Spectral theory with non-unitary twists

非酉扭曲谱理论

• 批准号: 441868048
• 负责人: Professorin Dr. Anke Pohl
• 金额: --
• 依托单位: Arbeitsgruppe Analysis
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/441868048?language=en
• 关键词: Spectral; theory; non; unitary; twists

The spectral theory of Riemannian manifolds is an important area of mathematics with strong influence on many other fields of mathematics (e.g., representation theory, number theory, harmonic analysis, and mathematical physics). During the last few years, a huge interest was risen in spectral theory of noncompact spaces (in particular of so-called open systems) as well as in a spectral theory for Riemannian locally symmetric spaces with non-unitary twists.With this project we will study the spectral properties of noncompact Riemannian locally symmetric spaces with twists of non-expanding cusp monodromy. This class of twists reaches far beyond the set of representations that are unitary at cusps, and it constitutes a frontier until which we might currently expect a twisted spectral theory. Our investigations will focus on hyperbolic spaces (thus, Riemannian locally symmetric spaces of noncompact type and of rank 1) but we will also carry out first steps towards an extension of the expected results to locally symmetric spaces of higher rank as well as to non locally symmetric spaces with hyperbolic ends. We will investigate how the geometry and dynamics at the various types of ends govern the properties of the resonances, proper eigenvalues, eigenfunctions, etc. of these spaces. Further we will study the asymptotic properties of these (sets of) spectral objects. In order words, we will understand here the leitmotiv "geometry at infinity" in two ways.This project is closely related to other projects in the SPP 2026 with regard to research interests, considered spaces and techniques applied.

黎曼流形的谱理论是数学的一个重要领域，对许多其他数学领域（例如，表示论、数论、调和分析和数学物理）。在过去的几年里，人们对非紧空间（特别是所谓的开系统）的谱理论以及具有非酉扭的黎曼局部对称空间的谱理论产生了极大的兴趣。在这个项目中，我们将研究具有非扩张尖点单值扭的非紧黎曼局部对称空间的谱性质。这类扭曲远远超出了在尖点处是酉的表示的集合，它构成了一个前沿，直到我们目前可能期待一个扭曲的谱理论。我们的研究将集中在双曲空间（因此，黎曼局部对称空间的非紧型和秩1），但我们也将进行第一步的预期结果扩展到局部对称空间的更高的秩以及非局部对称空间的双曲结束。我们将研究在各种类型的端部的几何和动力学如何支配这些空间的共振、固有本征值、本征函数等的性质。进一步，我们将研究这些（套）光谱对象的渐近性质。换句话说，我们将从两个方面理解“无穷大的几何”的主旨。该项目与SPP 2026的其他项目在研究兴趣，考虑空间和应用技术方面密切相关。