Vietnamese-Illini Math Guy

Research Description

My research focuses on Geometric Analysis. I am interested in various aspects of Geometry and its interplay with other fields, such as Analysis, Topology, and Physics. I used to study Geometric Flows, especially Ricci flow (RF) and Mean Curvature Flow (MCF), during my undergraduate studies.

These days, I have mostly been thinking about hypoelliptic operators on contact manifolds. I am trying to combine the so-called Heisenberg calculus developed by Beals–Greiner with the 0-calculus of Melrose–Mazzeo, thereby enabling study of boundary value problems for hypoelliptic operators. I aim to address some fundamental problems, such as Fredholmness of BVPs. If successful, I plan to apply these theories to the Hodge Laplacians of Rumin’s complex, an analogue of the Hogde theory for de Rham's complex, on contact manifolds. In the long run, I expect to manipulate them in studying invariants on contact manifolds.

Geometric Flows

Geometric Flows have emerged as powerful tools in Geometric Analysis. They are described in terms of Partial Differential Equations (PDEs), which evolve geometric structures of manifolds while still preserving other properties of manifolds, such as topological or smooth structures. Prominent applications of Geometric Flows are Perelman's resolution of Poincaré Conjecture and Schoen-Brendle's proof of Differentiable Sphere Theorem using Ricci flow, or Huisken-Illmanen's proof of Riemannian Penrose Inequality using Inverse Mean Curvature Flow.