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The interplay of geometry, spectral theory and stochastics has a long and fruitful history, and is the driving force behind many developments in modern mathematics. Bringing together contributions from a 2017 conference at the University of Potsdam, this volume focuses on global effects of local properties. Exploring the similarities and differences between the discrete and the continuous settings is of great interest to both researchers and graduate students in geometric analysis. The range of survey articles presented in this volume give an expository overview of various topics, including curvature, the effects of geometry on the spectrum, geometric group theory, and spectral theory of Laplacian and Schrödinger operators. Also included are shorter articles focusing on specific techniques and problems, allowing the reader to get to the heart of several key topics.
Infinite Planar Graphs with Non-negative Combinatorial Curvature
Curvature Calculations for Antitrees
Gromov–Lawson Tunnels with Estimates
Norm Convergence of the Resolvent for Wild Perturbations
Manifolds with Ricci Curvature in the Kato Class: Heat Kernel Bounds and Applications
Multiple Boundary Representations of λ-Harmonic Functions on Trees
Internal DLA on Sierpinski Gasket Graphs
Universal Lower Bounds for Laplacians on Weighted Graphs
Critical Hardy Inequalities on Manifolds and Graphs
Neumann Domains on Graphs and Manifolds
On the Existence and Uniqueness of Self-Adjoint Realizations of Discrete (Magnetic) Schrödinger Operators
Box Spaces: Geometry of Finite Quotients
Ramanujan Graphs and Digraphs
From Partial Differential Equations to Groups
Spectral Properties of Limit-Periodic Operators
Uniform Existence of the IDS on Lattices and Groups