# David Ayala

I live in Bozeman, Montana, as an assistant professor of mathematics at Montana State University. Previously, I was a postdoc at the University of Southern California, Harvard University, and the University of Copenhagen. I received my PhD through Stanford University under the supervision of Ralph Cohen

See my abbreviated Curriculum Vitæ (2 pages) for specifics.

## Research

My research concerns the differential topology of locally defined entities, such as manifolds and links, and their moduli as informed by conceptual and computational techniques from (derived) algebraic geometry and (higher) category theory. I'm compelled by factorization homology as a basic maneuver for transforming geometric data into categorical data.

## Photo Gallery

Click on an image to view as a slideshow.

## Projects

1. The bordism hypothesis (with John Francis; to appear).
We give a proof of the bordism hypothesis, using factorization homology.
2. Factorization homology with adjoints (with John Francis and Nick Rozenblyum; to appear).
We show that, in the presence of adjoints, factorization homology is naturally defined on framed stratified manifolds.
3. The orthogonal group and adjoints (with John Francis; to appear).
We construct an action of the combinatorial orthogonal group on n-categories.
4. The combinatorial orthogonal group (with John Francis; to appear).
We amalgamate the Schubert stratifications of Grassmannians to combinatorialize the orthogonal group, as a group.
5. Factorization homology from higher categories (with John Francis and Nick Rozenblyum).
We define vari-framings, and develop factorization homology over vari-framed stratified manifolds with coefficients in higher categories. We prove that this construction embeds higher categories as invariants of vari-framed stratified manifolds.
6. A stratified homotopy hypothesis (with John Francis and Nick Rozenblyum).
We give a geometrically convenient model for $\infty$-categories using stratified spaces, and introduce some universal examples.
7. Poincar'e/Koszul duality (with John Francis).
We articulate a duality among certain topological field theories that exchanges perturbative and non-perturbative ones; this duality generalizes Poincar'e duality as well as Koszul duality.
8. Zero-pointed manifolds (with John Francis).
We give a convenient category of manifolds that is the home for dualities. We see the Bar-coBar construction in this way, and prove a general version of non-abelian Poincar'e duality.
9. Factorization homology of stratified spaces (with John Francis and Hiro Lee Tanaka).
We define factorization homology of structured stratified spaces, and characterize such through excision.
10. Local structures on stratified spaces (with John Francis and Hiro Lee Tanaka).
We develop a theory of stratified spaces and their moduli. We characterize local structures on them.
11. Factorization homology of topological manifolds (with John Francis).
We classify excisive invariants of topological manifolds by way of factorization homology of disk-algebras.
12. Cyclotomic trace via factorization homology (with Nick Rozenblyum; in preparation).
We find the cyclotomic category $\Lambda$ in geometric terms, and construct the cyclotomic trace through derivatives of factorization homology.
13. Configurations spaces and $\Theta_n$ (with Richard Hepworth; Proceedings of the AMS).
We explain that the category $\Theta_n$ encodes configuration spaces of points in Euclidean n-space.
14. The Ran complex and $\Theta_n$ (with John Francis; to appear).
We show that the Ran space of Euclidean n-space is a localization of $\Theta_n$. This implies k-monoidal n-categories are precisely the k-reduced pointed (n+k)-categories.
15. Geometric cobordism categories
(PhD thesis -- 82 pages)
16. Stable topology of moduli spaces of curves in complex projective space
(19 pages)
17. Counting bitangents with stable maps (with Renzo Cavalieri -- Expositiones Mathematicae, volume 24, no. 4, pages 307-335)