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These notes, based on lectures delivered in Saint Flour, provide an easy introduction to the authors' 2007 Springer monograph "Random Fields and Geometry." While not as exhaustive as the full monograph, they are also less exhausting, while still covering the basic material, typically at a more intuitive and less technical level. They also cover some more recent material relating to random algebraic topology and statistical applications. The notes include an introduction to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness. This is followed by a quick review of geometry, both integral and Riemannian, with an emphasis on tube formulae, to provide the reader with the material needed to understand and use the Gaussian kinematic formula, the main result of the notes. This is followed by chapters on topological inference and random algebraic topology, both of which provide applications of the main results.
List of contents
1 Introduction.- 2 Gaussian Processes.- 3 Some Geometry and Some Topology.- 4 The Gaussian Kinematic Formula.- 5 On Applications: Topological Inference.- 6 Algebraic Topology of Excursion Sets: A New Challenge
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From the reviews:
"These little lecture notes are a rare delight. The authors succeed in an impressive manner to combine a writing style that focuses on the main ideas and intuitions while still stating all the results in full mathematical rigor. They take the reader on an exciting journey through the theories of Gaussian processes and differential topology and geometry and then show how fascinating mathematics arises when combining these fields not to speak about the wide range of applications." (H. M. Mai, Zentralblatt MATH, Vol. 1230, 2012)
"This concise book is written for graduate students as well as researchers who want to learn the state of the art of geometry of smooth Gaussian (and Gaussian-related) random fields and their significant applications. ... The authors have done an excellent job in showing not only the mathematical beauty and the essence of the 'Gaussian Kinematic Formulae', but also their powerful applicability. The book is very interesting to read." (Yimin Xiao, Mathematical Reviews, Issue 2012 h)
