Research

Lagrangian Hofer–Zehnder capacities and energy-capacity inequalities

With Antonio Rieser. Preprint, 2025.

In our earlier paper, Rieser and I defined a Hofer–Zehnder capacity relative to a coisotropic submanifold. Here we return to the case of a Lagrangian. Using a persistent homology approach to filtered Floer theory, following Polterovich and Shelukhin, we obtain two energy-capacity inequalities for displaceable Lagrangians, a local one and a global one. To do this we modify the capacity by imposing topological conditions on the admissible Hamiltonians, and we show by constructing an example that one of the inequalities breaks down without them. This suggests that our capacity is closely related to the real Gromov radius of Barraud and Cornea.

On symplectic fillings of spinal open book decompositions, I and II

With Jeremy Van Horn-Morris and Chris Wendl. Preprints, 2018 and 2020.

An open book decomposition is the structure that appears on the boundary of a Lefschetz fibration over a disk. A spinal open book is what appears on the boundary of a Lefschetz fibration over a general compact oriented surface with boundary. Part I develops the part of the theory that does not need punctured J-holomorphic curves: spinal open books support contact structures, Lefschetz fibrations carry compatible symplectic and Stein structures, and there is a new symplectic cobordism, spine removal surgery, which generalizes constructions of Eliashberg, of Gay and Stipsicz, and of Wendl.

Part II uses punctured J-holomorphic curves to show that, for a large class of contact 3-manifolds supported by spinal open books with planar pages, the symplectic fillings are classified up to deformation by the Lefschetz fibrations that fill the spinal open book. This recovers most of the known classification results for fillings in dimension four, and gives new ones, for example for \(S^1\)-invariant contact structures on circle bundles.

Spine removal surgery and the geography of symplectic fillings

With Chris Wendl. Michigan Math. J., 2021.

This note is a spin-off of the spinal open book project. Baykur and Van Horn-Morris found contact 3-manifolds whose Stein fillings have arbitrarily large Euler characteristic. We show that this cannot happen in the planar case: for any contact 3-manifold supported by a spinal open book with planar pages, there is a universal bound on the Euler characteristic and signature of its minimal symplectic fillings. The proof uses spine removal surgery and a small amount of J-holomorphic curve theory, with no appeal to mapping class groups.

Coisotropic Hofer–Zehnder capacities and non-squeezing for relative embeddings

With Antonio Rieser. J. Symplectic Geom., 2020.

We introduce the notion of a symplectic capacity relative to a coisotropic submanifold, and construct two examples by modifying the Hofer–Zehnder capacity. If a symplectic capacity measures the width of a symplectic manifold, a coisotropic capacity measures the symplectic size of a coisotropic submanifold inside it. As consequences, we obtain a non-squeezing theorem for symplectic embeddings that respect coisotropic constraints, and existence results for leafwise chords on energy surfaces.

Symplectic homology of complements of smooth divisors

With Luís Diogo. J. Topol., 2019.

Symplectic homology carries a great deal of structure and is usually very hard to compute. We consider the complement of a smooth symplectic divisor \(\Sigma\) in a closed symplectic manifold \(X\). A large family of examples come from looking at a smooth ample divisor in a projective variety. Under monotonicity assumptions, we construct a chain complex that computes the symplectic homology of the complement, with a differential given by counts of pseudoholomorphic spheres in \(\Sigma\) and in \(X\), that is, by absolute and relative Gromov–Witten invariants. The strategy is inspired by the Bourgeois–Oancea exact sequence relating symplectic homology to contact homology.

Morse–Bott split symplectic homology

With Luís Diogo. J. Fixed Point Theory Appl., 2019.

This paper develops the technical machinery used in our paper on divisor complements. It also is of interest in its own right. For a Liouville domain whose boundary is a prequantization space, we define a chain complex whose differential counts Floer cylinders with cascades, in the spirit of Morse–Bott homology. We show that the moduli spaces of simple cascades are cut out transversally for generic auxiliary data, and index estimates then show that very few combinatorial types of cascades contribute to the differential.

On properties of Bourgeois contact structures

With Aleksandra Marinković and Klaus Niederkrüger. Algebr. Geom. Topol., 2019.

The Bourgeois construction takes a contact open book on a manifold \(V\) and produces a contact structure on \(V \times T^2\). We study properties of \(V\) that are inherited by \(V \times T^2\) and some that are not. The appendix reviews Giroux's recent formalism for contact open books and works through a few classical examples in that language.

Stein structures on Lefschetz fibrations and their contact boundaries

With Chris Wendl. Appendix to a paper of Baykur and Van Horn-Morris, J. Differential Geom., 2015.

Wendl and I prove that a 4-dimensional Lefschetz fibration over a general surface with boundary admits a Stein structure, canonically up to Stein homotopy. Furthermore, the contact structure on the smoothed boundary is uniquely determined up to isotopy. We also explain the relationship to spinal open book decompositions.

Symplectic embeddings of polydisks

With Richard Hind. Selecta Math., 2015.

A polydisk \(D^2(a) \times D^2(b)\) is a product of disks of areas \(a\) and \(b\), and \(B^4(c)\) is the ball of cross-sectional area \(c\). We prove that \(D^2(1) \times D^2(2)\) embeds symplectically in \(B^4(a)\) if and only if \(a \ge 3\). In other words, no symplectic embedding does better than the obvious inclusion.

Dividing sets as nodal sets of an eigenfunction of the Laplacian

Algebr. Geom. Topol., 2011.

I show that the neighbourhood of a convex surface in a contact 3-manifold can be described in terms of eigenfunctions of an associated Laplacian on the surface. This answers a question of Komendarczyk. The proof uses only soft methods of symplectic topology; the main idea is that Stein structures on Riemann surfaces with boundary depend only on the orientation. Earlier work on this question had used hard tools from the theory of partial differential equations.

The result is useful for constructing local pseudoholomorphic foliations, and a generalization of this idea enters into the technical parts of the spinal open book papers.

Homoclinic orbits and Lagrangian embeddings

Int. Math. Res. Not. IMRN, 2008.

I prove the existence of orbits homoclinic to a rest point in a Hamiltonian system, under various geometric hypotheses, in both the autonomous and time-dependent cases. This generalizes results of Coti-Zelati, Ekeland and Séré, of Hofer and Wysocki, and of Séré. It also partially generalizes work of Cieliebak and Séré, proving a weaker result than theirs under weaker hypotheses.

The proof uses only techniques from symplectic topology, notably Chekanov's theorem relating the displacement energy of a Lagrangian to the area of a disk with boundary on it. In retrospect, this can be understood as a foreshadowing of the later work of Bourgeois, Ekholm and Eliashberg on the effect of Legendrian surgery on symplectic homology.