Preface

This volume contains the proceedings of the workshop "Harmonic Functions

on Graphs" held at the Graduate Center of CUNY, October 3G-November 3, 1995.

The workshop had a double purpose, which is reflected in the present volume.

The first purpose is served by the four minicourses, consisting of four one-hour

lectures each, which were intended as introductions to some wider contexts of the

subject for people with some knowledge about (and maybe actively working on)

harmonic functions on trees from a direct combinatorial viewpoint.

The direct viewpoint has achieved some notable successes. Its first important

landmark is P. Cartier's 1971 work, which laid the foundations and opened the way

to a large number of investigations about finer points of the theory of harmonic

functions. The second landmark is the application of the eigenfunctions of the

combinatorial Laplacian to harmonic analysis on homogeneous trees and through

that to the representation theory of free groups and some other groups ( cf. the well-

known monographs of Fig8.-Talamanca & Picardello, 1983, and Figa-Talamanca &

Nebbia, 1991).

While it is always dangerous to make general statements about the importance

of various trends in mathematics, it can perhaps be said with some confidence that

the most important future developments will now consist of the exploration of the

wider connections of the subject. The most important wider contexts seem to be

the probabilistic aspects, the extension of the investigations from trees to buildings

(which are the discrete analogues of general Riemannian symmetric spaces, while

semihomogeneous trees are the analogues of such spaces of rank one), and the

connections with p-adic analysis. It was these subjects that the authors of the

minicourses were asked to address.

In the editor's not ·unbiased opinion they have done this, and more, with great

success. They all stl¥1 essentially from zero and manage to give readable but sub-

stantial introductions to their subjects. The minicourse of S. Sawyer on Martin

boundary theory for trees contains some serious simplifications relative to the stan-

dard treatments; it is by far the simplest coherent exposition of this subject in the

literature. The course of A. Fig8.-Talamanca actually contains new results about

the crucially important connections between trees and p-adic numbers. The subtly

coordinated texts of D. Cartwright and T. Steger make buildings appear as truly

natural objects (one from the combinatorial, the other from the p-adic side) and

will undoubtedly save countless man-hours for people approaching the subject from

the side of discrete harmonic analysis and geometry.

Reflecting the second purpose of the workshop, this volume also presents a

cross-section of current activity in the field. At the workshop, which was well

attended by researchers in the field, there were twenty lectures by participants

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