## Research

I work at the intersection of stable homotopy theory and mirror symmetry. In particular, a good chunk of my time is spent thinking about how to enrich the Fukaya category (an invariant of a symplectic manifold) over spectra (where one can do far richer algebra than just using rings or chain complexes). The approach I've thought about longest involves the cobordism theory of Lagrangian submanifolds, and more recently I've started thinking about derived geometry approaches to Floer theory. I also used to think a lot about factorization algebras, especially as they relate to topological field theories and invariants of embedding spaces. Other buzz-words that pop up a lot in my research conversations are: Geometric Langlands, (topological) chiral homology, configuration spaces, E_n algebras and n-fold loop spaces, Khovanov homology, Chern-Simons theory, deformation theory, quantum field theories, and infinity-categories.

Here is a list of my arXiv preprints. My non-arXiv papers are all in "Proceedings" type books. You can find a list of my publications here.

## Lagrangian Cobordisms and Fukaya categories

Building on previous work with David Nadler, I am trying to understand Lag(M), an oo-category of Lagrangian cobordisms inside a symplectic manifold M, which we conjecture to be equivalent to a version of the Fukaya category of M. One can construct non-trivial functors from Lag(M) to modules over the Fukaya category, and construct natural actions of the sphere spectrum on Lag(M) by relating framed cobordism theory to Lagrangian cobordism theory. Unlike Floer theory, it is far easier to deal with correspondences, bimodules, and immersed objects in Lag(M), though--as usual--the inherent symplectic geometry of Lagrangians still exhibits the dual properties of rigidity and flexibility.

A Stable Infinity-Category of Lagrangian Cobordisms, with David Nadler. (Uploaded Sept 23, 2011.)
We define an oo-category Lag(M) whose objects are Lagrangian branes, and whose morphisms are Lagrangian cobordisms. We prove Lag is stable in the sense of Lurie, which means that its homotopy category is triangulated, just as the (triangulated envelope of the) Fukaya category, or the (derived) category of coherent sheaves. This also means that Lag is enriched over spectra. Moreover, the shift functor is identical to the familiar shift functor from the Fukaya category -- one simply shifts the gradings of a brane. There is also a variant Lag_Lambda(M) for any choice of subset Lambda inside of M.

In simply-connected cotangent bundles, exact Lagrangian cobordisms are h-cobordisms. (Uploaded Apr 24, 2014.)
We show that if Q is simply connected, every exact Lagrangian cobordism between compact, exact Lagrangians in the cotangent bundle of Q is an h-cobordism. The result is an exercise in basic algebraic topology once one invokes the Abouzaid-Kragh theorem.

Functors (between oo-categories) that aren't strictly unital. (Uploaded June 17, 2016.)
Let C and D be quasi-categories (aka oo-categories). We show that if a semisimplicial set map F: C -> D respects identity maps up to homotopy, then there exists an honest functor F': C -> D which is homotopic to F.

The Fukaya category pairs with Lagrangian cobordisms. (Uploaded July 18, 2016.)
Fix a suitably convex, exact symplectic manifold M. We consider the stable oo-category Lag(M) of non-compact Lagrangians whose (higher) morphisms are (higher) Lagrangian cobordisms between them. We show that this oo-category pairs with the Fukaya category Fuk(M) of compact branes. In fact, we also show that there is a subcategory of Lag(M) which pairs with the wrapped Fukaya category of M. This is a first step in a project to enrich wrapped Fukaya categories over cobordism spectra. As a corollary, we show that cobordant compact branes are equivalent in the Fukaya category. We will also mention several other applications (without proof) of the oo-categorical approach: One can realize Seidel's representation as a \pi_0-level consequence of a map of spaces; stable cobordism groups of non-compact branes map to Floer cohomology groups; some of Biran-Cornea's results can be recovered from the colored planar operad associated to the s-dot constructions of each category; and there is an Eoo map of spectra from exact Lagrangian cobordisms in Euclidean space to the integers.

