Symmetries and Laplacians: Introduction to Harmonic Analysis, Group Representations and Applications

Designed as an introduction to harmonic analysis and group representations,
this book covers a wide range of topics rather than delving deeply into any
particular one. In the words of H. Weyl ...it is primarily meant for
the humble, who want to learn as new the things set forth therein, rather than
for the proud and learned who are already familiar with the subject and merely
look for quick and exact information....

The main objective is to
introduce the reader to concepts, ideas, results and techniques that evolve
around symmetry-groups, representations and Laplacians. More
specifically, the main interest concerns geometrical objects and structures
{X}, discrete or continuous, that possess sufficiently large symmetry
group G, such as regular graphs (Platonic solids), lattices, and
symmetric Riemannian manifolds. All such objects have a natural Laplacian
&Dgr;, a linear operator on functions over X, invariant under
the group action. There are many problems associated with Laplacians on
X, such as continuous or discrete-time evolutions, on X,
random walks, diffusion processes, and wave-propagation. This book contains
sufficient material for a 1 or 2-semester course.