I am an applied mathematician working at the interface of mathematics with physics; more specifically, the physics of materialsâ€™ electronic properties. These properties arise from the complex quantum-mechanical dynamics of materialsâ€™ 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). 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°.
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 pre-print 12 (see below), 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.
Wavepackets in inhomogeneous periodic media: Propagation through a one-dimensional band crossing (with M. I. Weinstein); Communications in Mathematical Physics363 655-698 (2018) [article, preprint]
Dirac operators and domain walls (with J. Lu, M. I. Weinstein); SIAM Journal on Mathematical Analysis52 2 1115-1145 (2020) [article, preprint]
The Iterated Projected Position Algorithm for Constructing Exponentially Localized Generalized Wannier Functions for Periodic and Non-Periodic Insulators in Two Dimensions and Higher (with K. D. Stubbs, J. Lu); Physical Review B103 075125 (2021) [article, preprint]
Computing edge states without hard truncation (with K. Thicke, J. Lu); SIAM Journal on Scientific Computing43 2 B323-B353 (2021) [article, preprint]
Wave-packet propagation in a finite topological insulator and the spectral localizer index (with J. Michala, A. Pierson, T. Loring); Involve14 2 209-239 (2021) [article, preprint]
Existence of the first magic angle for the chiral model of bilayer graphene (with M. Luskin); Journal of Mathematical Physics62 091502 (2021) [article, preprint]
Defect state resonances of truncated crystal structures (with J. Lu and J. Marzuola); SIAM Journal on Applied Mathematics82 1 (2022) [article, preprint]
Existence and computation of generalized Wannier functions for non-periodic systems in two dimensions and higher (with K. D. Stubbs, J. Lu); Archive for Rational Mechanics and Analysis243 3 (2022) [article, preprint]
Computing spectral properties of topological insulators without artificial truncation or supercell approximation (with M. J. Colbrook, A. Horning, K. Thicke); IMA Journal of Mathematics (2023) [preprint]
Bistritzer-MacDonald dynamics in twisted bilayer graphene (with T. Kong, A. H. MacDonald, M. Luskin); Journal of Mathematical PhysicsEditor's Pick (2023) [preprint]
Locality of the windowed local density of states (with T. Loring, J. Lu); submitted [preprint]
On the Su-Schrieffer-Heeger model of electron transport: low-temperature optical conductivity by the Mellin transform (with D. Margetis, M. Luskin); submitted [preprint]
Other
Wave dynamics in locally periodic structures by multiscale analysis Columbia (APAM) PhD thesis, supervised by M. I. Weinstein (2017) [pdf]