# Research

Here is a list of my papers, with links and a brief description. Please email me if you’d to learn more.

[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] and [9], below.

[3] From one Reeb orbit to two [with M. Hutchings], Journ. of Diff. Geom. 102.1 (2016), 25-36.

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], Journ. of Top. 7.4 (2014), 1054-1076.

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 studied further 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.

[5] Symplectic embeddings of products [with R. Hind], Comm. Math. Helv., 93 (2018), 1-32.

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 symplectically 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.

[6] Ehrhart polynomials and symplectic embeddings of ellipsoids [with A. Kleinman] Revised version to appear in Journ. Lon. Math Soc.

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.

[7] New examples of period collapse [with T. Li and R. Stanley], Disc. and Comp. Geo. 61(2): 227-246 (2019).

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.

[8] Symplectic embeddings from concave toric domains into convex ones, J. Diff. Geom. 112 (2019), 199-232.

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.

[9] Symplectic embeddings of four-dimensional ellipsoids into integral polydiscs [with D. Frenkel and F. Schlenk], Alg. Geom. Top. 17 (2017) 1189-1260.

Description: Another curious aspect of the McDuff-Schlenk computation mentioned in [6] is that when the ellipsoid is sufficiently stretched, all embedding obstructions into a ball vanish except for the classical volume obstruction. Here we show that when the target is sufficiently stretched, the problem becomes much more regular as well.

[10] Torsion contact forms in three dimensions have two or infinitely many Reeb orbits [with M. Hutchings and D. Pomerleano], to appear in Geom. Top.

Description: In [3], we showed that every Reeb vector field on a closed three-manifold has at least two distinct closed embedded orbits. Examples exist with exactly two orbits, but we show here that for any nondegenerate contact form, if there are more than two distinct orbits then there are infinitely many, as long as the Chern class of the associated contact structure is torsion. Combined with previous results, this gives many examples of three-manifolds for which every nondegenerate contact form has infinitely many Reeb orbits.

[11] The ghost stairs stabilize to sharp symplectic embedding obstructions [with R. Hind and D. McDuff], Journ. of Top. 11.2 (2018), 309-378.

Description: This paper continues to study higher dimensional symplectic embedding problems. In [6], we began the study of the stabilized symplectic embedding problem, and we showed that the McDuff-Schlenk infinite staircase persists under stabilization. Here, we show that a different kind of phenomenon occurs. Namely, in the McDuff-Schlenk computation there is a curious sequence of symplectic obstructions, determined by the even index Fibonacci numbers, called the “ghost stairs”. These ghost obstructions are weaker then the volume obstruction, and so are not active in dimension four. We show that, in contrast, the ghost obstructions are optimal after we stabilize.

[12] The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed [with M. Mazzucchelli], to appear in Comm. Math. Helv.

Description: A classical theorem due to Wadsley implies that, on a contact manifold all of whose Reeb orbits are closed, there is a common period for the Reeb orbits. In this paper we show that, for any Reeb flow on a closed 3-manifold, the following conditions are actually equivalent: (1) every Reeb orbit is closed; (2) all closed Reeb orbits have a common period; (3) the action spectrum has rank 1. We also show that, on a fixed closed 3-manifold, a contact form with an action spectrum of rank 1 is determined (up to pull-back by diffeomorphisms) by the set of minimal periods of its closed Reeb orbits.

[13] Sub-leading asymptotics of ECH capacities [with N. Savale], to appear in Selecta Math.

Old updates: updated preprint available here (I put it here rather than on the arxiv so as to avoid spamming the arxiv): Sub-leading asymptotics of ECH capacities

Description: This paper continues the study of the asymptotics of the ECH spectrum. In [2], we showed that the leading asymptotics of the ECH spectrum recover the contact volume. Our main theorem here is a new bound on the subleading asymptotics.

[14] Proof of the simplicity conjecture [with V. Humiliere and S. Seyfaddini]

Description: We show that the group of compactly supported area-preserving homeomorphisms of the two-disc is not simple; in fact, we prove the apriori stronger statement that this group is not perfect. This settles what is known as the “simplicity conjecture” in the affirmative. As a corollary, we find that for closed surfaces, the kernel of the “mass-flow” homomorphism is not simple. This answers a question of Fathi, and implies that the connected component of the identity in the group of area-preserving homeomorphisms of the two-sphere is not simple; the two-sphere is the only closed manifold for which the question of simplicity of the component of the identity in the group of volume-preserving homeomorphisms remained open — for other closed manifolds, this was settled by Fathi in 1980.

