📋 Tender Overview
Integrating Spectral And Geometric Data On Moduli Space
Exploring How Vibration In Hyperbolic Surfaces Affects Their Geometryspectral Geometry Studies How An Object’S Vibration Frequencies (Eigenfrequencies) And Their Associated Modes (Eigenfunctions) Relate To Its Geometric Shape. Although This Research Field Is Applied In Various Areas, Including Quantum Mechanics And Seismic Waves, It Poses Several Unresolved Theoretical Questions In Mathematics. The Erc-Funded Inspegmos Project Will Deal With Hyperbolic Surfaces, Exploring How Adjusting These Surface Parameters Affects Their Geometry And Spectrum. Researchers Will Seek To Identify Spectral And Geometric Phenomena That Occur In 99 % Of Cases From A Probabilistic Perspective. The Team Will Develop New Integration Techniques On Moduli Space, Seek New Coordinates, Generalise Mirzakhani’S Study Of Volume Functions, And Employ Random Graph Theory To Advance Probabilistic Methods In The Spectral Theory Of Random Surfaces.
Funded Under
European Research Council (Erc)
Total Cost
€ 1 686 575,00
Eu Contribution
€ 1 686 575,00
Coordinated By
Universite De Strasbourg
France
Objective
Each physical object possesses specific frequencies of vibrations, called its eigenfrequencies , at which it enters in 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 is concerned with understanding how the spectrum of an object, as well as the modes of vibration (eigenfunctions) associated to each eigenfrequency, are related to its geometric shape. This is a wide area of research, with applied and interdisciplinary aspects (electromagnetic waves, vibrating solids, seismic waves, wave functions in quantum mechanics... ), but also involving very theoretical mathematics, with many natural questions still open: What can we learn about the topology or geometry of an object by observing its spectrum? 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 ? 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 ? Can there be objects of arbitrarily large size, with no small eigenfrequencies ? 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. 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: try to exhibit spectral and geometric phenomena that happen in 99$% of cases. The project is focussed on developing new integration techniques on Moduli Space. We shall look for new coordinates, generalize Mirzakhani’s
