Discrete Geometric Analysis

This is a volume of lecture notes based on three series of lectures given by visiting professors of RIMS, Kyoto University during the year-long project "Discrete Geometric Analysis," which took place in the Japanese academic year 20122013. The aim of the project was to make comprehensive research on topics related to discreteness in geometry, analysis and optimization. Discrete geometric analysis is a hybrid field of several traditional disciplines, including graph theory, geometry, discrete group theory, and probability. The name of the area was coined by Toshikazu Sunada, and since being introduced, it has been extending and making new interactions with many other fields. This volume consists of three chapters: (I) Loop Erased Walks and Uniform Spanning Trees, by Martin T Barlow; (II) Combinatorial Rigidity: Graphs and Matroids in the Theory of Rigid Frameworks, by Tibor Jordn; (III) Analysis and Geometry on Groups, by Andrzej Zuk. The lecture notes are useful surveys that provide an introduction to the history and recent progress in the areas covered. They will also help researchers who work in related interdisciplinary fields to gain an understanding of the material from the viewpoint of discrete geometric analysis.