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### Review

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Developing good bounds for the distribution of the suprema of a Gaussian field $f$, i.e., for the quantity $\Bbb{P}\{\sup_{t\in M}f(t)\ge u}$, has been for a long time both a difficult and an interesting subject of research. A thorough presentation of this problem is the main goal of the book under review, as is stated by the authors in its preface. The authors develop their results in the context of smooth Gaussian fields, where the parameter spaces $M$ are Riemannian stratified manifolds, and their approach is of a geometrical nature. The book is divided into three parts. Part I is devoted to the presentation of the necessary tools of Gaussian processes and fields. Part II concisely exposes the required prerequisites of integral and differential geometry. Finally, in part III, the kernel of the book, a formula for the expectation of the Euler characteristic function of an excursion set and its approximation to the distribution of the maxima of the field, is precisely established. The book is written in an informal style, which affords a very pleasant reading. Each chapter begins with a presentation of the matters to be addressed, and the footnotes, located throughout the text, serve as an indispensable complement and many times as historical references. The authors insist on the fact that this book should not only be considered as a theoretical adventure and they recommend a second volume where they develop indispensable applications which highlight all the power of their results. (José Rafael León for Mathematical Reviews)

“This book presents the modern theory of excursion probabilities and the geometry of excursion sets for … random fields defined on manifolds. … The book is understandable for students … with a good background in analysis. … The interdisciplinary nature of this book, the beauty and depth of the presented mathematical theory make it an indispensable part of every mathematical library and a bookshelf of all probabilists interested in Gaussian processes, random fields and their statistical applications.” (Ilya S. Molchanov, Zentralblatt MATH, Vol. 1149, 2008)

### From the Back Cover

This monograph is devoted to a completely new approach to geometric problems arising in the study of random fields. The groundbreaking material in Part III, for which the background is carefully prepared in Parts I and II, is of both theoretical and practical importance, and striking in the way in which problems arising in geometry and probability are beautifully intertwined.

The three parts to the monograph are quite distinct. Part I presents a user-friendly yet comprehensive background to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities. Part II gives a quick review of geometry, both integral and Riemannian, to provide the reader with the material needed for Part III, and to give some new results and new proofs of known results along the way. Topics such as Crofton formulae, curvature measures for stratified manifolds, critical point theory, and tube formulae are covered. In fact, this is the only concise, self-contained treatment of all of the above topics, which are necessary for the study of random fields. The new approach in Part III is devoted to the geometry of excursion sets of random fields and the related Euler characteristic approach to extremal probabilities.

“Random Fields and Geometry” will be useful for probabilists and statisticians, and for theoretical and applied mathematicians who wish to learn about new relationships between geometry and probability. It will be helpful for graduate students in a classroom setting, or for self-study. Finally, this text will serve as a basic reference for all those interested in the companion volume of the applications of the theory. These applications, to appear in a forthcoming volume, will cover areas as widespread as brain imaging, physical oceanography, and astrophysics.

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