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Modern approaches to discrete curvature. (English) Zbl 1380.53007

Lecture Notes in Mathematics 2184. Cham: Springer (ISBN 978-3-319-58001-2/pbk; 978-3-319-58002-9/ebook). xxvi, 351 p. (2017).

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Publisher’s description: This book provides a valuable glimpse into discrete curvature, a rich new field of research which blends discrete mathematics, differential geometry, probability and computer graphics. It includes a vast collection of ideas and tools which will offer something new to all interested readers. Discrete geometry has arisen as much as a theoretical development as in response to unforeseen challenges coming from applications. Discrete and continuous geometries have turned out to be intimately connected. Discrete curvature is the key concept connecting them through many bridges in numerous fields: metric spaces, Riemannian and Euclidean geometries, geometric measure theory, topology, partial differential equations, calculus of variations, gradient flows, asymptotic analysis, probability, harmonic analysis, graph theory, etc. In spite of its crucial importance both in theoretical mathematics and in applications, up to now, almost no books have provided a coherent outlook on this emerging field.
The articles of this volume will be reviewed individually.
Indexed articles:
Bauer, Frank; Hua, Bobo; Jost, Jürgen; Liu, Shiping; Wang, Guofang, The geometric meaning of curvature: local and nonlocal aspects of Ricci curvature, 1-62 [Zbl 1388.53081]
Saucan, Emil, Metric curvatures revisited: a brief overview, 63-114 [Zbl 1401.53053]
Mémoli, Facundo, Distances between datasets, 115-132 [Zbl 1386.53045]
Chazal, Frédéric; Cohen-Steiner, David; Lieutier, André; Mérigot, Quentin; Thibert, Boris, Inference of curvature using tubular neighborhoods, 133-158 [Zbl 1388.53007]
Maas, Jan, Entropic Ricci curvature for discrete spaces, 159-174 [Zbl 1388.53037]
Keller, Matthias, Geometric and spectral consequences of curvature bounds on tessellations, 175-209 [Zbl 1391.53006]
Baird, Paul, The geometric spectrum of a graph and associated curvatures, 211-258 [Zbl 1382.05018]
Bobenko, Alexander I.; Bücking, Ulrike; Sechelmann, Stefan, Discrete minimal surfaces of Koebe type, 259-291 [Zbl 1392.53014]
Lachaud, Jacques-Olivier; Coeurjolly, David; Levallois, Jérémy, Robust and convergent curvature and normal estimators with digital integral invariants, 293-348 [Zbl 1386.52015]

MSC:

53-06 Proceedings, conferences, collections, etc. pertaining to differential geometry
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
52C99 Discrete geometry
00B15 Collections of articles of miscellaneous specific interest
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