General theory of topological groups --
Basic topological group theory --
Some examples --
Integration and convolution --
Representation theory and compact groups --
Basic representation theory --
Representations of compact groups --
Compact Lie groups and homogeneous spaces --
Discrete co-compact subgroups --
Introduction to commutative spaces --
Basic theory of commutative spaces --
Spherical transforms and Plancherel formulae --
Special case: Commutative groups --
Structure and analysis for commutative spaces --
Riemannian symmetric spaces --
Weakly symmetric and reductive commutative spaces --
Structure of commutative nilmanifolds --
Analysis on commutative nilmanifolds --
Classification of commutative spaces.

This book starts with the basic theory of topological groups, harmonic analysis, and unitary representations. It then concentrates on geometric structure, harmonic analysis, and unitary representation theory in commutative spaces. Those spaces form a simultaneous generalization of compact groups, locally compact abelian groups, and riemannian symmetric spaces. Their geometry and function theory is an increasingly active topic in mathematical research, and this book brings the reader up to the frontiers of that research area with the recent classifications of weakly symmetric spaces and of Gelfand pairs.