Collaborative research: computational methods for extremal eigenvalue problems with geometric constraints

Federal Project Grant Summary

Claremont McKenna College received a $147,731 Project Grant award dated July 15, 2025, from the National Science Foundation's Division of Mathematical Sciences under the Mathematical and Physical Sciences program (CFDA 47.049). The grant funds collaborative computational research focused on developing and analyzing new mathematical methods for solving extremal eigenvalue problems with geometric constraints. The research addresses two primary objectives: (1) studying the Steklov eigenvalue problem on compact Riemannian surfaces with boundary and optimizing eigenvalues across smooth metrics, and (2) designing topological photonic crystals by identifying materials with large shared spectral bandgaps and prescribed topological invariants. These computational advances have applications to physical phenomena including light propagation, mechanical vibrations, and liquid sloshing.

The project integrates educational activities with research outcomes across R1 and primarily undergraduate institutions through four mechanisms: student training at K-12, undergraduate, and graduate levels; researcher participation and engagement; public dissemination of research results; and organization of international workshops on the research topics. The award supports basic research in computational mathematics and mathematical physics with anticipated completion by June 30, 2028, providing undergraduate students at the primarily undergraduate Claremont McKenna College with hands-on research experience in computationally intensive scientific fields.

Abstract

In a variety of real-world applications, eigenvalues of linear partial differential operators describe physical phenomena of interest, e.g., light propagation, mechanical vibrations, and liquid sloshing. IT is of practical and fundamental interest to study the dependence of an eigenvalue on a control variable, such as the material coefficient or the domain shape, and to engineer/design/optimize control variables to enhance relevant spectral properties. This project will develop and analyze new computational methods for solving extremal eigenvalue problems, especially involving challenging geometric constraints. The research activities will advance discovery and understanding in computational mathematics and mathematical physics, as well as more general areas of science and engineering through applications. Educational activities are integrated with research activities in four specific ways: (i) training of students (including k-12, undergraduate, and graduate students across different schools) and junior researchers at different levels, (II) encouraging participation of researchers in the area (III) dissemination and sharing of research results publicly, and (IV) organization of international workshops on proposed research topics. Due to the collaborative nature of this proposal, students will engage in activities across R1 and primarily undergraduate institutions. The aim of this project is to tackle two canonical extremal eigenvalue problems from the mathematical and engineering communities: (1) study the steklov eigenvalue problem on a compact riemannian surface with boundary and seek to maximize of an eigenvalue over the class of smooth metrics; (2) address a key challenge in the design of topological photonic crystals (tpcs): find materials that have large shared spectral bandgaps where the adjacent dispersion surfaces have prescribed topological invariants (e.g., the chern number, a topological invariant obtained from berry curvature). A technical challenge in these problems is to handle geometric constraints - either stemming from topological constraints on riemannian surfaces or topological invariants of dispersion surfaces. The proposed research activities will develop analytical and computational tools to tackle this challenge. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the foundation's intellectual merit and broader impacts review criteria.- subawards are not planned for this award.