Research

Partial Differential Equations

My research aims to understand quantitative and qualitative aspects of solutions to PDEs. Most of my work concerns PDEs of elliptic or parabolic type, usually with some degeneracy or nonlinearity.  These include:

• the porous medium equation

• the Stefan problem

• the insulated and perfect conductivity problems

• constant rank theorems for nonlinear elliptic equations

Geometric Analysis

Most of my work in geometric analysis is in complex geometry.  Many of the fundamental results in complex and Kähler geometry involve highly non-linear partial differential equations, such as the complex Monge-Ampère equation.  My work investigates nonlinear PDEs in complex geometry, including:

• the Kähler-Ricci flow

• the J-flow (and Donaldson's J-equation)

• Kähler-Einstein metrics

• Hermitian and almost-Kähler extensions of Yau's theorem

• the Chern-Ricci flow

• (n-1)-plurisubharmonic functions and the Gauduchon conjecture

Publications

All of my articles are available to download on the arXiv.  They are also listed on the AMS MathSciNet.