Integrating Spectral and Geometric data on Moduli Space

Introduction

Each physical object possesses specific frequencies of vibrations, called its “eigenfrequencies,” at which it enters into resonance under an external stimulus. In mathematical terms, these frequencies are the “eigenvalues” of a linear operator; they form the “spectrum” of the object.

Spectral Geometry

Spectral geometry is concerned with understanding how the spectrum of an object, as well as the modes of vibration (eigenfunctions) associated with each eigenfrequency, are related to its geometric shape. This is a wide area of research, with applied and interdisciplinary aspects, including:

• Electromagnetic waves
• Vibrating solids
• Seismic waves
• Wave functions in quantum mechanics

It also involves very theoretical mathematics, with many natural questions still open:

1. What can we learn about the topology or geometry of an object by observing its spectrum?
2. Can we predict if the vibrations will be localized in a small part of the object or, on the contrary, if they will take place everywhere?
3. Can we construct an object and be sure that certain frequencies are in the spectrum, or, on the opposite, be sure to avoid certain sets of frequencies?
4. Can there be objects of arbitrarily large size, with no small eigenfrequencies?

Project Overview

Project InSpeGMoS deals with a specific mathematical model: hyperbolic surfaces. The Moduli Space is a space of parameters of these surfaces that we can tune, and observe how the geometry and the spectrum vary.

Semiclassical Regime

In the semiclassical regime (when the wavelength is small compared to the size of the object), it is expected that certain spectral features are universal. We will adopt a probabilistic point of view: trying to exhibit spectral and geometric phenomena that happen in 99% of cases.

Research Focus

The project is focused on developing new integration techniques on Moduli Space. We shall look for new coordinates, generalize Mirzakhani’s study of volume functions, and seek inspiration in Random Graph Theory to develop new probabilistic methods in the spectral theory of random surfaces.