Lectures on Nonsmooth Differential Geometry - Nicola Gigli, Enrico Pasqualetto

Lectures on Nonsmooth Differential Geometry

Buch | Hardcover
XI, 204 Seiten
2020 | 1st ed. 2020
Springer International Publishing (Verlag)
978-3-030-38612-2 (ISBN)
96,29 inkl. MwSt

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.


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.

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.

"This is an excellent starting point for entering this highly active field of research, written by two of its leading researchers." (M. Kunzinger, Monatshefte für Mathematik, Vol. 200 (2), 2023)

"The lecture notes in this book are a gentle introduction to the topic, intended for readers with no prior exposure to non-smooth analysis, and hint towards the most recent developments of the theory. ... This book is written in a clear and accessible way. It is definitely a recommended introduction to a quickly developing subject." (Daniele Semola, Mathematical Reviews, February, 2023)

"The book is written in a clear and precise style. The notions are well motivated and many examples are given. In the reviewer's opinion, this monograph will be of great interest to Ph.D. students and researchers working in the field of nonsmooth differential geometry." (Gabriel Eduard Vilcu, zbMATH 1452.53002, 2021)

Erscheinungsdatum
Reihe/Serie SISSA Springer Series
Zusatzinfo XI, 204 p. 8 illus.
Verlagsort Cham
Sprache englisch
Maße 155 x 235 mm
Gewicht 485 g
Themenwelt Mathematik / Informatik Mathematik Analysis
Mathematik / Informatik Mathematik Geometrie / Topologie
Schlagworte Differential Calculus • Differential Geometry • Geometric Ananlysis • Lower Ricci Curvature Bounds • nonsmooth calculus
ISBN-10 3-030-38612-0 / 3030386120
ISBN-13 978-3-030-38612-2 / 9783030386122
Zustand Neuware
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