I am interested in developing and applying tecniques from microlocal and semi-classical analysis to study operators on sub-riemannian manifolds. Sub-Riemannian geometry is a vast generalization of Riemannian geometry, providing a framework for gauge theory, control theory, and quantum mechanical phenomena, among other things. Singular objects in sub-Riemannian geometry have no Riemannian counterpart, and appear to dominate spectral behavior. Representation-theoretic tecniques play a big role in my research. The main fields involved in my work are microlocal analysis, semi-classical analysis, sub-Riemannian geometry, inverse problems, representation theory. Some general research dierctions include:
Microlocal and Semi-classical analysis of sub-elliptic operators:
- Develop a robust symbol calculus on sub-Riemannian manifolds using representation theory.
- Characterize quantum limits for subelliptic operators on sub-Riemannian manifolds. Relate their invariance properties to the underlying geodesic flow.
Integral geometry and X-ray transforms on Carnot Groups:
- Proving injectivity and stability of X-ray transforms on Carnot Lie groups.
Dynamics of a particle in a Yang-Mills field:
- Study the scattering map and the X-ray transform for a particle in a Yang-Mills field on surfaces.
- Localization of a quantum particle in a Yang-Mills field and its energy asymptotics.