David Ayala

I live in Bozeman, Montana, and work as an Associate Professor of mathematics at Montana State University. 

Previously, I was a postdoc at the Mathematical Sciences Research Institute (MSRI, atop UC Berkeley), 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 for specifics.

Research

My research is ambiently framed by quantum topology: using physically motivated algebraic data to construct global invariants invariants that abstract physical invariants/observables of confined quantum systems in ground states. 

My work uses a blend of homotopy theory and algebra/category theory to achieve such invariants through universal properties, thereby endowing them with expected continuous symmetries and functorialities as well as local-to-global formulae.  This contrasts more traditional hands-on approaches that construct invariants upon a choice of presentation, which are then verified to be independent of the choice of presentation by scrutenizing a comprehensive list of moves that relate such presentations.

In broad strokes, my research follows this paradigm. 

• Invariants of locally standard entities -- such as manifolds, links, or (derived) schemes with very few points -- that possess a nuanced local-to-global expression can be constructed from (higher-)algebraic, or (higher-)categorical, data.  The result, factorization homology, is an association of an object, such as a vector space, from such an entity and such a (higher-)algebra/category.
• Deformations of the algebraic/categorical input are organized as a likewise algebra/category -- this is Koszul duality.  Through factorization homology, Poincare' duality intertwines with Koszul duality, thereby offering unforseen identities of, and within, the values of factorization homology. 
• (Scores of classical, and more recent, identities can be recovered as instances of this paradigm.)

Articles

1. Symmetries of the cyclic nerve (with Aaron Mazel-Gee and Nick Rozenblyum; last revised May 2024).
   We parametrize (\infty,1)-categories by quivers, generalizing their characterization as complete Segal spaces as the case of linear quivers.  Specializing to cyclic quivers supplies a construction of the Hochschild homology of an (\infty,1)-category.  This approach manifestly reveals that each positive-degree self-map of the circle determines a (not-necessarily-invertible) symmetry of this Hochschild homology, which we show is a (non-stable) cyclotomic structure on Hochschild homology.  Along the way, we regognize the cyclic, the paracyclic, and the epicyclic categories in terms of quivers.  
2. Additivity for factorization algebras (with Eric Berry; last revised December 2023).
   Factorization algebras rigorously abstract the observables of a perturbative quantum field theories.  We prove an "additivity'' result for factorization algebras: a factorization algebra on a product is a factorization algebra in one factor valued in factorization algebras on the other.  This identification restricts to locally constant factorization algebras, which implies a conceptual proof of Additivity for E_n-algebras.  
3. Symmetries of a rigid braided category (with John Francis; last revised May 2022).
   We construct action of the continous group \Omega O(n) on each E_{n-1}-monoidal category \cal{R} in which each object is dualizable, functorially in \cal{R}.  For each such \cal{R}, this action descends as an action of the continuous group \Omega RP^{n-1} on the moduli space Obj(\cal{R}) of its objects.
4. Natural Symmetries of secondary Hochschild homology (with John Francis and Adam Howard; to appear in Algebraic and Geometric Topology).
   

   We explicitly identify the continuous group of framed diffeomorphisms of the torus as well as the continuous monoid of framed isogenies of the torus.  We explicate an action of this group on secondary Hochschild homology, which we extend as an action of this monoid in certain cases, and abstract as an (unstable) secondary cyclotomic structure.

5. Derived Mackey functors and C_{p^n}-equivariant cohomology (with Aaron Mazel-Gee and Nick Rozenblyum; last revised March 2023).
   For p an odd prime, we identify the Picard group of genuine C_{p^n}-Z-modules, as well as the C_{p^n}-equivariant cohomology of a point.  These identifications are achieved through application of stratifications of noncommutative stacks.  
6. Trace for factorization homology in dimension 1 (with John Francis; last revised December 2021).
   We construct circle-equivariant trace map, and prove a conjecture of Toën-Vezzosi, using 1-dimensional factorization homology.
7. Stratified noncommutative geometry (with Aaron Mazel-Gee and Nick Rozenblyum; to appear in Memoirs of the AMS).
   We develop a rather comprehensive theory of stratifications of stable presentable categories, and give several examples.  As an appendix, we consolidate some general definitions and basic results in lax category theory.
8. A factorization homology primer (with John Francis; to appear in The Homotopy Handbook).
   We give a consolidated account of the theory of, examples/instances of, and general features of factorization homology (“alpha”).  This includes discussing relationships with factorization algebras, defects, filtrations, and Poincare’/Koszul duality.  
9. Flagged higher categories (with John Francis; to appear in CBMS proceedings).
   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. 
10. 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.
11. 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. 
12. Factorization homology of enriched $(\infty,1)$-categories (with John Francis, 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).
13. The bordism hypothesis (with John Francis; last revised August 2017).
    We supply a proof of the bordism hypothesis, using factorization homology. 
14. Fibrations of $\infty$-categories (with John Francis; to appear in Higher Structures).
    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. 
15. Factorization homology I: higher categories (with John Francis and Nick Rozenblyum; to appear in Advances of Mathematics). 
    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.
16. A stratified homotopy hypothesis (with John Francis and Nick Rozenblyum; to appear in Journal of the European Mathematical Society.). 
    We give a geometrically convenient model for $\infty$-categories using stratified spaces, and introduce some universal examples.
17. Poincar'e/Koszul duality (with John Francis; to appear in Communications in Mathematical Physics). 
    We articulate a duality among certain topological field theories that exchanges perturbative sigma-models and reduced state-sum models; this duality generalizes Poincar'e duality as well as Koszul duality.
18. Zero-pointed manifolds (with John Francis;  to appear in Journal of the Institute of Mathematics Jussieu). 
    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.
19. 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.
20. 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.
21. 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.
22. 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.
23. 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. 
24. Geometric cobordism categories
    (PhD thesis -- 82 pages)
25. 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

