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.
1. Infinite planar graphs with nonnegative combinatorial curvature Bobo Hua and Yanhui Su; 2. Curvature calculations for antitrees David Cushing, Shiping Liu, Florentin Munch and Norbert Peyerimho; 3. Gromov-Lawson tunnels with estimates Józef Dodziuk; 4. Norm convergence of the resolvent for wild perturbations Colette Anné and Olaf Post; 5. Manifolds with Ricci curvature in the Kato class: heat kernel bounds and applications Peter Stollmann and Christian Rose; 6. Multiple boundary representations of λ – harmonic functions on trees Massimo A. Picardello and Wolfgang Woess; 7. Internal DLA on Sierpinski gasket graphs Joe P. Chen, Wilfried Huss, Ecaterina Sava-Huss and Alexander Teplyaev; 8. Universal lower bounds for Laplacians on weighted graphs Daniel Lenz and Peter Stollmann 9. On Hardy inequalities on graphs and manifolds Matthias Keller, Yehuda Pinchover and Felix Pogorzelski; 10. Neumann domains on manifolds and graphs Lior Alon, Michael Bersudsky, Sebastian Egger and Rami Band; 11. On the existence and uniqueness of self-adjoint realizations of discrete (magnetic) Schrödinger operators Marcel Schmidt; 12. Box spaces: geometry of finite quotients Ana Khukhro and Alain Valette; 13. Ramanujan graphs and digraphs Ori Parzanchevski; 14. From partial differential equations to groups Andrzej Żuk; 15. Spectral properties of limit-periodic operators David Damanik and Jake Fillman; 16. Uniform existence of the IDS on lattices and groups Christoph Schumacher, Fabian Schwarzenberger and Ivan Veselic.
Matthias Keller is a Professor of Mathematics at the University of Potsdam. He has contributed to books including Random Walks, Boundaries and Spectra (2011) and Modern Approaches to Discrete Curvature (2017). Daniel Lenz is a Professor of Mathematics at the Friedrich-Schiller University Jena. He has written numerous papers and previously contributed to Groups, Graphs and Random Walks (Cambridge, 2017). Radoslaw K. Wojciechowski is an Associate Professor at the City University of New York and a member of the American Mathematical Society. He currently has a Simons Collaboration Grant for Mathematicians and previously was awarded a National Science Foundation Conferences and Workshops in the Mathematical Sciences Grant. He has contributed to the book Random Walks, Boundaries and Spectra (2011).
