Topological Complexity of Smooth Random Functions: École d'Été de Probabilités de Saint-Flour XXXIX-2009 (Lecture Notes in Mathematics Book 2019)

Annotation. These notes, based on lectures delivered in Saint Flour, provide an easy introduction to the authors⁰₉ 2007 Springer monograph ⁰Random Fields and Geometry.⁰₊ 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 Annotation. These notes, based on lectures delivered in Saint Flour, provide an easy introduction to the authors09 2007 Springer monograph 0Random Fields and Geometry.0+ 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 Front Matter....Pages i-viii Introduction....Pages 1-12 Gaussian Processes....Pages 13-35 Some Geometry and Some Topology....Pages 37-58 The Gaussian Kinematic Formula....Pages 59-85 On Applications: Topological Inference....Pages 87-106 Algebraic Topology of Excursion Sets: A New Challenge....Pages 107-114 Back Matter....Pages 115-122 Robert J. Adler, Jonathan E. Taylor. Based On Lectures Delivered In Saint Flour. Includes Bibliographical References (p. 115-118) And Index.