An important step in our proof involves verifying in a special case a conjecture of Hutchings concerning recovering the Calabi invariant from the asymptotics of spectral invariants defined using periodic Floer homology. Another key step involves proving that these spectral invariants extend continuously to area-preserving homeomorphisms of the disc. These two properties of PFH spectral invariants are potentially of independent interest. Our strategy is partially inspired by the approach of Oh towards the simplicity question.

[15] On infinite staircases in toric symplectic 4-manifolds [with T. Holm, A. Mandini, and A. Pires]

An influential result of McDuff and Schlenk asserts that the function that en- codes when a four-dimensional symplectic ellipsoid can be embedded into a four-dimensional ball has a remarkable structure: the function has infinitely many corners, determined by the odd-index Fibonacci numbers, that fit together to form an infinite staircase.

This work has recently led to considerable interest in understanding when the ellipsoid embedding function for other symplectic 4-manifolds is partly described by an infinite stair- case. We provide a general framework for analyzing this question for a large family of targets, called finite type convex toric domains, which we prove generalizes the class of closed toric symplectic four-manifolds. When the target is of finite type, we prove that any infinite staircase must have a unique accumulation point a0, given as the solution to an explicit quadratic equation. Moreover, we prove that the embedding function at a0 must be equal to the classical volume lower bound. In particular, our result gives an obstruction to the existence of infinite staircases that we show is strong.

In the special case of rational convex toric domains, we can say more. We conjecture a complete answer to the question of existence of infinite staircases, in terms of six families that are distinguished by the fact that their moment polygon is reflexive. We then provide a uniform proof of the existence of infinite staircases for our six families, using two tools. For the first, we use recursive families of almost toric fibrations to find symplectic embeddings into closed symplectic manifolds. In order to establish the embeddings for convex toric domains, we prove a result of potentially independent interest: a four-dimensional ellipsoid embeds into a closed toric symplectic four-manifold if and only if it can be embedded into a corresponding convex toric domain. For the second tool, we find recursive families of convex lattice paths that provide obstructions to embeddings. We conclude by reducing our conjecture that these are the only infinite staircases among rational convex toric domains to a question in number theory related to a classic work of Hardy and Littlewood.

[16] Special eccentricities of rational four-dimensional ellipsoids, to appear in Alg. and Geom. Top.

Description: A striking result of McDuff and Schlenk asserts that in determining when a four-dimensional symplectic ellipsoid can be symplectically embedded into a four-dimensional symplectic ball, the answer is governed by an “infinite staircase” determined by the odd-index Fibonacci numbers and the Golden Mean. Here we study embeddings of one four-dimensional symplectic ellipsoid into another, and we show that if the target is rational, then the infinite staircase phenomenon found by McDuff and Schlenk is quite rare. Specifically, in the rational case, there is an infinite staircase in precisely three cases — when the target has “eccentricity” 1, 2, or 3/2; in all other cases the answer is given by the classical volume obstruction except on finitely many compact intervals on which it is linear. This verifies in the special case of ellipsoids a conjecture by Holm, Mandini, Pires, and the author.

[17] PFH spectral invariants on the two-sphere and the large scale geometry of Hofer’s metric [with V. Humiliere and S. Seyfaddini], to appear in JEMS

Description: We resolve three longstanding questions related to the large scale geometry of the group of Hamiltonian diffeomorphisms of the two-sphere, equipped with Hofer’s metric. Namely: (1) we resolve the Kapovich-Polterovich question by showing that this group is not quasi-isometric to the real line; (2) more generally, we show that the kernel of Calabi over any proper open subset is unbounded; and (3) we show that the group of area and orientation preserving homeomorphisms of the two-sphere is not a simple group. Central to all of our proofs are new sequences of spectral invariants over the two-sphere, defined via periodic Floer homology.

[18] Contact three-manifolds with exactly two simple Reeb orbits [with M. Hutchings, H. Liu, and U. Hryniewicz], to appear in Geom. Topol.

Description: It is known that every contact form on a closed three-manifold has at least two simple Reeb orbits, and a generic contact form has infinitely many. We show that if there are exactly two simple Reeb orbits, then the contact form is nondegenerate. Combined with a previous result, this implies that the three-manifold is diffeomorphic to the three-sphere or a lens space, and the two simple Reeb orbits are the core circles of a genus one Heegaard splitting. We also obtain further information about the Reeb dynamics and the contact structure. For example the contact struture is universally tight; and in the case of the three-sphere, the contact volume and the periods and rotation numbers of the simple Reeb orbits satisfy the same relations as for an irrational ellipsoid.

