Contemporary Mathematics 1997; 181 pp; softcover Volume: 206 ISBN10: 082180605X ISBN13: 9780821806050 List Price: US$43 Member Price: US$34.40 Order Code: CONM/206
 This volume presents the proceedings of the workshop "Harmonic Functions on Graphs" held at the Graduate Center of CUNY in the fall of 1995. The main papers present material from four minicourses given by leading experts: D. Cartwright, A. FigàTalamanca, S. Sawyer and T. Steger. These minicourses are introductions which gradually progress to deeper and less known branches of the subject. One of the topics treated is buildings, which are discrete analogues of symmetric spaces of arbitrary rank; buildings of rank are trees. Harmonic analysis on buildings is a fairly new and important field of research. One of the minicourses discusses buildings from the combinatorial perspective and another examines them from the \(p\)adic perspective. The third minicourse deals with the connections of trees with \(p\)adic analysis. And the fourth deals with random walks, i.e., with the probabilistic side of harmonic functions on trees. The book also contains the extended abstracts of 19 of the 20 lectures given by the participants on their recent results. These abstracts, well detailed and clearly understandable, give a good crosssection of the present state of research in the field. Readership Graduate students and research mathematicians interested in potential theory. Table of Contents Part 1. Minicourses  A. FigàTalamanca  Local fields and trees
 S. A. Sawyer  Martin boundaries and random walks
 D. I. Cartwright  A brief introduction to buildings
 T. Steger  Local fields and buildings
Part II. Abstracts of Lectures  D. Bednarchak  Heat kernel for regular trees
 D. I. Cartwright, M. G. Kuhn, and P. M. Soardi  Tensor product of spherical representations of the group of automorphisms of a homogeneous tree
 E. C. Tarabusi  The horocyclic Radon transform on trees
 J. M. Cohen and F. Colonna  Eigenfunctions of the Laplacian on a homogeneous tree
 F. Di Biase  Exotic convergence in theorems of Fatou type
 Y. Guivarcl'h  A spectral gap property for transfer operators
 V. A. Kaimanovich  Harmonic functions on discrete subgroups of semisimple Lie groups
 R. Lyons  Biased random walks and harmonic functions on the Lamplighter group
 A. M. Mantero and A. Zappa  Characterization of the eigenfunctions of the Laplace operators for an affine building of rank 2
 W. Młotkowski  Free product of representations
 T. Nagnibeda  The Jacobian of a finite graph
 S. Northshield  Flows and harmonic functions on graphs
 M. Pagliacci  Applications of diffusion processes on trees to mathematical finance
 M. A. Picardello  Characterizing harmonic functions by mean value properties on trees and symmetric spaces
 J. Ramagge and G. Robertson  Factors from buildings
 M. Rigoli, M. Salvatori, and M. Vignati  Harnack and Liouville properties on graphs
 G. Robertson  The spectrum of a directed Cayley graph of a free group
 M. H. Taibleson  Factorization of the Green's kernel for nonnearest neighbor random walks
 W. Woess  Harmonic functions for groupinvariant random walks
