PRIMES: Hear the Boundary via Investigating Steklov Eigenvalues

This project explores the fundamental connections between the shape of a boundary and certain types of vibrations, known as Steklov eigenvalues. These unique mathematical quantities provide insights into characteristics like a boundary's perimeter and curvature. This area of study has diverse real-world applications, from designing "cloaking" devices that make objects invisible, to improving medical imaging techniques, and enhancing our understanding of fluid dynamics, such as how water sloshes in a tank. By exploring these mathematical relationships, this project aims to provide foundational knowledge that could lead to new technologies and more efficient designs. This project also aims to build and grow research capacity in the mathematical sciences at California State University Long Beach (CSULB), through a formal partnership that enables the PI's research and participation in programs and workshops at the American Institute of Mathematics (AIM), specifically through AIM's SQuaRE program. Through specialized courses, tailored research projects, and mentorship opportunities open and available to all students, as well as outreach events for high schoolers, the project seeks to inspire and prepare the next generation of scientific talent, fostering an innovative workforce that benefits society.

The proposed research investigates the dependence of Steklov eigenvalues on the boundary geometry, addressing two primary objectives. The first objective focuses on estimating Steklov eigenvalue counting functions based on boundary geometry, specifically tackling weakened versions of the Pólya conjecture within the Steklov context. This will involve employing the frequency method to establish fundamental inequalities in Euclidean domains. The investigation will also extend this approach to differential forms and sloshing problems, refining estimates and aiming to prove weakened versions of the Pólya Conjecture. The second objective examines the effects of symmetric boundary perturbations on Steklov eigenvalues. This includes investigating regions with symmetry and addressing the Arnol'd conjecture for Steklov eigenvalues, with an intent to confirm this conjecture in special cases. Additionally, the research seeks to uncover generic properties of Steklov eigenfunctions in symmetric regions. These efforts address the unique challenges posed by boundary conditions using innovative mathematical approaches.

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.

Award Number: 2524167
Principal Investigator: Lihan Wang
Funds Obligated: $400,000
State: CA