My research is in symplectic and contact topology and geometry. I am currently working on several projects, all of which can be grouped under the banner of using pseudoholomorphic curves to study the topology of contact manifolds, of symplectic manifolds and to understand Hamiltonian flows and diffeomorphisms.

Generally speaking, I study the behaviour of pseudoholomorphic curves. Gromov introduced them in 1985, proving a number of striking results and opening the field. Pseudoholomorphic curves are solutions to a certain non-linear, first order elliptic partial differential equation. The key point is that these curves retain some of the local properties of holomorphic curves from complex analysis, while also exhibiting global properties of a topological and dynamical nature. A good understanding of the space of pseudoholomorphic curves can then be used to extract topological information about symplectic and contact manifolds and about Hamiltonian systems.

One of the features that makes symplectic geometry/topology very exciting is the interplay between rigidity results and flexibility results. Unlike the situation in Riemannian geometry, there are no local invariants of a symplectic manifolds, i.e. all symplectic manifolds are locally the same. Despite this, there are many surprising global rigidity results -- for instance, Lagrangian immersions satisfy an h-principle (i.e. the space of immersions is homotopy equivalent to a space of formal Lagrangian immersions) whereas (by Gromov) there are no embedded Lagrangian spheres in $\mathbb{R}^{2n}$. The main focus of my work is to explore this boundary between rigidity and flexibility, mostly from the rigidity side, though not always.

Symplectic Embeddings of Polydisks:
joint work with Richard Hind
(submitted).

The
question of existence of symplectic embeddings is a typical example of
the interaction between flexibility and rigidity results in symplectic
geometry.
Darboux's classical theorem tells us that we can
symplectically embed a standard symplectic ball into any
symplectic manifold, provided the radius is small enough. Gromov's spectacular
'85 paper tells us there are global obstructions to embedding big balls.
In subsequent work (notably of Ekeland and Hofer), it became clear that
the shape of the ball also matters for embedding questions.

In our paper, we consider the question of embedding a polydisk into a ball,
in dimension 4.
A polydisk is a product of disks, $D^2(a) \times D^2(b)$, where the parameters
a and b denote the areas of the respective factors (thus describing a product symplectic form). Heuristically, this is a very blocky shape, compared to the roundness of a Euclidean ball $B^4(c)$, where we take the parameter to denote $\pi r^2$ (i.e. the cross-sectional area of the ball).

Our theorem is that $D^2(1) \times D^2(2)$ embeds in $B^4(a)$
if and only if $a \ge 3$. In other words, we prove that a non-linear
symplectic embedding doesn't do any better than the obvious inclusion
obtained by considering these as sub-domains of $\mathbb{R}^4$.

The result is interesting because a long and thin bidisk $D^2(1)
\times D^2(a)$ for $a \gt 2$ can be twisted around to obtain a
better embedding than the obvious inclusion. As we take the parameter
$a$ larger and larger, Schlenk is able to approach the volume
constraint. In other words, long and thin polydisks are "close" to
flexible. In the opposite direction, Ekeland and Hofer showed that
for $D^2(1) \times D^2(1)$, the best embedding is the obvious
inclusion. Our example case of $D^2(1) \times D^2(2)$ is precisely
the borderline between these two regimes, and it was unknown which
behaviour would dominate: is it a very "blocky" and "square" polydisk,
or is it a "long, thin" polydisk? Our work shows that it is a
"blocky" polydisk.

Our method of proof is novel. The two
key ingredients are a careful analysis of finite energy foliations
(originally developped by Hofer, Wysocki and Zehnder), using the
intersection theory for punctured holomorphic curves (as developed
extensively by Siefring
and Wendl and Hutchings)
and an application of an idea originally from Hind and Kerman
where we extract information from high dimensional moduli spaces of
curves in the quadric surface.

Stein structures on Lefschetz fibrations and their contact boundaries joint with Chris Wendl; appearing as an appendix to "Families of contact 3-manifolds with arbitrarily large Stein fillings" by Baykur and Van Horn-Morris.

