Prerequisites: ordinary differential equations, partial differential equations, elementary probability theory and stochastic processes.
This research topic lies at the interface of applied mathematics and theoretical physics. A goal of this research is to develop mathematical methods that help us quantify and understand real-world phenomena. For example, questions of interest are: How can we describe the motion of electrons in extremely thin materials such as graphene and black phosphorus? How can we describe the interaction of light with particles (electrons and phonons) in a solid-state device? What are the consequences of the existence of atomic defects in crystals for devices such as novel transistors used in last-generation computers? The proposed projects for undergraduates are inspired by the above research objectives.
Quantum mechanics project. This project concerns the existence of quantum states known as "edge states" in atomically thin materials (e.g., graphene) that have corrugated boundaries. The main theme is to describe the motion of an electron along the boundary via partial differential equations (PDEs) such as the Schroedinger or Dirac equation coupled with the Poisson equation. This project would require the student to become familiar (as part of his/her training in this REU) with particular solutions of PDEs, aspects of perturbation theory, and other mathematical concepts of quantum dynamics.
Classical mechanics project. This project concerns the erratic motion of atomic defects is crystalline surfaces. It turns out that defects move randomly at small scales but, according to the law of large numbers, their mesoscale or macroscale description in many experimental situations can appear deterministic. In some cases, the mesoscale description involves local geometric laws at interfaces between the solid material and its environment. This project would focus on formal derivations of such geometric laws from atomistic motion in simple geometries with smooth defect curves. The students can be trained in aspects of statistical mechanics, asymptotics for several types of differential equations, and applied probability theory.