# 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

## Papers

- Flagged higher categories (with John Francis; last revised January 2018).

We introduce flagged $(\infty,n)$-categories, as a model-independent characterization of Segal sheaves on Joyal's category $\bTheta_n$. We indicate some compelling examples of such. This result can be interpreted as a non-linear instance of iterated Koszul duality. - The geometry of the cyclotomic trace (with Aaron Mazel-Gee and Nick Rozenblyum; last revised October 2017).

Using enriched factorization homology and linearization, we give a construction of the cyclotomic trace from algebraic K-theory to topological cyclic homology. Essentially, for each derived scheme X we organize the derived loop space LX as a quasi-coherent sheaf over the stratified algebraic stack of the previous paper; for each vector bundle over X, "trace of monodromy" defines a global function on LX. Much of this paper surveys its supporting papers. - A naive approach to genuine G-spectra and cyclotomic spectra (with Aaron Mazel-Gee and Nick Rozenblyum; last revised October 2017).

We interpret the $\infty$-category of cyclotomic spectra as that of quasi-coherent sheaves on a stratified algebraic stack over the sphere spectrum, with TC being global functions; we do similarly for genuine G-spectra. We do this by organizing Tate constructions as recollements, and use Glassman's definition of stratified stable categories. - Factorization homology of enriched $(\infty,1)$-categories (with Aaron Mazel-Gee and Nick Rozenblyum; last revised October 2017).

We define factorization homology of enriched $(\infty,1)$-categories over oriented 1-manifolds (as well as directed graphs). - The bordism hypothesis (with John Francis; last revised August 2017).

We supply a proof of the bordism hypothesis, using factorization homology. - Fibrations of $\infty$-categories (with John Francis; last revised February 2017).

We give a model-independent account of fibrations among $\infty$-categories, with exponentiable fibrations playing a central role. We show that each notion of fibration is classified by an $\infty$-category. This offers an operationally practical technique for making constructions among $\infty$-category theory, in a model-independent manner. - Factorization homology I: higher categories (with John Francis and Nick Rozenblyum; accepted).

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. - A stratified homotopy hypothesis (with John Francis and Nick Rozenblyum; to appear, Journal of the European Mathematical Society.).

We give a geometrically convenient model for $\infty$-categories using stratified spaces, and introduce some universal examples. - Poincar'e/Koszul duality (with John Francis; last revised November 2015).

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. - Zero-pointed manifolds (with John Francis; last revised July 2017).

We give a convenient category of manifolds that is the home for dualities. We recover the Bar-coBar construction in this way, and prove a general version of non-abelian Poincar\'e duality. - Factorization homology of stratified spaces (with John Francis and Hiro Lee Tanaka; Selecta Mathematica (N.S.) 23 (2017), no. 1, 293-362).

We define factorization homology over structured stratified spaces, and characterize such through excision. - Local structures on stratified spaces (with John Francis and Hiro Lee Tanaka; Advances in Mathematics 307 (2017), 903-1028).

We develop a theory of stratified spaces and their moduli. We characterize local structures on them. - Factorization homology of topological manifolds (with John Francis; Journal of Topology 8 (2015), no. 4, 1045-1084).

We classify excisive invariants of topological manifolds by way of factorization homology of disk-algebras. - Configurations spaces and $\Theta_n$ (with Richard Hepworth; Proceedings of the American Mathematical Society 142 (2014), no.7, 2243-2254).

We explain that the category $\Theta_n$ encodes configuration spaces of points in Euclidean n-space. - Counting bitangents with stable maps (with Renzo Cavalieri; Expositiones Mathematicae, volume 24, no. 4, pages 307-335).

We use ideas from Gromov-Witten theory to do some enumerative geometry. - Geometric cobordism categories

(PhD thesis -- 82 pages) - Stable topology of moduli spaces of curves in complex projective space

(Warning: this paper contains an error -- the abstract points you to a remark detailing this error.)

## Other projects

- Factorization homology II: adjoints (with John Francis and Nick Rozenblyum; to appear).

We show that, in the presence of adjoints, factorization homology is naturally defined on solidly n-framed stratified manifolds. - The orthogonal group and adjoints (with John Francis; to appear).

We amalgamate the Schubert stratifications of Grassmannians to combinatorialize the orthogonal group, as a group. We construct a lax-action of this combinatorial orthogonal group on n-categories. This action is implemented by adjoining adjoints. - 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.

## Other Stuff

- Some juggling videos
- Some photos of outrageous natural features
- Some modest comics
- Tom Hayes' compilation of nearby trails to run along

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