LEAPS-MPS: Investigation on Spectral Geometry of Steklov Eigenvalues

California State University-Long Beach Foundation, Long Beach CA

Investigators

Abstract

Already in the Middle Ages bell makers knew how to detect invisible cracks by sounding a bell on the ground before lifting it up to the belfry. Since then, the relations between sounds and geometric structures have been a central topic of interest in mathematics with important applications in the broader scientific community. Steklov eigenvalues play an important role in this context and, for example, are essential in electrical impedance tomography for medical imaging, and in the study of linear water waves in physics. In this project, the PI will reveal new relations between Steklov eigenvalues and the underlying geometry from which they are derived. Funds for the project will also support the PI’s efforts to make the research accessible and beneficial to students and young scholars and to promote the participation of underrepresented groups in mathematics, through mentoring and conference organization. This project involves two research directions tied to spectral geometry of Steklov eigenvalues. In one direction, the PI will study generic properties of Steklov eigenvalues and eigenfunctions and will investigate domain perturbations, differential forms and their applications to equidistribution theorems. In another direction, the PI will investigate Steklov eigenvalues on smooth metric spaces by studying collapsing manifolds. The PI will promote the participation of early career, female and other underrepresented group in mathematics as organizer of AMS special sessions and distinguished lectures series at Cal State, Long Beach. The PI will also organize career panels for undergraduate math majors, and a “Math Day at the Beach” for high school students. 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.