Random matrix theory and random processes, integrable and exactly solvable lattice models, Painlevé transcendents, orthogonal polynomials, and special function and nonlinear wave theory.
Random holomorphic functions, Gaussian processes, nodal sets of Laplace eigenfunctions, Bergman- or Fock-type spaces of functions.
Nonlinear partial differential equations, free boundary problems, general relativity and mathematical physics.
Spectral perturbation theory of selfadjoint operators in Hilbert space, spectral theory of Hankel and Toeplitz operators, scattering theory and spectral shift function theory.
Geometric analysis, index theory, elliptic boundary value problems, microlocal analysis (pseudodifferential and fourier integral operators) and topological quantum field theory.
Operator theory, partial differential and integral equations, complex variables and applications to fluid dynamics and elasticity theory.