Project IV 2024-25

Can we hear the shape of a drum?

Katie Gittins

Description

Imagine a 2-dimensional drumhead \(D\) with fixed boundary \(\partial D\) in the Euclidean plane. The frequencies at which this drum can vibrate are closely related to the eigenvalues of the Dirichlet Laplacian on the domain \(D\). That is, the real values \(\lambda\) for which there exists a function \(u(x,y)\) such that \[-\frac{\partial^2}{\partial x^2} u(x,y) -\frac{\partial^2}{\partial y^2} u(x,y) = \lambda u(x,y), (x,y) \in D; \quad u(x,y) = 0, (x,y) \in \partial D.\] In this framework, we can reformulate the question can we hear the shape of a drum? as does the collection of all frequencies of the drum tell us what the shape of the drum is?

Although interesting links between the eigenvalues and the shape of the drum had been identified since the work of Lord Rayleigh in the early 1900s, it was the fascinating paper of M. Kac entitled Can one hear the shape of a drum? that enticed mathematicians from various research fields to uncover further intriguing results about this relationship.

Questions like what can the analytic quantities tell us about the shape? and what can the shape tell us about the analytic quantities? are at the heart of an exciting field of mathematics called Spectral Geometry.

The goal of this project is to use analytic tools to investigate some links between the eigenvalues and eigenfunctions of a drum and its shape. Some possible directions for the project include (but are not limited to) the following:

  • There is an asymptotic formula for the eigenvalues of the Laplacian corresponding to high frequencies called Weyl's Asymptotic Law. Roughly speaking, it asserts that if we know all the frequencies of a drum, then we can hear its area.
  • If we fix an orchestra of drums all satisfying some geometric condition (e.g. they have the same area), is there a unique drum that vibrates with lowest frequency? This type of question comes from the intriguing field of shape optimisation.
  • While the drum is vibrating at a given frequency, there will be some parts that are stationary (that is \(u(x,y) = 0\)). The stationary parts (called the nodal set) chop the drum into smaller pieces. How many smaller pieces are there? What does the geometry of this splitting look like?
  • What if the boundary of the drum is free to move instead of being fixed? We can ask similar questions when the drum is subject to other boundary conditions (e.g. Neumann).
  • What if we consider a different orchestra of drums? We aren’t restricted to drums in the Euclidean plane and can even consider drums in hyperbolic space, Riemannian manifolds, and beyond!
The tools used to investigate these directions come from Analysis. A plan for the project would be that we first study the eigenvalues and eigenfunctions of the Dirichlet Laplacian. After laying these foundations together, you would be free to choose your preferred direction in which to continue your investigation.

Prior and Companion modules

Essential prior modules: Analysis III.

The project involves some PDEs but don't worry, you do not need to have taken the module PDE III to work on this project. We will work together on any aspects of PDEs that we require for this project.

Depending on your preferred direction of the project, some of the following modules could be useful companions but are not essential: Functional Analysis and Applications IV, Riemannian Geometry IV.

Some Resources

P. Buser Geometry and Spectra of Compact Riemann Surfaces. Modern Birkhauser Classics. Birkhauser, reprint of the 1992 edition.

I. Chavel Eigenvalues in Riemannian Geometry. Pure and Applied Mathematics. Volume 115. (1984) Academic Press.

A. Henrot Shape optimization and spectral theory. De Gruyter (2017).

A. Henrot and M. Pierre Shape variation and optimization a geometrical analysis. European Mathematical Society Publishing House (2018).

R. S. Laugesen Spectral Theory of Partial Differential Equations. Spring (2017).

M. Levitin, D. Mangoubi, and I. Polterovich , Topics in Spectral Geometry, preliminary version dated May 29, 2023.

Spoiler alert! This article contains an answer to the question "can we hear the shape of a drum?"

If you would like more information about this project, then please feel free to contact me via email: K Gittins