Geometry and spectra of large objects

Symposium in English.
Symposium en anglais. Read the presentation in English below.

Presentation

A fundamental challenge common to mathematics and physics is to understand how geometric features - whether on a large scale or at a very fine level of detail - influence spectral properties, and how various objects can be compared through these properties.

Finding criteria for comparing shapes, geometries and spectra is a cross-cutting theme in mathematics, whether studying two given objects, large families of objects of increasing size, or a single geometric object observed at different scales.

Several analytical frameworks approach these questions from complementary angles. Semiclassical analysis, for example, describes wave propagation in a system of fixed size as wavelengths become very small. Homogenization theory studies the spectral behavior of objects with periodic or highly oscillating structures on a fine scale. In both cases, the challenge is to understand how microscopic geometric details influence macroscopic spectral properties, such as the behavior of eigenvalues of the Laplacian or more general elliptic operators. The theory of measured metric spaces offers a natural framework for studying the limits of geometric objects that can develop singularities.

Another approach is proposed by Benjamini-Schramm topology (or weak local topology), which focuses on large-scale geometric objects by comparing their geometry at intermediate, or mesoscopic, scales. In this framework, the local geometric environment is described statistically, and this information is reflected in empirical spectral measurements. Beyond the global distribution of eigenvalues, an important challenge is to understand finer spectral features, such as the presence and size of spectral holes. The concept of strong convergence, derived from free probability theory, addresses this issue. In recent years, powerful new techniques have made it possible to obtain strong spectral convergence results for various random geometric models, making it possible to identify limiting spectral holes.

The talk will also explore the construction of geometric objects with prescribed spectral properties, with a particular focus on arithmetic hyperbolic surfaces.

Coffee will be served at 9.15 a.m. in rooms 7 and 8.

Geometry and spectra of large objects

A fundamental challenge, shared by mathematics and physics, is to understand how geometric features-both at large scales and at very fine levels of detail-influence spectral properties, and how different objects can be meaningfully compared through them.

Finding criteria to compare shapes and geometries, as well as spectra, is thus a transversal theme in mathematics. Such comparisons arise when studying two given objects, large families of objects of increasing size, or a single geometric object observed at different scales.

Several analytical frameworks address these questions from complementary perspectives. Semiclassical analysis, for example, describes how waves propagate in a system of fixed size when wavelengths are very small. Homogenization theory studies the spectral behavior of objects with highly oscillatory or fine-scale periodic structures. In both cases, one seeks to understand how microscopic geometric features influence macroscopic spectral properties, such as the behavior of eigenvalues of Laplace or more general elliptic operators. The theory of metric measure spaces offers a natural setting to study limits of geometric objects that may develop singularities.

Another approach is provided by the Benjamini-Schramm (or local weak) topology, which focuses on large geometric objects by comparing their geometry at intermediate, or mesoscopic, scales. In this framework, the local geometric environment is described statistically, and this information is reflected in corresponding empirical spectral measures. Beyond global eigenvalue distributions, an important challenge is to understand finer spectral features, such as the presence and size of spectral gaps. The concept of strong convergence, originating in free probability theory, addresses this issue. In recent years, powerful new techniques have led to strong spectral convergence results in various random geometric models, making it possible to identify limiting spectral gaps.

The conference will also explore the construction of geometric objects with prescribed spectral properties, with a particular emphasis on arithmetic hyperbolic surfaces.