Geometry of discrete spaces and spectral theory of non-local operators

几何的研究及其对拉普拉斯算子及其半群的谱和随机特征的影响在数学的许多领域中起着核心作用，如度量几何，概率和算子理论。尽管流形上的拉普拉斯算子和图上的拉普拉斯算子之间有着众所周知的类比，但近年来的各种发现表明离散空间和连续空间之间存在着明显的差异。特别是，这导致了对基本几何的极大兴趣，这些几何编码在离散空间的距离和曲率等概念中。该项目的目的是发展对离散空间基本几何的深入理解，并研究拉普拉斯算子在图上的应用。这些应用主要涉及全局性质，特别涉及以下主题：-谱理论（谱估计，基本谱的缺乏，独特的延续，谱类型的稳定性）。热方程（随机完整性，长期行为，基态的唯一性和存在性）。自伴扩展（本质自伴性，边界可忽略性）。我们认为图拉普拉斯算子作为非局部正则狄利克雷形式的关键例子，我们的目标是发展理论的相应部分，考虑到一般的非局部狄利克雷形式。鉴于建立良好的理论强地方狄利克雷形式，这应该作为一个重要的一步，统一处理所有经常狄利克雷形式。

The study of geometry and its impact on spectral and stochastic features of Laplacians and their semigroups plays a central role in many areas of mathematics such as metric geometry, probability and operator theory. Despite the well known analogies between Laplacians on manifolds and Laplacians on graphs, various discoveries were made in recent years that show a clear disparity between discrete and continuum spaces. This, in particular, lead to significant interest in basic geometry as encoded in notions such as distance and curvature for discrete spaces. The aim of this project is to develop a deep understanding of basic geometry on discrete spaces and study applications for Laplacians on graphs. These applications concern mostly global properties and involve in particular the following topics:- Spectral theory (spectral estimates, absence of essential spectrum, unique continuation,stability of spectral types).- The heat equation (stochastic completeness, long term behavior, uniqueness and existence of ground states).- Selfadjoint extensions (essential selfadjointness, negligibility of boundary).We consider graph Laplacians as the key example of non-local regular Dirichlet forms and we aim at developing the corresponding parts of the theory with general non-local Dirichlet forms in mind. Given the well established theory for strongly local Dirichlet forms this should serve as an important step towards a unified treatment for all regular Dirichlet forms.