Alejandro Uribe

I am generally interested in global questions in analysis, as they relate to geometry and also to mathematical physics. I have worked 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.

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.

Currently I am working on propagation of wave packets (or coherent states) under Zeno and non self-adjoint Hamiltonians.