This is a new version of a set of lecture notes by the same author [Publ. Mat. IMPA, Rio de Janeiro: Instituto de Matemática Pura e Aplicada (IMPA) (2005; Zbl 1076.37005)]. The subject is the study of infinite dimensional groups from the point of view of geometric functional analysis. These “large” groups arise as groups of automorphisms of various structures. Examples include the unitary group of an infinite-dimensional Hilbert space, unitary groups of operator algebras, groups of automorphisms of measure spaces, and groups of homeomorphisms of topological spaces. The book focuses on the fascinating interactions between the dynamical properties of actions of such groups, the geometry of high-dimensional structures, and combinatorial theorems of Ramsey type.
This is a very well-written and lively exposition, with a number of basic examples worked out in detail. In comparison to the original version, while the set of topics treated is essentially the same, some chapters have been reorganized, updated and largely expanded.