# Research

I am an applied mathematician working at the interface of mathematics with physics; more specifically, the physics of materials’ electronic properties. These properties arise from the complex quantum-mechanical dynamics of materials’ negatively-charged electrons and positively-charged atomic ions.

The goals of my research are to develop rigorous analysis and numerical methods to meet basic challenges in understanding this system, especially in novel materials that have been the subject of theoretical and experimental interest in recent years. Mathematically, my work relates to partial differential equations (PDE), mathematical physics, and numerical analysis.

Left: the atomic structure of TBG is aperiodic for generic twist angles, but shows an approximate large-scale periodicity known as the moiré pattern (hexagon shows moiré unit cell). Figure by Tianyu Kong.

My most recent works have focused on twisted bilayer graphene (TBG): two layers of the 2D material graphene, stacked with a relative interlayer twist. The atomic structure of TBG is aperiodic for generic twists, but nevertheless has approximate large-scale periodicity known as the bilayer moiré pattern.

In publication 11 (see Publications), I rigorously justified the Bistritzer-MacDonald PDE model, which captures the electronic properties of TBG on the moiré scale and allows for computation of an approximate band structure/dispersion relation for electrons in TBG. This model led to predictions of many-body quantum phases such as superconductivity at the "magic" twist angle 1°, which were recently experimentally observed.

My PhD advisor at Columbia was Michael I. Weinstein. I was William E. Elliott Assistant Research Professor at Duke working primarily with Jianfeng Lu, and then Postdoctoral Associate at UMN working primarily with Mitchell Luskin.

Right: the dispersion relation for electron propagation in TBG; first without, and then with, interlayer tunneling turned on. Interlayer tunneling causes the Dirac cones of monolayer graphene to form nearly-flat bands, leading to many-body quantum phases such as superconductivity at the "magic" twist angle 1°. Figure by Tianyu Kong.