**Current Projects**

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)*

*Slides for Indiana GSTGC*

**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].

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.

**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] [NYJM].(with Igor Kriz)

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