Topological Complexity of Smooth Random Functions

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)

• 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