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Metric Structures for Riemannian and Non-Riemannian Spaces (Progress in Mathematics)

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Formally speaking, this is the second edition of a set of Paris lecture notes published by Gromov two decades ago in the French language. However, such a wealth of entirely new material has been added that in essence we are talking of a new book.

Among the additions, the bulky new chapter 3 1/2+ stands out, dealing with the phenomenon of concentration of measure on high-dimensional structures. This is a relatively recent discovery of modern analysis and geometry, tracing its origin to the work of Paul Levy and especially Vitali Milman. The essence of the phenomenon is that on many multidimensional structures, every `nice' function is constant with high probability. The manifestations of the phenomenon are many - from geometric functional analysis (Dvoretzky theorem) through information theory (blowing-up lemma) and probability (law of large numbers) to graph theory (superconcentrators) and topological dynamics. As Gromov stresses in his book, even deeper aspects of the concentration phenomenon have been long since discovered and are constantly explored in statistical physics in the context of phase transitions of various kind, and some of the first known examples where phase transitions appear in the context of geometry have been discovered by Gromov himself, e.g. for hyperbolic groups. Finding and exploring more instances of phase transitions in mathematics might well become a unifying heuristic principle across a large number of disciplines.

The mathematical setting for dealing with concentration and related issues is the concept of a metric space equipped with finite measure, what Gromov calls an mm-space. Apart from concrete objects (such as for instance spheres and cubes), there are `higher-level' examples of mm-spaces, for instance those whose elements are isomorphism classes of mathematical objects themselves (e.g. Riemanning manifolds or finitely generated groups). This leads to a probabilistic treatment of such objects. Of course Gromov's strength is that his treatment is always concrete and he never theorizes without having particular objects and applications in mind.

It is quite safe to claim that the full range and power of applications of the interaction between metric and measure are yet to be discovered, which is what makes this book so important. It is rich in open questions and suggested new research directions, but more than that, it helps the reader to develop a good intuitive feeling of where things are going these days, what things ought to be done, and what constitutes proper mathematics.

Even though I unexpectedly found myself among the privileged ones who received a copy of the book as a gift from the author, I would have certainly purchased it otherwise, as I firmly believe that every mathematical library in the world, be it that of a top-class University or just a modest, lovingly selected office collection of a humble mathematician, will be wanting without a copy of the monograph under review, which might well become one of the most important books in mathematical sciences for the early XXIst century.

This book (originally published in French and improved here) is a fundamentally important book opening up an entire field of mathematics. For a textbook based on this material and related topics try Burago-Burago-Ivanov's textbook on the subject which can be taught to first year graduate students. For mathematicians and advanced graduate students, Gromov's book is a masterpiece.

The french version (written by Pierre Pansu, based on Gromov's lectures) is a jewel which gives a perfect introduction to the subject (not too much detail, a lot of insight, etc). The new parts (chapter 3.5, Semmes' supplement) really unbalance the book, though if you think of them as Volume 2, the changes might be palatable...