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

研究拉普拉斯半群及其半群的几何及其对谱和随机特征的影响在度量几何、概率论和算子论等许多数学领域中占有重要地位。尽管流形上的拉普拉斯空间和图上的拉普拉斯空间之间有众所周知的相似之处，但近年来的各种发现表明，离散空间和连续空间之间存在明显的差异。这尤其引起了人们对基本几何学的浓厚兴趣，这些几何学以离散空间的距离和曲率等概念进行编码。这个项目的目的是加深对离散空间上的基本几何的理解，并研究拉普拉斯在图上的应用。这些应用主要涉及全局性质，尤其涉及以下主题：-谱理论(谱估计、本质谱的缺失、唯一连续性、谱类型的稳定性)。-热方程(随机完备性、长期行为、唯一性和基态的存在)。-自伴扩展(本质自伴性、边界可忽略)。我们将图拉普拉斯作为非局部正则Dirichlet形式的关键例子，我们的目的是在考虑一般非局部Dirichlet形式的情况下发展该理论的相应部分。考虑到强局部Dirichlet形式的成熟理论，这应该是统一处理所有正则Dirichlet形式的重要一步。

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.