Website updated March 5, 2023
Since 2017 I have been a Fellow of the American Mathematical Society.
Here is my math genealogy. I received my PhD from Columbia University in 2004 under the supervision of D.H. Phong.
Much of my recent work investigates PDEs which describe a physical process. The Porous Medium Equation is a degenerate nonlinear equation of parabolic type, and is used to describe diffusion of a gas within a porous medium. In a recent preprint with Albert Chau we showed that concavity of the pressure is not preserved by this flow, answering an open problem of Vazquez. In an earlier related paper we investigated concavity of solutions of the one-phase Stefan Problem, which is a free boundary problem used to model phase transitions (such melting ice). We also showed a convexity breaking result for the quasi-static droplet model, answering a question of Feldman-Kim.
There are many interesting open questions even about simple and well-studied equations. For the classical Laplace equation on domains, I have studied optimal gradient estimates of the solution in narrow regions - see this recent paper and this even more recent one with Morgan Sherman. Physically this is about estimating the magnitude of the electric field in a narrow region between insulators.
I'm interested in obtaining simpler and more illuminating proofs of known results: see here and here. In these papers with Gábor Székelyhidi, we found a new proof of Bian-Guan's constant rank theorem for nonlinear elliptic equations. This theorem says that convex solutions of a large class of nonlinear elliptic PDEs have Hessian of constant rank.
My earlier work investigated PDEs arising in complex geometry. These equations are highly nonlinear. They include the Kähler-Ricci flow, the Chern-Ricci flow, and extensions of Yau's theorem such as here and here.
All of my articles are available to download on the arXiv.
My research is supported in part by the National Science Foundation grant "Nonlinear Partial Differential Equations and Geometry," DMS-2005311.