PDE Analysis

• Qualitative study of solutions to linear and nonlinear partial differential equations: existence and regularity of solutions, description of singularities in fluid mechanics, nonlinear hyperbolic equations, existence and stability of solitary waves, kinetic models
• Mixed problems and wave-structure interactions
• Control and inverse problems: control for Navier–Stokes and wave equations, observability and stabilization of Schrödinger equations on manifolds, inverse scattering problems
• General relativity: Einstein equations, relativistic constraint equations, stability of black holes
• Harmonic analysis and parabolic PDEs: spectral multipliers, Riesz transforms on manifolds, heat kernel estimates, Strichartz estimates, free boundary problems
• Calculus of variations: study of stationary solutions, critical points in infinite dimensions (construction and description), Euler–Lagrange equations

Mathematical Physics

• Spectral asymptotics: Schrödinger and Dirac operators, non-self-adjoint operators, hypoelliptic and subelliptic operators, semiclassical limit, distribution of eigenvalues and resonances, magnetic operators, spectral gaps for stochastic processes, metastability, concentration of high-energy eigenfunctions, semiclassical measures and quantum limits, quantum KAM theory
• Classical and quantum scattering theory: inverse spectral problems, dynamical zeta functions
• Spectral and microlocal analysis of PDE systems: hyperbolic systems, transmission problems, matrix Schrödinger operators

Modeling and Numerical Analysis

• Population dynamics: epidemic models, predator–prey systems, existence and stability of traveling waves
• Modeling, analysis, and simulations in coastal oceanography: modeling, theoretical analysis, numerical analysis
• Modeling, analysis, and simulations in fluid mechanics and plasma physics: fluid models, moment models, kinetic models for complex gases, dissipative or dispersive perturbations of hyperbolic systems, Lattice-Boltzmann methods