Problems in microlocal analysis with an emphasis on spectral asymptotics and scattering theory

In this project, you will gain exposure to a variety of ideas from microlocal analysis and how they are used in problems initiated with the field of quantum mechanics, more specifically quantum chaos.

Microlocal analysis originated in the study of partial differential equations through the lens of phase space methods, that is, by combining ideas from symplectic geometry and Fourier analysis to investigate qualitative and quantitative properties of PDEs. In particular, the philosophy has lead to major advances in the understanding of linear PDEs in the last 50 years, as can be seen from the recent breakthroughs of Bourgain-Dyatlov on spectral gaps and Hintz-Vasy on the global nonlinearity stability of certain spacetimes, for example.

Our research goal is to combine microlocal techniques along with those from a variety of other disciplines (such as dynamical systems, differential geometry, and number theory) to give a novel approach to problems which are somewhat out of reach by using solely microlocal methods. Some specific projects involve random spherical harmonics and the use of geometric measure theory in the restriction theory of eigenfunctions.

A component of the programme will be focused on obtaining numerics for semi-classical asymptotics in certain models, such as compact hyperbolic surfaces. Quantized toral automorphisms can be considered as well.

The skills to be developed include, but are not limited to, the following: distribution theory, basic Fourier analysis on $\R^n$, calculus on smooth manifolds, spectral theory of self-adjoint and non-self adjoint operators, basic harmonic analysis on $\R^n$, elements of dynamical systems, elements of geometric measure theory, and some analytic number theory.