Spectral Shape Optimization: Extremality and Curvature

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Mathematics successfully models the natural and human worlds by exploiting hidden phenomena that underpin seemingly unrelated scenarios. Wave motion through water, sound waves in air, seismic waves through the earth’s crust, radiation traveling in the vacuum of outer space (satellite communications), pollutants diffusing through groundwater, heat spreading through a machine part, and quantum waves or particles moving at subatomic scales — all these and more are analyzed by equations involving the same mathematical object, a "second order rate of change" operator called (in honor of the 18th century scientist Pierre-Simon Laplace) simply the Laplacian. The "eigenvalues" of the Laplacian represent frequencies of wave motion or radiation, rates of spreading for pollutant, speed of heat transfer, and energy levels of particles. This research project promotes the progress of science by discovering new mathematical properties of eigenvalues of the Laplacian for waves, diffusions and particles living in spaces that are not flat like our usual three-dimensional space but are instead curved positively like a sphere or else negatively like a saddle. Problems include the identification of shapes supporting the fastest vibrations, an explanation of why a mysterious "zeta function" of eigenvalues depends increasingly on the curvature, understanding how scaling invariance persists in curved spaces through an eigenvalue monotonicity relation, and finding the largest eigenvalues among deformed-sphere surfaces. The project provides research training opportunities for undergraduate and graduate students.The project advances mathematical knowledge at the intersection of analysis, partial differential equations and geometric analysis. Conjectures are addressed for extremal frequencies, energies and diffusions, offering theoretical insights and practical applicability. Conceptual and technical challenges are overcome by synthesizing modern analytic techniques into a suite of tools including variational principles, metric perturbation and monotonicity, conformal mapping, probabilistic coupling, and elliptic partial differential equations (PDE) symmetrization, applied on regions of positive, zero or negative curvature. Free membrane frequencies on the sphere are maximized by differing techniques depending on the presence or absence of holes. Curvature-driven monotonicity of the spectral zeta function is to be derived from new properties of the heat kernel on the spatial diagonal in hyperbolic space and the sphere. On higher dimensional projective space, the third eigenvalue is conjecturally maximized by constructing appropriate trial functions, while for the fourth-order biharmonic landscape function, extremality is investigated by a new kind of two-ball optimization under elliptic symmetrization. The unifying emphasis on maximization of eigenvalues is designed to yield both computable bounds and qualitative insights into the behavior of waves, diffusions and particles in curved geometric spaces.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.