For a long time mathematics was divided in two main areas, those disciplines
who could be applied in physics, and this was mainly the field of
differential equations, and the more exoteric disciplines of pure mathematics.
Among pure mathematics the purest of all was considered number theory. In the
last ten years, there have been discovered many connections between number
theory and physics, mainly on the road of recent new insights in number
theoretical aspects of topology. Some purely mathematical results were
obtained by physicists.
A big book about the aspects of number theory and function theory relevant to
these applications is
Waldschmidt/Moussa/Luck/Itzkyson (ed.): From number theory to
physics. Springer 1992, 680 p. ISBN 3-540-53342-7. DM 133.
Although originating from a meeting at the Centre de Physique in Les Houches
in 1989, the book contains 14 long expository articles on the field, which
usually start from scratch and are well written.
A companion volume from the same conference is
Luck/Moussa/Waldschmidt (ed.): Number theory and physics.
Springer 1990, 300 p. ISBN 3-540-52129-1. DM 99. *4779.
It contains those seminars on the conference which concern the more
concreteley physical aspects:
Conformally invariant field theories and quantum groups;
Quasicrystals;
Spectral problems, automata, and substitutions;
Dynamical and stochastic systems;
Further arithmetical problems.
From the preface: "Number theory, or arithmetic, sometimes referred to as the
queen of mathematics, is often considered as the purest branch of
mathematics. It also has the false reputation of being without any
application to other areas of knowledge. Nevertheless, throughout their
history, physical and natural sciences have experienced numerous unexpected
relationships to number theory. [...]
The most recent developments of theoretical physics have involved more and
more questions related to number theory, and in an increasingly direct way.
This new trend is especially visible in two broad families of physical
problems. The first class, dynamical systems and quasiperiodicity, includes
classical and quantum chaos, the stability of orbits in dynamical systems,
KAM theory, and problems with small denominators, as well as the study of
incommensurate structures, aperiodic tilings, and quasicrystals. The second
class, which includes the string theory of fundamental interactions,
completely integrable models, and conformally invariant two-dimensional
field theories, seems to involve modular forms and p-adic numbers in a
remarkable way."