The goal of this course is to prepare students to enter research in hyperbolic PDE, nonlinear dispersive equations, and mathematical general relativity.

The course should also be useful to students working in applied areas where nonlinear evolution, multiscale behavior, and wave propagation play a central role. Techniques such as Fourier analysis, wave-packet decomposition, dispersive estimates, and concentration compactness have important connections with problems in fluid dynamics, nonlinear optics, imaging, signal processing, geophysics, and other areas involving complex wave phenomena and scale interactions.

We will study both conservative evolution equations, including the wave, Klein–Gordon, and Schrödinger equations, and dissipative models such as the heat equation.

A central theme will be concentration compactness and profile decomposition, with emphasis on how these methods capture loss of compactness in large-data problems. We will develop the necessary analytic tools, including Sobolev and Strichartz estimates, and study their connection with Fourier restriction, the Kakeya problem, and wave-packet/tube decompositions in nonlinear dispersive PDE.

The final part of the course will focus on applications to low-regularity Einstein equations, including the phenomenon that weak limits of vacuum solutions may carry a nontrivial effective stress-energy tensor generated by high-frequency gravitational oscillations. The course is intended for graduate students and advanced undergraduates with basic preparation in ODEs, functional analysis, and PDE.