P r e f a c e
This volume contains papers written by some of the participants in the program
"Analysis on Graphs and its Applications" (AGA) that ran at the Isaac Newton
Institute for Mathematical Sciences (INI), Cambridge, from January 8th until June
29th 2007. The organizers of the program were M. Brown (UK), P. Exner (Czech
Republic), P. Kuchment (USA), and T. Sunada (Japan).
Let us start by explaining the title of the program (and hence also of this
volume). The term "analysis on graphs" was used to cover a variety of problems
relating to graphs that have a strongly analytic flavor, either as regards their formu-
lation or the techniques used to approach them. Examples include graph analogs
of differential equations, spectral theory, differential geometry, and geometric anal-
ysis. Problems of this kind have proliferated recently through many branches of
Mathematics, the Natural Sciences, and Engineering, including, for instance, num-
ber theory, the theory of discrete groups, probability theory, various branches of
computer science, optics, quantum mechanics, waveguide theory, nanophysics and
nanotechnology, chemistry, microelectronics, materials science, and biological mod-
eling (e.g. models of lungs). In many cases, maybe even most, researchers in each
area were not aware of the developments, methods and techniques available in oth-
ers. The aim of the program was to bring together experts from various disciplines
to share and advance their knowledge.
In addition to new problems and techniques, novel graph-like objects have also
been introduced. For instance, metric and quantum graphs have been the subject
of intense recent interest (see Chapters 3 and 4), and there has been considerable
attention paid to analysis on fractals (self-similar structures). Thus the program
and this volume naturally split into the following intertwined parts: analysis on
graphs in the traditional sense (i.e. on combinatorial graphs), on quantum graphs,
and on fractals.
The AGA program at the INI was structured as follows (for further details see
http://www.newton.cam.ac.uk/programmes/AGA/index.html). There were around
140 long term participants, each of whom visited the INI for a prolonged period,
anywhere from two weeks to six months. These participants engaged in formal dis-
cussions at the weekly seminars and, perhaps more importantly, informal meetings
in personal offices and at the coffee machine and blackboards distributed through-
out the splendid INI building. In addition, there were four formal workshops (which
brought the total number of participants to about 200), as detailed below.
A tutorial "Analysis on Graphs and its Applications" (designated as a London
Mathematical Society Short Course) was held during the period 10-1 5 January
2007 as a Satellite Meeting at Gregynog, Newtown, Wales. Most attendees were stu-
dents and postdocs. Lecture courses were delivered by T. Sunada (Meiji University,
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