The Fukaya category pairs with Lagrangian cobordisms exactly. (Uploaded Sep 27, 2016.)
We prove that the pairing (from above) between the Fukaya category and the oo-category of Lagrangian cobordisms respects mapping cones. This is another step toward constructing a lift of Fukaya categories to the level of spectra (in the sense of stable homotopy theory). As corollaries, we show that the map in our previous work from cobordism groups to Floer cohomology lifts to the level of spectra, and one also recovers some results of Biran and Cornea for what we call "vertically collared" cobordisms.

Lagrangian cobordisms are linear over L, in preparation.
We prove that when M is a cotangent bundle, Lag(M) is generated by a single object, the cotangent fiber. When M equals a point, or R^n, we prove that this object is the unit of a symmetric monoidal structure on Lag(M). This further implies that the endomorphisms of a point, L:=End(pt), is an Eoo ring spectrum, and that--after idempotent completion--Lag(M) is equivalent to the category of finitely generated modules over L. We in turn prove that for any Liouville M, and any choice of skeleton Lambda, Lag(M,Lambda) is linear over L. This is the main structural result for the algebraic theory of non-characteristic Lagrangian cobordisms.

The integers are linear over Lagrangian cobordisms, in preparation.
We prove that when M is a point, the functor constructed above lifts to a symmetric monoidal one---in other words, one has an Eoo ring map from L to the integers. All told, this finally makes the conjecture stated by me and David Nadler in our previous work well-defined: That it makes sense to tensor Lag(M,Lambda) by the integers, and ask whether it is equivalent to the partially wrapped Fukaya category of M.

## Factorization Homology, Stratified Spaces, and Link Invariants

This is joint work with David Ayala and John Francis on factorization homology for stratified manifolds.

Factorization homology is also called topological chiral homology by Lurie. It is a topologist's version of Beilinson and Drinfeld's chiral homology, in that it gives rise to invariants of manifolds (not algebraic curves) starting from algebraic gadgets called E_n algebras (which are like chiral algebras for topologists). In our paper, instead of using the usual term, E_n algebra,' we use the term Disk_n^fr algebra.

One point of our papers is that we construct invariants for singular manifolds' (not just smooth manifolds) out of basic algebraic building blocks, for instance E_n-algebras acting on E_k-algebras. The applicable class of singular manifolds is quite large, and includes many stratified spaces -- examples include graphs, n-manifolds with embedded k-manifolds, and manifolds with corners. One application is to construct link invariants. The main results of the paper are a generalization of Lurie and Salvatore's non-abelian Poincare duality, a classification of `homology theories for singular manifolds,' and proofs that the category of singular manifolds is amenable to many classical tools from ordinary differential topology. The big picture of all this, however, leads to Ayala-Francis and Ayala-Francis-Rozenblyum's later work. In a very concrete way, the geometry of stratified spaces captures the algebra of higher categories and higher algebra.

Local structures on stratified spaces with David Ayala and John Francis. Adv. Math. 307 (2017), 903--1028.

Factorization homology of stratified spaces with David Ayala and John Francis. Selecta Mathematica New Series (2016), 1--70.

## Primitives and filtered forms on symplectic manifolds

This is joint work with Li-Sheng Tseng. Given any symplectic manifold M, the symplectic form omega classifies a complex line bundle, hence a circle bundle. (One also gets higher-dimensional sphere bundles by examining direct sums of the line bundle.) Because differential forms on M become a polynomial ring over omega, one can filter the forms on M by their degree as a polynomial in omega. It turns out this defines a sequence of Aoo algebras, which (as Aoo algebras) are equivalent to the usual differential forms on the sphere bundles E mentioned above. Then, via Hodge theory, one can play a game: Given cohomology classes on E, try to find harmonic forms on M (or Poincare dual submanifolds) representing these classes. Playing this game amounts to looking for certain co/isotropic submanifolds of M, and when M is compact, this game seems to define an interesting intersection pairing between isotropics, and coisotropics with boundary. On the algebraic/formal side, the identification of these Aoo algebras with forms on E allow us to prove several functorial properties.

Odd sphere bundles, symplectic manifolds, and their intersection theory with Li-Sheng Tseng. (Uploaded Feb 11, 2017.)