Date of Thesis

Spring 2024

Description

We consider the question asked by Wyman and Xi [WX23]: ``Can you hear your location on a manifold?” In other words, can you locate a unique point x on a manifold, up to symmetry, if you know the Laplacian eigenvalues and eigenfunctions of the manifold? In [WX23], Wyman and Xi showed that echolocation holds on one- and two-dimensional rectangles with Dirichlet boundary conditions using the pointwise Weyl counting function. They also showed echolocation holds on ellipsoids using Gaussian curvature.

In this thesis, we provide full details for Wyman and Xi's proof for one- and two-dimensional rectangles and we show that echolocation also holds on many three-dimensional boxes. We also prove that echolocation holds on rectangles with certain mixed boundary conditions using a similar approach. Secondly, we explore echolocation via Gaussian curvature and we focus on two categories of manifolds, namely surfaces of revolution and minimal surfaces. We provide counterexamples to two conjectures of necessary conditions for echolocation to hold on surfaces of revolution. We also show that Gaussian curvature is not enough for us to echolocate on Enneper's surface and Henneberg's surface by constructing pairs of non-symmetric points that have identical Gaussian curvatures.

Keywords

Spectral geometry, PDEs, Differential geometry, Minimal surfaces, Laplacian eigenvalue problems

Access Type

Honors Thesis

Degree Type

Bachelor of Science

Major

Applied Mathematical Sciences

Second Major

Economics

First Advisor

Emily Dryden

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