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Lectures on nonsmooth differential g...

Gigli, Nicola.

Lectures on nonsmooth differential geometry

Record Type:	Electronic resources : Monograph/item
Title/Author:	Lectures on nonsmooth differential geometryby Nicola Gigli, Enrico Pasqualetto.
Author:	Gigli, Nicola.
other author:	Pasqualetto, Enrico.
Published:	Cham :Springer International Publishing :2020.
Description:	xi, 204 p. :ill., digital ;24 cm.
Contained By:	Springer eBooks
Subject:	Geometry, Differential.
Online resource:	https://doi.org/10.1007/978-3-030-38613-9
ISBN:	9783030386139$q(electronic bk.)

Lectures on nonsmooth differential geometry
Gigli, Nicola.

Lectures on nonsmooth differential geometry[electronic resource] /by Nicola Gigli, Enrico Pasqualetto. - Cham :Springer International Publishing :2020. - xi, 204 p. :ill., digital ;24 cm. - SISSA Springer series,v.22524-857X ;. - SISSA Springer series ;v.1..

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 ﬂow on metric measure spaces -- 6. Second-order calculus on RCD spaces -- 7. Appendix A: Functional analytic tools -- 8. Appendix B: Solutions to the exercises.

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 a comprehensive 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.

ISBN: 9783030386139$q(electronic bk.)

Standard No.: 10.1007/978-3-030-38613-9doiSubjects--Topical Terms:

182610
Geometry, Differential.

LC Class. No.: QA641 / .G545 2020

Dewey Class. No.: 516.36

Lectures on nonsmooth differential geometry
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505 0 $a 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 ﬂow on metric measure spaces -- 6. Second-order calculus on RCD spaces -- 7. Appendix A: Functional analytic tools -- 8. Appendix B: Solutions to the exercises.
520 $a 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 a comprehensive 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.
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