My research interests include soliton and vortex existence/stability, well-posedness theory for quasilinear and degenerate models, linear scattering theory, spectral theory, Strichartz/dispersive estimates, microlocal analysis, properties of eigenfunctions on billiards, classical dynamics descriptions of Analysis & PDEs solutions, graph Laplacian spectral problems, spectral theory of combinatorial and quantum graphs, numerical simulations, asymptotic analysis, degenerate dispersive/diffusive equations with applications in nonlinear optics, fluid dynamics, topological physics, material science, quantum mechanics, general relativity, quantum chemistry, statistical physics, and weak turbulence.