In this appendix, Wendl and I prove that a 4-dimensional
Lefschetz fibration over a general surface 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 our notion of Spinal Open Book decomposition
(which we develop more thoroughly in our work-in-progress with Van Horn-Morris,
and discussed further later on this page.)

Dividing sets as nodal sets of an eigenfunction of the Laplacian (Algebraic and Geometric Topology 11 (2011), no. 3, also available at arXiv:1004.0238).

In this paper, 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.

The methods of my proof are especially interesting, since I address the question using only soft methods of symplectic topology.
Perhaps the main idea is that Stein structures on Riemann surfaces
with boundary only depend on the orientation. Prior to my work, this question
had been considered by using hard tools from the theory of PDE.

The theorem itself is of interest in constructing local pseudoholomorphic
foliations. Unpublished computations of Richard
Siefring and of Joe Coffey show that whenever one has a neighbourhood
described in terms of these eigenfunctions, the characteristic foliation
can be lifted to a (local) foliation by pseudoholomorphic curves. This idea,
generalized a fair bit, enters into some of the technical parts of
my work-in-progress with Wendl and Van Horn-Morris on
Spinal Open Book Decompositions.

Homoclinic orbits and Lagrangian embeddings. (International Mathematics Research Notices 2008, article id rnm151) (Also available at arXiv:math/0608801)

In this paper, 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. In particular, this generalizes
some results of Coti-Zelati, Ekeland and Séré Hofer-Wysocki,
and of Séré. This also partially generalizes some work of
Cieliebak and Séré, proving a weaker result than they do,
but under weaker hypotheses.

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

My PhD thesis is also available, in PDF format and in in PS format.

The main topic is a weaker version of my
theorem on existence of homoclinic orbits. I also use the same techniques
to recover previously known results on existence of periodic orbits.

On symplectic fillings of spinal open books, joint work with Jeremy Van Horn-Morris and Chris Wendl.

In this paper, we introduce a generalization of the notion of an open book
decomposition for a contact 3-manifold. This arises naturally as the description
of the boundary of a Lefschetz fibration over a general surface with boundary.
(Compare this to an open book, which arises naturally as the boundary of
a Lefschetz fibration over a disk.) Using foliation techniques for punctured
pseudoholomorphic curves, we are able to classify symplectic fillings
for a class of contact manifolds that include all of Lutz's circle bundles
and many graph manifolds.

Symplectic homology for affine algebraic varieties. Joint work with Luis Diogo.

Our goal is to compute Symplectic Homology (with its ring structure)
for affine algebraic varieties, by exploiting its relationship to
Contact Homology. Symplectic Homology, introduced by Floer and Hofer
and extensively developped by Viterbo, is a version of Floer Homology
for non-compact symplectic manifolds, and carries a very rich
structure. For example, the Symplectic Homology of a cotangent
bundle recovers the homology of the loop space. A related project
is to compute Floer Homology with its action filtration for a special
family of Hamiltonians on projective varieties. As a first step,
Diogo and I provide an explicit description of the relevant moduli
spaces in terms of holomorphic spheres in the projective variety,
giving a formula in terms of genus 0 Gromov-Witten invariants. Our
eventual goal is to compute symplectic quasistates, which in turn
would enable us to determine Hamiltonian non-displaceability of
certain sets. (This ongoing project is joint work with Borman,
Diogo, Eliashberg and Polterovich.)

Coisotropic Hofer-Zehnder capacities. Joint work with Tony Rieser.

We introduce a notion of a capacity relative to a coisotropic submanifold,
and introduce a modification of the Hofer-Zehnder capacity to obtain
one in this setting. In the case of a Lagrangian, we relate this
capacity to the Barraud-Cornea real Gromov width. We also obtain
energy-capacity inequalities for a class of Lagrangians that include
all weakly exact ones. We note that Rizell observes that certain
Lagrangians in $\mathbb{C}^n$ constructed by Ekholm, Eliashberg,
Murphy and Smith have infinite real Gromov width, so it seems that
it is essential that the Lagrangian be unobstructed in order for
the energy capacity inequality to hold.