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Dynamics of Infinite-dimensional Groups: The Ramsey-Dvoretzky-Milman Phenomenon
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University Lecture Series
2006; 192 pp; softcover
Volume: 40
ISBN-10: 0-8218-4137-8
ISBN-13: 978-0-8218-4137-2
List Price: US$40 Member Price: US$32
Order Code: ULECT/40

The "infinite-dimensional groups" in the title refer to unitary groups of Hilbert spaces, the infinite symmetric group, groups of homeomorphisms of manifolds, groups of transformations of measure spaces, etc. The book presents an approach to the study of such groups based on ideas from geometric functional analysis and from exploring the interplay between dynamical properties of those groups, combinatorial Ramsey-type theorems, and the phenomenon of concentration of measure.

The dynamics of infinite-dimensional groups is very much unlike that of locally compact groups. For instance, every locally compact group acts freely on a suitable compact space (Veech). By contrast, a 1983 result by Gromov and Milman states that whenever the unitary group of a separable Hilbert space continuously acts on a compact space, it has a common fixed point.

In the book, this new fast-growing theory is built strictly from well-understood examples up. The book has no close counterpart and is based on recent research articles. At the same time, it is organized so as to be reasonably self-contained. The topic is essentially interdisciplinary and will be of interest to mathematicians working in geometric functional analysis, topological and ergodic dynamics, Ramsey theory, logic and descriptive set theory, representation theory, topological groups, and operator algebras.

Graduate students and research mathematicians interested in representation theory, dynamical systems, geometric functional analysis, Ramsey theory, and descriptive set theory.

Reviews

"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."

-- Zentralblatt MATH