Spectral Correspondences for Trees and Buildings

Projekter: ProjektForskning



In theoretical physics, one often compares classical mechanics with quantum mechanics. While the laws of classical mechanics can be used to describe the movement of macroscopic particles in terms of position and momentum, quantum mechanics provides a model for the behavior of microscopic particles where only the probability of observable quantities is described.
While these models are conceptually very different, they can be put into a common mathematical framework which allows to compare quantities from both sides. This framework is based on the geometry of the surrounding space on which the particles live, and the mathematical model for this space is a Riemannian manifold M, generalizing the notion of curves and surfaces to higher dimensions. To every Riemannian manifold M, one can associate a set of classical resonances and a set of quantum resonances, both sets describing the long time behavior of the corresponding physical system. A spectral correspondence is an explicit relationship between classical and quantum resonances.
For general Riemannian manifolds M, spectral correspondences are difficult to establish, so one often focuses on manifolds M which are particularly symmetric, also called (locally) symmetric spaces. For locally symmetric spaces of rank one, surprisingly explicit spectral correspondences were recently obtained and are considered a major breakthrough in the field. It seems natural to try to extend these results to locally symmetric spaces of higher rank. However, higher rank spaces are significantly more complicated, and there are both conceptual, geometric and analytical difficulties to overcome.
The key idea of this project is to study spectral correspondences in a different geometric framework that is analytically simpler than the one of Riemannian manifolds, but still has analogous geometric features. More precisely, we propose to consider classical and quantum resonances for affine buildings. In contrast to manifolds which require tools from geometry and analysis, affine buildings are defined and studied in terms of algebra and combinatorics and are therefore easier to handle. The
simplest affine buildings are trees, i.e. collections of points connected with edges. In this setting, we expect to overcome the difficulties described above, first for trees and then for general affine buildings, and establish spectral correspondences in this framework. We expect that many ideas and techniques can be transferred from buildings to manifolds and vice versa. Moreover, affine buildings are themselves interesting mathematical objects, allowing applications to geometric group theory and number theory, and we further hope to establish new connections to these topics with our results.
Effektiv start/slut dato01/04/202331/03/2026


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