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Research in recent years has highlighted the deep connections between the algebraic, geometric, and analytic structures of a discrete group. New methods and ideas have resulted in an exciting field, with many opportunities for new researchers. This book is an introduction to the area from a modern vantage point. It incorporates the main basics, such as Kesten's amenability criterion, Coulhon and Saloff-Coste inequality, random walk entropy and bounded harmonic functions, the Choquet–Deny Theorem, the Milnor–Wolf Theorem, and a complete proof of Gromov's Theorem on polynomial growth groups. The book is especially appropriate for young researchers, and those new to the field, accessible even to graduate students. An abundance of examples, exercises, and solutions encourage self-reflection and the internalization of the concepts introduced. The author also points to open problems and possibilities for further research.
The main theme of this book is the interplay between the behaviour of a class of stochastic processes (random walks) and discrete structure theory. The author considers Markov chains whose state space is equipped with the structure of an infinite, locally finite graph, or as a particular case, of a finitely generated group. The transition probabilities are assumed to be adapted to the underlying structure in some way that must be specified precisely in each case. From the probabilistic viewpoint, the question is what impact the particular type of structure has on various aspects of the behaviour of the random walk. Vice-versa, random walks may also be seen as useful tools for classifying, or at least describing the structure of graphs and groups. Links with spectral theory and discrete potential theory are also discussed. This book will be essential reading for all researchers working in stochastic process and related topics.
An up-to-date, panoramic account of the theory of random walks on groups and graphs, outlining connections with various mathematical fields.
This handbook is volume II in a series collecting mathematical state-of-the-art surveys in the field of dynamical systems. Much of this field has developed from interactions with other areas of science, and this volume shows how concepts of dynamical systems further the understanding of mathematical issues that arise in applications. Although modeling issues are addressed, the central theme is the mathematically rigorous investigation of the resulting differential equations and their dynamic behavior. However, the authors and editors have made an effort to ensure readability on a non-technical level for mathematicians from other fields and for other scientists and engineers. The eighteen surveys collected here do not aspire to encyclopedic completeness, but present selected paradigms. The surveys are grouped into those emphasizing finite-dimensional methods, numerics, topological methods, and partial differential equations. Application areas include the dynamics of neural networks, fluid flows, nonlinear optics, and many others.While the survey articles can be read independently, they deeply share recurrent themes from dynamical systems. Attractors, bifurcations, center manifolds, dimension reduction, ergodicity, homoclinicity, hyperbolicity, invariant and inertial manifolds, normal forms, recurrence, shift dynamics, stability, to namejust a few, are ubiquitous dynamical concepts throughout the articles.
This volume contains the proceedings of the CRM Workshops on Probabilistic Methods in Spectral Geometry and PDE, held from August 22–26, 2016 and Probabilistic Methods in Topology, held from November 14–18, 2016 at the Centre de Recherches Mathématiques, Université de Montréal, Montréal, Quebec, Canada. Probabilistic methods have played an increasingly important role in many areas of mathematics, from the study of random groups and random simplicial complexes in topology, to the theory of random Schrödinger operators in mathematical physics. The workshop on Probabilistic Methods in Spectral Geometry and PDE brought together some of the leading researchers in quantum chaos, semi-classical theory, ergodic theory and dynamical systems, partial differential equations, probability, random matrix theory, mathematical physics, conformal field theory, and random graph theory. Its emphasis was on the use of ideas and methods from probability in different areas, such as quantum chaos (study of spectra and eigenstates of chaotic systems at high energy); geometry of random metrics and related problems in quantum gravity; solutions of partial differential equations with random initial conditions. The workshop Probabilistic Methods in Topology brought together researchers working on random simplicial complexes and geometry of spaces of triangulations (with connections to manifold learning); topological statistics, and geometric probability; theory of random groups and their properties; random knots; and other problems. This volume covers recent developments in several active research areas at the interface of Probability, Semiclassical Analysis, Mathematical Physics, Theory of Automorphic Forms and Graph Theory.
A collection of papers from leading researchers in algebra and geometric group theory.
Die jüngsten Entwicklungen zeigen, dass sich Wahrscheinlichkeitsverfahren zu einem sehr wirkungsvollen Werkzeug entwickelt haben, und das auf so unterschiedlichen Gebieten wie statistische Physik, dynamische Systeme, Riemann'sche Geometrie, Gruppentheorie, harmonische Analyse, Graphentheorie und Informatik.
The present volume contains the transactions of the lOth Oberwolfach Conference on "Probability Measures on Groups". The series of these meetings inaugurated in 1970 by L. Schmetterer and the editor is devoted to an intensive exchange of ideas on a subject which developed from the relations between various topics of mathematics: measure theory, probability theory, group theory, harmonic analysis, special functions, partial differential operators, quantum stochastics, just to name the most significant ones. Over the years the fruitful interplay broadened in various directions: new group-related structures such as convolution algebras, generalized translation spaces, hypercomplex systems, and hypergroups arose from generalizations as well as from applications, and a gradual refinement of the combinatorial, Banach-algebraic and Fourier analytic methods led to more precise insights into the theory. In a period of highest specialization in scientific thought the separated minds should be reunited by actively emphasizing similarities, analogies and coincidences between ideas in their fields of research. Although there is no real separation between one field and another - David Hilbert denied even the existence of any difference between pure and applied mathematics - bridges between probability theory on one side and algebra, topology and geometry on the other side remain absolutely necessary. They provide a favorable ground for the communication between apparently disjoint research groups and motivate the framework of what is nowadays called "Structural probability theory".
This book provides a detailed exposition of a wide range of topics in geometric group theory, inspired by Gromov’s pivotal work in the 1980s. It includes classical theorems on nilpotent groups and solvable groups, a fundamental study of the growth of groups, a detailed look at asymptotic cones, and a discussion of related subjects including filters and ultrafilters, dimension theory, hyperbolic geometry, amenability, the Burnside problem, and random walks on groups. The results are unified under the common theme of Gromov’s theorem, namely that finitely generated groups of polynomial growth are virtually nilpotent. This beautiful result gave birth to a fascinating new area of research which is still active today. The purpose of the book is to collect these naturally related results together in one place, most of which are scattered throughout the literature, some of them appearing here in book form for the first time. In this way, the connections between these topics are revealed, providing a pleasant introduction to geometric group theory based on ideas surrounding Gromov's theorem. The book will be of interest to mature undergraduate and graduate students in mathematics who are familiar with basic group theory and topology, and who wish to learn more about geometric, analytic, and probabilistic aspects of infinite groups.