Lectures on Nonsmooth Differential Geometry

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Sprache: Englisch

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Beschreibung

This book provides an introduction to some aspects of the flourishing field of nonsmooth geometric analysis. In particular, a quite detailed account of the first-order structure of general metric measure spaces is presented, and the reader is introduced to the second-order calculus on spaces ¿ known as RCD spaces ¿ satisfying a synthetic lower Ricci curvature bound. Examples of the main topics covered include notions of Sobolev space on abstract metric measure spaces; normed modules, which constitute a convenient technical tool for the introduction of a robust differential structure in the nonsmooth setting; first-order differential operators and the corresponding functional spaces; the theory of heat flow and its regularizing properties, within the general framework of ¿infinitesimally Hilbertian¿ metric measure spaces; the RCD condition and its effects on the behavior of heat flow; and second-order calculus on RCD spaces. The book is mainly intended for young researchers seeking acomprehensive and fairly self-contained introduction to this active research field. The only prerequisites are a basic knowledge of functional analysis, measure theory, and Riemannian geometry.

Über den Autor

Nicola Gigli studied Mathematics at the Scuola Normale Superiore of Pisa and is Professor of Mathematical Analysis at SISSA, Trieste. He is interested in calculus of variations, optimal transport, and geometric and nonsmooth analysis, with particular focus on properties of spaces with curvature bounded from below.

Enrico Pasqualetto earned his PhD degree in Mathematics at SISSA (Trieste) in 2018 and is currently a postdoctoral researcher at the University of Jyväskylä. His fields of interest consist in functional analysis and geometric measure theory on non-smooth metric structures, mainly in the presence of synthetic curvature bounds. Within this framework, his research topics include rectifiability properties, Sobolev spaces, and sets of finite perimeter.

Zusammenfassung

Provides a fairly self-contained introduction to the field

Offers content highly relevant to current research trends

Includes numerous exercises, with solutions

Inhaltsverzeichnis

1. Preliminaries.- 2. Sobolev calculus on metric measure spaces.- 3. The theory of normed modules.- 4. First-order calculus on metric measure spaces.- 5. Heat ¿ow on metric measure spaces.- 6. Second-order calculus on RCD spaces.- 7. Appendix A: Functional analytic tools.- 8. Appendix B: Solutions to the exercises.

Details

Erscheinungsjahr:	2020
Fachbereich:	Geometrie
Genre:	Mathematik , Medizin , Naturwissenschaften , Technik
Rubrik:	Naturwissenschaften & Technik
Medium:	Buch
Inhalt:	xi 204 S. 8 s/w Illustr. 204 p. 8 illus.
ISBN-13:	9783030386122
ISBN-10:	3030386120
Sprache:	Englisch
Einband:	Gebunden
Autor:	Gigli, Nicola Pasqualetto, Enrico
Hersteller:	Springer Springer International Publishing AG
Verantwortliche Person für die EU:	Springer Verlag GmbH, Tiergartenstr. 17, D-69121 Heidelberg, juergen.hartmann@springer.com
Maße:	241 x 160 x 18 mm
Von/Mit:	Nicola Gigli (u. a.)
Erscheinungsdatum:	11.02.2020
Gewicht:	0,494 kg