Given an ordinary differential equation A(x,y)dx + B(x,y)dy = 0, its solutions f(x,y) define a decomposition of the plane outside the zeros of A(x,y) and B(x,y) into regular curves. This is a prototype of a foliation, the leaves being the solutions of the given differential equation. In general, a foliation will be a generalization of this concept, i.e. instead of taking one equation, we take a system of equations, and to have solutions we demand an integrability condition. In this talk, I will introduce the concept of holomorphic foliation and give a characterization of regular foliations on rational surfaces.

In dimension 3, the theory of codimension 2 contact submanifolds is better known as the transverse knot theory of a contact manifold, a theory which has a complete description in terms of braid theory. In higher dimensions, almost nothing is known, but there is a small (but growing) list of results. I will explain a method developed with A. Kaloti to use open books and Lefschetz fibrations to study codimension 2 contact embeddings. I will present as some initial applications and some interesting behaviors.

Wave packets describe the quantum vibrations of a molecule. They are highly oscillatory, highly localized and move in high dimensional configuration spaces. The governing equation is the time-dependent Schr\"odinger equation in the semiclassical regime. The talk addresses three meshless numerical methods for catching wave packets: single Gaussian beams, superpositions of them, and the so-called linearized initial value representation.

Open Gromov-Witten (OGW) invariants should count pseudoholomorphic maps from Riemann surfaces with boundary to a symplectic manifold, with boundary conditions and various constraints on boundary and interior marked points. The presence of boundary leads to bubbling phenomena that pose a fundamental obstacle to invariance. In a joint work with J. Solomon, we developed a general approach to defining genus zero OGW invariants. For real symplectic manifolds in dimensions 2 and 3, these invariants are strongly related to Welschinger's invariants.