**Current Projects**

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…

*2.1 A Primer to Unstable Motivic Homotopy Theory. (with Benjamin Antieau)*

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].

*2.2 Modules over MW-Motivic Cohomology. **(with Håkon Andreas Kolderup) (preliminary version, comments welcome!)
*

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.

*2.4 Motivic Infinite Loop Spaces. (with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson). *

The recognition principle for infinite loop spaces in motivic homotopy theory in terms of “Gysin transfers along finite syntomic morphisms.”

Marc’s talk at UIUC conference

*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.

**Older Project:**

**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)