. Khan

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2019
A gluing result for weight structures along semi-orthogonal decompositions, which is applied to give some new examples of regular ring spectra. Joint with Vladimir Sosnilo. Preprint (submitted), 12 pages, January 2019.
2018
Proves topological invariance in SH[1/p]. Joint with Elden Elmanto. Preprint, 9 pages, December 2018.
Proves projective bundle and blow-up formulas in K-theory for derived stacks. Also gives a direct new proof of cdh descent for homotopy invariant K-theory. This version extends the cdh descent criterion to stacks. Preprint, 18 pages, last updated November 2018 (changelog).
Relates the theory of motivic fundamental classes with the recognition principle for infinite P1-loop spaces. Joint with Elden Elmanto, Marc Hoyois, Vladimir Sosnilo, and Maria Yakerson. Preprint, 38 pages, September 2018.
Intersection theory à la Fulton, in the motivic homotopy category. Joint with Frédéric Déglise and Fangzhou Jin. Preprint (submitted), 43 pages, last updated December 2018 (changelog).
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.
2017
Develops a motivic version of the theory of (grouplike) E-spaces. Joint work with Elden Elmanto, Marc Hoyois, Vladimir Sosnilo, and Maria Yakerson. Preprint (submitted), 77 pages, last updated September 2018.
A1-homotopy invariance in spectral algebraic geometry
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, and deduces some consequences for the homotopy invariant K-theory of commutative ring spectra. Joint work with Denis-Charles Cisinski, in preparation.
2016
A revised version of an older preprint called Brave new motivic homotopy theory I, featuring a greatly condensed exposition. The main result is an analogue of Kashiwara's lemma for A1-homotopy invariant Nisnevich sheaves over spectral algebraic spaces. 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.
2014
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.