2017 Simon Prize Winner: James Lee-Thorp
Dr. James Patrick Lee-Thorp, a former Applied Mathematics student in Prof. Michael I. Weinstein's group, is the recipient of this year's Robert Simon Memorial Prize in recognition of his outstanding dissertation.
We study Schrödinger operators perturbed by non-compact (spatially extended) defects. We consider two models: a one-dimensional (1D) dimer structure with a global phase shift, and a two-dimensional (2D) honeycomb structure with a line-defect or "edge''. In both the 1D and 2D settings, the non-compact defects are modeled by adiabatic, domain wall modulations of the respective dimer and honeycomb structures. Our main results relate to the rigorous construction of states via bifurcations from continuous spectra. These bifurcations are controlled by asymptotic effective (homogenized) equations that underlie the protected or non-protected character of the states. In 1D, the states we construct are localized solutions. In 2D, they are "edge states'' - time-harmonic solutions which are propagating (plane-wave-like) parallel to a line-defect or "edge'' and are localized transverse to it. The states are described as protected if they persist in the presence of spatially localized (even strong) deformations of the global phase defect (in 1D) or edge (in 2D). The protected states bifurcate from "Dirac points'' (linear/conical spectral band-crossings) in the continuous spectra and are seeded by an effective Dirac equation. The (more conventional) non-protected states bifurcate from spectral band edges are seeded by an effective Schrödinger equation. Our 2D model captures many aspects of the phenomenon of topologically protected edge states observed in honeycomb structures such as graphene and "artificial graphene''. The protected states we construct in our 1D dimer model can be realized as highly robust TM- electromagnetic modes for a class of photonic waveguides with a phase-defect. We present a detailed computational study of an experimentally realizable photonic waveguide array structure.
Dr. James Lee-Thorp grew up and attended college in South Africa at the University of Cape Town, where he graduated with a B.S. in Applied Mathematics in 2010.
In 2011, he moved to New York to start his doctoral studies at the Department of Applied Physics and Applied Mathematics at Columbia University under the supervision of Professor Michael I. Weinstein. James said, “I had never visited New York up until the few days before my first term started. However, I could not have been happier with the move and with the Department – my fellow Columbia and APAM students were not only bright and talented, but they were outgoing, friendly and very welcoming.”
His doctoral dissertation was a “Bifurcation perspective on topologically protected and non-protected states in continuous systems.” Together with Professor Weinstein and collaborators, he studied the mathematical properties of topologically protected edge states (in 2D) and localized solutions (in 1D). Their main results relate to the rigorous mathematical construction of solutions via bifurcations governed by effective equations that underlie the protected or non-protected character of the solutions. Their models capture many aspects of (a) the phenomenon of topologically protected edge states observed in honeycomb structures such as graphene and “photonic graphene”, and (b) TM- electromagnetic modes for a class of photonic waveguides.
James is currently a postdoctoral associate at the Courant Institute of Mathematical Sciences at New York University, where he is pursuing related work on the mathematical and numerical modeling of materials science problems.
“James is an outstanding choice for this year’s Simon Memorial Prize for top thesis. His work is first-rate, is having impact in the Mathematics (applied and fundamental) and Applied Physics communities, and is a model of the kinds of deep and cross-disciplinary work that our department strives to promote.”
- Prof. Michael Weinstein, advisor
The Robert Simon Memorial Prize is awarded annually by the Department of Applied Physics and Applied Mathematics to the graduate student who has completed the most outstanding dissertation. Should no graduate student’s dissertation qualify in any given year, the prize may be awarded to either the most outstanding student who has completed a master of science degree in the Department or to the most outstanding graduating senior in the Department. The Department chair in consultation with the Department faculty selects the awardee.
Robert Simon (December 25, 1919–February 11, 2001) received a B.A. degree cum laude in classics from the City College of New York in 1941, where he was elected to Phi Beta Kappa, and an M.A. in mathematics from Columbia University in 1949. Between 1941 and 1944, Mr. Simon was a lieutenant in the United States Armed Forces serving in England, France, and Italy. He participated in the D-Day operation as a navigator for a plane that dropped paratroopers in the vicinity of Omaha Beach. General Dwight Eisenhower personally shook his hand and wished him well the night before the D-Day assault.
Mr. Simon, who was born and lived in New York City, spent a lifetime making valuable contributions to the field of computer science. Starting in 1953, he worked for 15 years at Sperry's Univac Division in various capacities including marketing, planning, systems engineering, systems programming, and information services. He also spent a year working at the Fairchild Engine Division as director of the Engineering Computer Group. He personally directed the establishment of several company computer centers at sites throughout the United States. Between 1969 and 1973, he was a partner with American Science Associates, a venture capital firm. Mr. Simon was a founder and vice president of Intech Capital Corporation and served on its board from 1972 to 1981 and a founder and member of the board of Leasing Technologies International, Inc. from 1983 until his retirement in 1995.
The prize was established in 2001 by Dr. Jane Faggen with additional support from friends and relatives of Mr. Simon.