Research
My research areas are symplectic and contact topology/geometry. I am currently working on several projects, all of which can be grouped under the banner of using pseudoholomorphic curves to better understand contact manifolds or to better understand Hamiltonian dynamics.
Generally speaking, I study the behaviour of pseudoholomorphic curves. Gromov introduced them in 1985, proving a number of striking results and opening a 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 properties of holomorphic curves from complex analysis. A good understanding of the space of pseudoholomorphic curves can be used to extract information from symplectic and contact manifolds and also about Hamiltonian systems.
My work combines tools from contact topology and symplectic topology with tools and ideas from complex geometry, Morse theory, non-linear functional analysis and geometric PDE (specifically, the theory of first-order, non-linear, elliptic PDE on manifolds).
My Ph.D. thesis (2005) is available in PDF format and in in PS format.
An existence result for orbits homoclinic to a rest point in a Hamiltonian system : Homoclinic orbits and Lagrangian embeddings. (IMRN 2008, article id rnm151)
A generalization of the last chapter of my thesis will be coming soon.