# Research

Here is a list of my papers, with relevant links and a brief description.  I've tried to make the description a little bit more informal than an abstract.  Most of my research involves a variant of symplectic field theory, called "embedded contact homology".  For background on this, see Hutchings' excellent "Lecture notes on embedded contact homology".  The links in the descriptions also give some background which is hopefully useful.

[1] The absolute gradings on embedded contact homology and Seiberg-Witten Floer cohomology, Alg. and Geom. Topol. 13 (2013) 2239-2260.

Description:  Embedded contact homology (ECH) admits an absolute grading by homotopy classes of 2-plane fields.  This paper shows that this grading is a topological invariant by relating it to an analogous structure on Seiberg-Witten Floer cohomology.  The paper also contains a formula of potentially independent interest relating the expected dimension of the Seiberg-Witten moduli space over a completed symplectic cobordism to the ECH index of a corresponding relative homology class.

[2] The asymptotics of ECH capacities [with M. Hutchings and V. Ramos], Invent. Math. 199.1 (2015), 187-214.

Description: In "Quantitative embedded contact homology", Hutchings used embedded contact homology to define a sequence of obstructions to four-dimensional symplectic embeddings, called "ECH capacities".  We show that for a four-dimensional Liouville domain with all ECH capacities finite, the ECH capacities recover the classical symplectic volume in their asymptotic limit.  This follows from a more general theorem relating the volume of a contact three-manifold to the asymptotics of the amount of symplectic action needed to represent certain classes in ECH.  This latter result was used in [3], below.

[3] From one Reeb orbit to two [with M. Hutchings], to appear in Jour. of Diff. Geom.

Description: The Weinstein conjecture states that every Reeb vector field on a closed contact manifold has at least one closed periodic orbit.  In "The Seiberg-Witten equations and the Weinstein conjecture", Taubes proved the Weinstein conjecture in dimension 3.  Here we show that every (possibly degenerate) contact form on a closed three-manifold has at least two embedded Reeb orbits. We also show that if there are exactly two embedded Reeb orbits, then the product of their symplectic actions is less than or equal to the contact volume of the manifold.  Our proofs use the main theorem from [2].

[4]  Symplectic embeddings into four-dimensional concave toric domains [with K. Choi, D. Frenkel, M. Hutchings, and V. Ramos], to appear in Journ. of Top.

Description: While much is known about when one four-dimensional symplectic ellipsoid can be embedded into another, symplectic embedding problems involving other domains are in general poorly understood.  Here we begin the study of symplectic embeddings of four-dimensional "toric domains", which is further studied in [5] below.  More specifically, we compute the ECH capacities (see the description of [2]) of a large family of symplectic four-manifolds with boundary, called "concave toric domains".  Examples include the (nondisjoint) union of two ellipsoids in \R^4.  We use these calculations to find sharp obstructions to certain symplectic embeddings involving concave toric domains.  For example: (1) we calculate the Gromov width of every concave toric domain; (2) we show that many inclusions of an ellipsoid into the union of an ellipsoid and a cylinder are "optimal"; and (3) we find a sharp obstruction to ball packings into certain unions of an ellipsoid and a cylinder.

Description:  This paper continues the study of symplectic embeddings of toric domains.  In [4], we computed the ECH capacities (see the description in [2]) of all "concave" toric domains, and showed that these give sharp obstructions in several interesting cases.  This paper shows that these obstructions are sharp for all symplectic embeddings of concave toric domains into "convex" ones.  In an appendix with Choi, we prove a new formula for the ECH capacities of most convex toric domains, which shows that they are determined by the ECH capacities of a corresponding collection of balls.

[6] Symplectic embeddings of products [with R. Hind]

Description:  This paper is about higher dimensional symplectic embeddings; in general, much remains unknown about these kinds of problems.  In four-dimensions, a celebrated result is McDuff and Schlenk's computation of when precisely an ellipsoid can be embedded into a ball: they found that when the ellipsoid is close to round, the answer is given by an "infinite staircase" determined by the odd-index Fibonacci numbers.  We show that this result also holds true in all higher dimensions for the "stabilized" problem.

[7]  Ehrhart polynomials and symplectic embeddings of ellipsoids [with A. Kleinman]

Description:  Here we establish some new connections between symplectic geometry and combinatorics.  ECH capacities give an obstruction to symplectically embedding one four-dimensional ellipsoid into another, and McDuff showed that this obstruction is sharp.  ECH capacities of ellipsoids can be interpreted as lattice point counts in appropriate triangles, so this result links symplectic geometry and enumerative combinatorics.  Here, we use this point of view to give a new proof of the staircase of McDuff and Schlenk mentioned above.  To do this, we show that a special class of triangles --- the "Fibonacci triangles" --- give new examples of a combinatorial phenomenon of independent interest called period collapse.    The relevant combinatorics are further studied in [8], below; work in progress is developing symplectic applications.

[8] New examples of period collapse [with T. Li and R. Stanley]

Description: Most of the machinery for counting points in polygons is for rational polytopes.  The "Fibonacci triangles" from [7] suggest that some of these results can be extend to certain irrational polytopes.  We prove this for an interesting class of triangles, and a few higher dimensional polytopes are also explored.  We also show that there are connections with the even-index Fibonacci numbers.