Broadly speaking, I like to implement homotopy theoretic thinking to geometry/algebraic geometry.
1. 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].
2.1 Twisted Homotopy –theory and Twisted Cycle Class Maps. (in preparation)
Twists certain cycle class maps by an Azumaya algebra via motivic homotopy theory.
2.2 Motivic Landewber Exact Theories and Étale Cohomology. (with Paul Arne Østvær) (50 pages – available upon request)
A generalization of Thomason’s Bott-inverted -theory descent theorem to Landweber exact theories. We prove integral statements and give applications.
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. Offers a similar obstruction theory to “motivic” rational points.
Computations of Heegard-Floer homology
Some nontrivial examples of the BOS twisted spectral sequence. New York J. Math. 22 (2016) 363–378. [Arxiv] [NYJM].(with Igor Kriz)