The PDE group at UNC consists of the following faculty (click for more information):
- Eigenfunctions and Quantum Chaos, including scarring vs. non-concentration, ergodicity, restrictions of eigenfunctions.
- Scattering Theory, including resolvent estimates, trapping, local smoothing, Strichartz estimates and dispersion.
- Damped wave type equations, including Nonself-adjoint spectral theory, control theory, overdamping, advection assisted diffusion, microfluidics.
- Semiclassical analysis, including high-energy/frequency limit in quantum mechanical problems, exotic symbol calculi, Fourier integral operators, symplectic geometry.
Hans Christianson’s website.
- Nonlinear PDE in Fluids and Optics, including existence, dynamics, stability; internal waves in fluids with surface tension; Dirac equations.
- Diffraction of Waves off of Corners, including eigenstates of the Laplacian on domains with corners; waves on manifolds with corners; vortex formation in fluids around leaves.
- Nonlinear Bound States, including steady state solutions for problems in many body quantum mechanics; higher co-dimension phase transitions for vortex models; stability theory.
- PDE in Statistical Mechanics, including microscopic/macroscopic models related to crystal surfaces and magnetic systems; equilibrium states and dynamics.
Jeremy Marzuola’s website.
- General relativity, including waves near black holes, black hole stability, Price’s law, trapping.
- Trapping, including problems on surfaces of revolution, boundary value problems.
- Quasilinear equations, including problems showing that geometry depends on the solution and vice versa, Schr\”odinger equations, wave equations.
- Waves on hyperbolic space, including nonlinear problems and problems showing negative curvature aids dispersion.
Jason Metcalfe’s website.
- Harmonic analysis, including Fourier analysis, spectral theory, eigenfunctions, Lie groups and noncommutative harmonic analysis.
- Pseudodifferential operators, including basic calculi, Fourier integral operators, and microlocal analysis.
- Other tools, including singular integrals, geometric measure theory, and analysis on rough domains.
Michael Taylor’s website.
- Stability of multidimensional structures, including projects involving such structures arising in fluid mechanics, shocks, vortex sheets, and detonation fronts.
- Rigorous justification of multiscale expansions, including expansions used in the study of solutions of nonlinear PDEs.
- Development of new tools, including, pseudodifferential calculi, for use in studying the above problems.
Mark Williams’ website.