I work in the intersection of microlocal, semiclassical and harmonic analysis. Key problems I am interested in are:

• How do high energy solutions to PDE and pseudodifferential equations behave? What kind of concentration properties do they display? Typically these kinds of problems involved proving a growth bound for a function norm.

• How do these concentration properties fit in with the theories and hypothesis of Quantum Chaos?

• How can we use growth bounds for solutions to obtain mapping norm estimates for classical harmonic analysis operators (such as Bochner-Riesz means)?

• How can we use Fourier integral operators to produce effective analysis/synthesis systems?

• How do random waves behave, what can we expect at a small scale?