I am generally interested in global questions in analysis, as they relate to geometry and also to mathematical physics. I have been working on spectral problems for operators like the Laplacian or the Schrodinger operator, in situations where the spectrum is discrete. For the Laplacian, I am interested in the question of how its spectrum and eigenfunctions relate to the geometry of the given manifold; for the Schrodinger operator I am interested in the relationship between its spectrum and eigenfunctions and the corresponding classical mechanical system. The first of these problems belongs to the general question of can you hear the shape of a drum?, while the other arises in mathematical physics.
Technically, my field is microlocal analysis. This is a branch of partial differential equations that grew out of the method of stationary phase, and that has been very successful, particularly in linear equations. It establishes a link between PDE and symplectic geometry which is both very useful and very deep. This relationship is also apparent in representation theory of Lie groups and harmonic analysis.
There are many open questions along the lines outlined in the first paragraph that I find very interesting, and that are pretty much open. A first big question is to understand quantum chaos, which is a catchy term for the problem of understanding how, for example, the dynamical properties of the geodesic flow of a compact Riemannian manifold are reflected in its Laplacian. Very little is known rigorously about this. The types of issues raised above are terra incognita for systems, and for quantum field theories.