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The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces

2023-11-27 Mathematics
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Release : 2023-11-27
Genre : Mathematics
Kind : eBook
Book Rating : 961/5 ( reviews)

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Book Synopsis The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces by : Karl-Theodor Sturm

Download or read book The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces written by Karl-Theodor Sturm. This book was released on 2023-11-27. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Gradient Flows

2008-10-29 Mathematics
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Release : 2008-10-29
Genre : Mathematics
Kind : eBook
Book Rating : 22X/5 ( reviews)

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Book Synopsis Gradient Flows by : Luigi Ambrosio

Download or read book Gradient Flows written by Luigi Ambrosio. This book was released on 2008-10-29. Available in PDF, EPUB and Kindle. Book excerpt: The book is devoted to the theory of gradient flows in the general framework of metric spaces, and in the more specific setting of the space of probability measures, which provide a surprising link between optimal transportation theory and many evolutionary PDE's related to (non)linear diffusion. Particular emphasis is given to the convergence of the implicit time discretization method and to the error estimates for this discretization, extending the well established theory in Hilbert spaces. The book is split in two main parts that can be read independently of each other.

Alexandrov Geometry

2024-05-24 Mathematics
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Release : 2024-05-24
Genre : Mathematics
Kind : eBook
Book Rating : 367/5 ( reviews)

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Book Synopsis Alexandrov Geometry by : Stephanie Alexander

Download or read book Alexandrov Geometry written by Stephanie Alexander. This book was released on 2024-05-24. Available in PDF, EPUB and Kindle. Book excerpt: Alexandrov spaces are defined via axioms similar to those of the Euclid axioms but where certain equalities are replaced with inequalities. Depending on the signs of the inequalities, we obtain Alexandrov spaces with curvature bounded above (CBA) and curvature bounded below (CBB). Even though the definitions of the two classes of spaces are similar, their properties and known applications are quite different. The goal of this book is to give a comprehensive exposition of the structure theory of Alexandrov spaces with curvature bounded above and below. It includes all the basic material as well as selected topics inspired by considering Alexandrov spaces with CBA and with CBB simultaneously. The book also includes an extensive problem list with solutions indicated for every problem.

Gradient Flows

2006-03-30 Mathematics
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Release : 2006-03-30
Genre : Mathematics
Kind : eBook
Book Rating : 091/5 ( reviews)

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Book Synopsis Gradient Flows by : Luigi Ambrosio

Download or read book Gradient Flows written by Luigi Ambrosio. This book was released on 2006-03-30. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to a theory of gradient ?ows in spaces which are not nec- sarily endowed with a natural linear or di?erentiable structure. It is made of two parts, the ?rst one concerning gradient ?ows in metric spaces and the second one 2 1 devoted to gradient ?ows in the L -Wasserstein space of probability measures on p a separable Hilbert space X (we consider the L -Wasserstein distance, p? (1,?), as well). The two parts have some connections, due to the fact that the Wasserstein space of probability measures provides an important model to which the “metric” theory applies, but the book is conceived in such a way that the two parts can be read independently, the ?rst one by the reader more interested to Non-Smooth Analysis and Analysis in Metric Spaces, and the second one by the reader more oriented to theapplications in Partial Di?erential Equations, Measure Theory and Probability.

Modern Approaches to Discrete Curvature

2017-10-04 Mathematics
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Release : 2017-10-04
Genre : Mathematics
Kind : eBook
Book Rating : 027/5 ( reviews)

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Book Synopsis Modern Approaches to Discrete Curvature by : Laurent Najman

Download or read book Modern Approaches to Discrete Curvature written by Laurent Najman. This book was released on 2017-10-04. Available in PDF, EPUB and Kindle. Book excerpt: 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.

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