Spectral Shape Optimization: Extremality and Curvature

数学成功地建立了自然和人类世界的模型，利用隐藏的现象来支撑看似不相关的场景。水中的波动、空气中的声波、穿过地壳的地震波、外层空间真空中的辐射（卫星通信）、地下水中的污染物扩散、机器部件中的热量传播以及亚原子尺度上运动的量子波或粒子所有这些以及更多的东西都是通过涉及同一数学对象的方程来分析的，这是一个“二阶变化率”算子，简称为拉普拉斯算子（为了纪念18世纪的科学家皮埃尔-西蒙·拉普拉斯）。拉普拉斯函数的“特征值”表示波动或辐射的频率、污染物扩散的速率、热传递的速度和粒子的能级。这个研究项目通过发现拉普拉斯特征值的新数学性质来促进科学的进步，这些特征值适用于生活在不像我们通常的三维空间那样平坦的空间中的波、扩散和粒子，而是像球体一样正向弯曲，或者像马鞍一样负向弯曲。问题包括识别支持最快振动的形状，解释为什么特征值的神秘“zeta函数”越来越依赖于曲率，理解如何通过特征值单调关系在弯曲空间中保持尺度不变性，以及在变形球面中找到最大的特征值。本项目为本科生和研究生提供研究训练机会。该项目在分析、偏微分方程和几何分析的交叉领域推进数学知识。推测解决了极端频率，能量和扩散，提供理论见解和实际应用。通过将现代分析技术综合到一套工具中，包括变分原理，度量摄动和单调性，保角映射，概率耦合和椭圆偏微分方程（PDE）对称，应用于正曲率，零曲率或负曲率区域，克服了概念和技术上的挑战。球体上的自由膜频率根据孔的存在或不存在而通过不同的技术得到最大化。利用双曲空间和球面空间对角线上热核的新性质，导出了谱zeta函数的曲率驱动单调性。在高维投影空间上，通过构造适当的试函数来推测第三特征值的最大值，而对于四阶双调和景观函数，采用椭圆对称下的一种新的双球优化方法来研究其极值性。对特征值最大化的统一强调旨在产生可计算边界和定性见解，以了解弯曲几何空间中的波，扩散和粒子的行为。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.