• An approach to less climate-impactful conferences (with Lukas Brantner, Theo Johnson-Freyd, Andre Henriques, and Aaron Mazel-Gee; London Mathematical Society Newsletter, no. 480, pages 32-33).
  We organized a somewhat experimental conference — a “double-conference” — which was a single event held in two locations: the Perimeter Institute in Waterloo, Canada; the Max Planck Institute for Mathematics in Bonn, Germany.  The intention was to implement, and possibly set a model for, an international conference that avoided trans-Atlantic travel.  The spirit of doing so was two-fold:
  • To decrease environmental impact of cutting-edge mathematics conferences.
  • To decrease the financial (and other) burdens of travel to attend such conferences, thereby accessing a broader community of researchers. 
• Topology and Quantum Theory in Interaction (with Daniel Freed and Ryan Grady; American Mathematical Society, Contemporary Mathematics, 718).  Ryan and I organized a week-long workshop featuring 10 lectures by Daniel Freed, and accompanying lectures by other leaders in the area, on recent developments in a classification of invertible topological quantum field theories, accompanied by predictions & comparisons in condensed matter physics.  Daniel Freed compiled some comprehensive set of notes of his lectures: Lectures on Field Theory and Topology.
• Factorization homology of rigid braided categories (with John Francis; to appear).
  We prove the tangle hypothesis in the case of 1-dimensional tangles in n-dimensional space.  Using this, we construct factorization homology for rigid E_{n-1}-monoidal (\infty,1)-categories.  We recover Skein modules of 3-manifolds, and in doing so achieve a local-to-global expression for such.  

Graduated PhD Students

To date, these PhD students have graduated under my supervision.

• Anna Cepek. 
  • 2019 Dissertation: On Configuration Categories, in which the first main result identifies the configuration category (which is closely related to the exit-path \infty-category of the Ran space) of a circle in terms of the paracyclic category, and the second main result identifies the configuration category of Euclidean n-space in terms of the category \Theta_n.
• Daniel Perry. 
  • 2019 Dissertation: Homotopy Groups of Contact 3-Manifolds, in which the main result shows that a contact 3-manifold, when tangently-probed by spheres, is a K(\pi,1).  Consequently, each 3-manifold is the quotient of a (metric) tree by a pro-finite group.
• Eric Berry.
  • 2021 Dissertation: Additivity of Factorization Algebras and the Cohomology of Real Grassmannians, in which the first main result establishes additivity for factorization algebras, which thereafter implies Dunn's additivity for E_n-algebras, and the second main result is the most explicit-to-date computation of the cohomology groups of (real) Grassmannians over abitrary commutative rings.  
• Adam Howard. 
  • 2021 Dissertation: Immersions of Surfaces, in which the first main result computes the homotopy-type, and thereafter many of the homotopy-groups, of the space of immersions from (orientable connected non-negative genus) surfaces into framed manifolds, and the second main result identifies the continuous monoid, as well as group, of symmetries of a framed 2-torus in terms of the braid group on 3-strands.  
• Benjamin Moldstad.
  • 2024 Dissertation: Stable Circle Actions, in which the main result characterizes linear circle actions.  Specifically, in a stable \infty-category, a circle action on an object therein is a degree 1 operator \partial with an identification of the ratio \partial^2/\partial as the Hopf element.  In cases where the Hopf element vanishes, a circle action is therefore a degree 1 differential.  Other results in this work lay a groundwork for identifying the Morita-type of Reedy categories.  

Also, Aaron Mazel-Gee, who received his PhD through UC Berkeley under the supervisin of Peter Teichner, worked closely with me in the final years of his PhD work.

Other Stuff

• Some Fastest Known Times along some mountain ranges
• Some juggling videos (Stanford Juggling Research Institute)
• Some photos of outrageous natural features
• Some modest comics
• Tom Hayes' compilation of nearby trails to run along

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