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 &
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