Analysis & PDE

In applications, it is often impossible to measure the phase of a signal, and one must recover it from some additional physical "redundancy" present in the problem. Phase retrieval problems arise in diverse fields, ranging from crystallography to quantum mechanics. Mathematically, the analysis…

Abstract: It is now an observational fact that perturbed black holes produce radiation at fixed (complex) frequencies which are characteristic of the spacetime rather than the perturbation. The work of Vasy `13 shows how to identify these frequencies as part of the spectrum of some natural non-…

Abstract: Investigating mixtures of bosons and fermions is an extremely active area of research in experimental physics for constructing and understanding novel quantum bound states such as those in superconductors, superfluids, and supersolids. These ultra-cold Bose-Fermi mixtures are…

Abstract: Suppose we are given a globally hyperbolic spacetime (M,g) solving the Einstein vacuum equations, and a timelike geodesic in M. I will explain how to construct, on any compact subset of M, a solution g_\epsilon of the Einstein vacuum equations which is approximately equal to g far from…

Abstract: I will survey some recent developments in the theory of maximal functions that arise from averaging over families of curves in the plane. A central question in this field is to understand the L^p mapping properties of these maximal functions, and how this relates to the geometry of the…

We develop an equivariant index theorem for a twisted Dirac operator on a compact globally hyperbolic spacetime with spacelike boundary on which a group acts isometrically, subject to the Atiyah-Patodi-Singer boundary condition. Our analysis shows that the geometric formula is the same as in the…

Abstract: We present a series of works, joint with J. Bedrossian, S. He, F. Wang, in which we prove nonlinear inviscid damping, enhanced dissipation, and inviscid limit for the 2D Navier-Stokes equations near Couette. The domain is the periodic channel, \mathbb{T} \times [-1,1], and…

Abstract: Diffusion processes beyond Brownian motion have recently attracted significant interest from different communities in mathematics, the physical and biological sciences. They are described by nonlocal operators with singular non-integrable kernels, such as fractional Laplacians. The…

The Euler-Poisson system of partial differential equations describes the dynamics of a self-gravitating gas. For the energy-critical polytropic pressure law, there is an explicit steady-state solution describing an isolated star. I will discuss recent work which describes the nonlinear phase…

Abstract: We consider the focusing wave equation for all powers in all dimensions. It is well-known that the equation admits spatially homogeneous blow-up solutions, often dubbed ODE blow-up, terminating in a singular hypersurface at {t=T}. In this talk, we show both that we can construct…