École d'Été de Probabilités de Saint-Flour

Topological Complexity of Smooth Random Functions

École d'Été de Probabilités de Saint-Flour XXXIX-2009

Authors: Adler, Robert, Taylor, Jonathan E.

  • The 2007 monograph has been very well received, but does not make for easy reading. The current notes, while not as exhaustive, are much more readable. As such, they provide a comparatively easy entry into a difficult, but important, area of research This is the main contribution of these notes...
  • There are no real competitors to these notes, beyond our 2007 book. Recent reprintings of Adler's `Geometry of Random Fields' (1980) by SIAM and Vanmarke's `Random Fields: Analysis and Synthesis" (1983) are not competitors. Both are terribly out of date.
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About this book

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.

Reviews

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)


Table of contents (6 chapters)

  • Introduction

    Adler, Robert J. (et al.)

    Pages 1-12

  • Gaussian Processes

    Adler, Robert J. (et al.)

    Pages 13-35

  • Some Geometry and Some Topology

    Adler, Robert J. (et al.)

    Pages 37-58

  • The Gaussian Kinematic Formula

    Adler, Robert J. (et al.)

    Pages 59-85

  • On Applications: Topological Inference

    Adler, Robert J. (et al.)

    Pages 87-106

Buy this book

eBook $34.99
price for USA (gross)
  • ISBN 978-3-642-19580-8
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $49.95
price for USA
  • ISBN 978-3-642-19579-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Topological Complexity of Smooth Random Functions
Book Subtitle
École d'Été de Probabilités de Saint-Flour XXXIX-2009
Authors
Series Title
École d'Été de Probabilités de Saint-Flour
Series Volume
2019
Copyright
2011
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-19580-8
DOI
10.1007/978-3-642-19580-8
Softcover ISBN
978-3-642-19579-2
Series ISSN
0721-5363
Edition Number
1
Number of Pages
VIII, 122
Number of Illustrations and Tables
6 b/w illustrations, 9 illustrations in colour
Topics