The very formulation of many high-profile problems in PDEs, some of which go back to the leading mathematicians of the last century, entails objects such as curves, level sets, critical points, cusps or corners, which ultimately makes them of a strongly geometric nature. It is now folk wisdom that, despite the intrinsic complexity of this kind of problems, their dual analytic and geometric nature can be a blessing in disguise, for solutions to these problems often open up new vistas in very different areas of mathematics. In a way, unveiling some of the underlying connections between different areas of mathematics is the most fundamental objective of this project.
This project aims to go significantly beyond the state of the art in several fundamental questions in PDEs with a clear geometric flavor. We will consider three kinds of problems. First, those related to singular (that is, low regularity) solutions in incompressible fluid mechanics. In this setting, we are mostly interested in the 3D Euler and MHD equations, with or without a free boundary, although some 1D and 2D models also provide extremely interesting models for the expected three-dimensional behavior. Second, we are interested in smooth phenomena in PDE related to qualitative changes of topology. We will explore these questions in the setting of the 3D Euler equations (where, in particular, we want to analyze the emergence of chaotic patterns in fluid in equilibrium), of nonlinear Schrödinger equations, and in Taubes' system of modified Seiberg-Witten equations. The third kind of problems we will consider concern random and deterministic questions in spectral theory, and particularly the phenomenon of inverse localization for high energy eigenfunctions.
