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.
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.)
- 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.
- 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; 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.
- 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.)
- 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.
- 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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