[19] Higher symplectic capacities and the stabilized embedding problem for integral ellipsoids [with R. Hind and K. Siegel], to appear in Journ. Fixed Point Theory and App., special issue in honor of Claude Viterbo

Description: The third named author has been developing a theory of “higher” symplectic capacities. These capacities are invariant under taking products, and so are well-suited for studying the stabilized embedding problem. The aim of this note is to apply this theory, assuming its expected properties, to solve the stabilized embedding problem for integral ellipsoids, when the eccentricity of the domain is the opposite parity of the eccentricity of the target and the target is not a ball. For the other parity, the embedding we construct is definitely not always optimal; also, in the ball case, our methods recover previous results of McDuff, and of the second named author and Kerman. There is a similar story, with no condition on the eccentricity of the target, when the target is a polydisc: a special case of this implies a conjecture of the first named author, Frenkel, and Schlenk concerning the rescaled polydisc limit function. Some related aspects of the stabilized embedding problem and some open questions are also discussed.

Note: this article expands on and replaces a previous note that was posted here; that note was joint with Siegel

[20] Quantitative Heegaard Floer cohomology and the Calabi invariant [with V. Humiliere, C. Y. Mak, S. Seyfaddini, and I. Smith]

Description: We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: we show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple; we extend the Calabi homomorphism to the group of Hameomorphisms constructed by Oh-Müller; and, we construct an infinite dimensional family of quasimorphisms on the group of area and orientation preserving homeomorphisms of the two-sphere. Our invariants are inspired by recent work of Polterovich and Shelukhin defining and applying spectral invariants for certain classes of links in the two-sphere.

[21] The smooth closing lemma for area-preserving surface diffeomorphisms [with R. Prasad and B. Zhang]

Description: Spectral invariants arising from twisted periodic Floer homology have recently played a key role in resolving various open problems in two-dimensional dynamics. We resolve in great generality a conjecture of Hutchings regarding the relationship between the asymptotics of the twisted PFH spectral invariants and the Calabi homomorphism for area-preserving disk maps. Our result, by an argument similar to that of Irie in the setting of Reeb flows, allows us to prove the smooth closing lemma for area-preserving diffeomorphisms of a closed surface.

[22] A note on the existence of U-cyclic elements in periodic Floer homology [with D. Pomerleano, R. Prasad and B. Zhang], to appear in Proc. AMS (Series B: open access).

Description: Edtmair-Hutchings have recently defined, using periodic Floer homology, a U-cycle property for Hamiltonian isotopy classes of area-preserving diffeomorphisms of closed surfaces. They show that every Hamiltonian isotopy class satisfying the U-cycle property satisfies the smooth closing lemma and also satisfies a kind of Weyl law involving the actions of certain periodic points; they show that every rational isotopy class on the two-torus satisfies the U-cycle property. The purpose of this note is to explain why the U-cycle property holds for every rational Hamiltonian isotopy class.

[23] Subleading asymptotics of link spectral invariants and homeomorphism groups of surfaces [with V. Humiliere, C. Y. Mak, S. Seyfaddini, and I. Smith], to appear in Forum of Math., Pi.

Description: In previous work, we defined "link spectral invariants" for any compact surface and used these to study the algebraic structure of the group of area-preserving homeomorphisms; in particular, we showed that the kernel of Fathi's mass-flow homomorphism is never simple. A key idea for this was a kind of Weyl law, showing that asymptotically the link spectral invariants recover the classical Calabi invariant.

In the present work, we use the subleading asymptotics in this Weyl law to learn more about the algebraic structure of these homeomorphism groups in the genus zero case. In particular, when the surface has boundary, we show that the kernel of the Calabi homomorphism on the group of hameomorphisms is not simple, answering an old question of Oh and Müller; this contrasts the smooth case, where the kernel of Calabi is simple. We similarly show that the group of hameomorphisms of the two-sphere is not simple. Related considerations allow us to extend the Calabi homomorphism to the full group of compactly supported area-preserving homeomorphisms, answering a longstanding question of Fathi. In fact, we produce infinitely many distinct extensions.

Central to the applications is that we show that the subleading asymptotics for smooth, possibly time-dependent, Hamiltonians are always O(1), and for certain autonomous maps recover the Ruelle invariant. The construction of a hameomorphism with "infinite Ruelle invariant" then shows that a normal subgroup with O(1) subleading asymptotics is proper.

[24] Generic higher asymptotics of holomorphic curves and applications [with M. Hutchings and B. Zhang]

6/30/19 Draft available here: Higher asymptotics draft

Description: We study the higher asymptotic behavior of a generic, somewhere injective, J-holomorphic curve in the symplectization of a contact 3- manifold. Our main theorem is that for generic J, a generic curve has “regular” positive and negative ends. As applications: (1) we provide new obstructions to the existence of J-holomorphic curves whose im- age is close to a holomorphic building containing trivial cylinders; (2) we verify a conjecture by the second author and Nelson and extend the definition of cylindrical contact homology to more general cases; and (3) we show that generically, the refined ECH index inequality is an equality.