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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 for specifics.

Research

Much of my research is framed by the following 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.)

 

Click on an image to view as a slideshow.

Papers

  1. 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.  
  2. 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. 
  3. 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.
  4. 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. 
  5. 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).
  6. The bordism hypothesis (with John Francis; last revised August 2017).
    We supply a proof of the bordism hypothesis, using factorization homology. 
  7. 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. 
  8. 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.
  9. 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.
  10. 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.
  11. 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.
  12. 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.
  13. 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.
  14. 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.
  15. 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.
  16. 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. 
  17. Geometric cobordism categories
    (PhD thesis -- 82 pages)
  18. 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. 
  • 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.

Other Stuff

 

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