Broadly speaking, I like to implement homotopy theoretic thinking to geometry/algebraic geometry.
Factorization Algebras and Motivic Homotopy Theory
I am working on incorporating ideas and, more concretely, filtrations from the theory of chiral/factorization algebras to study homotopy types (e.g. of certain mapping spaces)…
1.1 Factorization Algebras in Unstable Motivic Homotopy Theory I: Contractibility of Ran Spaces (draft – comments welcome!)
Proves contractibility of a version of the Ran space in unstable motivic homotopy, with a view towards nonabelian Poincare duality.
1.2 Factorization Algebras in Unstable Motivic Homotopy Theory II: Nonabelian Poincare Duality. (in preparation)
2. Motivic Homotopy Theory
I also like to think about motivic homotopy theory from the point of view algebraic geometry and -theory…
To appear in “Surveys on Recent Developments in Algebraic Geometry (Edited with Izzet Coskun, and Tommaso de Fernex), in the Proceedings of Symposia in Pure Mathematics” [Arxiv:1605.00929].
Proves that modules over MW-motivic cohomology is the Déglisé-Fasel spectrum using Barr-Beck-Lurie [Arxiv:1708.05651].
2.3 Motivic Landewber Exact Theories and Étale Cohomology. (with Paul Arne Østvær) (60 pages – available upon request)
The localization of a Landweber exact theory at étale motivic cohomology is a universal way of imposing étale descent. This is a refinement and generalization of the work of Thomason in the 80’s and Quick’s thesis for algebraically closed field.
The recognition principle for infinite loop spaces in motivic homotopy theory in terms of “Gysin transfers along finite syntomic morphisms.”
2.5 Twisted Homotopy –theory and Twisted Cycle Class Maps. (in preparation)
Twists certain cycle class maps by an Azumaya algebra via motivic homotopy theory.
3. Étale Homotopy Theory
and I like the point of view that étale homotopy types are shapes of (higher) topoi.
3.1 Relative étale realization of motivic spaces. (with David Carchedi)
A motivic version of the theory of relative étale realization of Barnea-Schlank.
Computations of Heegard-Floer homology
Some nontrivial examples of the BOS twisted spectral sequence. New York J. Math. 22 (2016) 363–378. [Arxiv:1604.04260] [NYJM].(with Igor Kriz)