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

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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.