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Author : Stephen Keith
Publisher :
ISBN 13 :
Total Pages : 182 pages
Book Rating : 4.3/5 (91 download)
Download or read book A Differentiable Structure for Metric Measure Spaces written by Stephen Keith and published by . This book was released on 2002 with total page 182 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Nicola Gigli
Publisher : American Mathematical Soc.
ISBN 13 : 1470414201
Total Pages : 104 pages
Book Rating : 4.4/5 (74 download)
Download or read book On the Differential Structure of Metric Measure Spaces and Applications written by Nicola Gigli and published by American Mathematical Soc.. This book was released on 2015-06-26 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main goals of this paper are: (i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without calling into play any sort of analysis in charts, our assumptions being: the metric space is complete and separable and the measure is Radon and non-negative. (ii) To employ these notions of calculus to provide, via integration by parts, a general definition of distributional Laplacian, thus giving a meaning to an expression like , where is a function and is a measure. (iii) To show that on spaces with Ricci curvature bounded from below and dimension bounded from above, the Laplacian of the distance function is always a measure and that this measure has the standard sharp comparison properties. This result requires an additional assumption on the space, which reduces to strict convexity of the norm in the case of smooth Finsler structures and is always satisfied on spaces with linear Laplacian, a situation which is analyzed in detail.
Author : Juha Heinonen
Publisher : Cambridge University Press
ISBN 13 : 1107092345
Total Pages : 447 pages
Book Rating : 4.1/5 (7 download)
Download or read book Sobolev Spaces on Metric Measure Spaces written by Juha Heinonen and published by Cambridge University Press. This book was released on 2015-02-05 with total page 447 pages. Available in PDF, EPUB and Kindle. Book excerpt: This coherent treatment from first principles is an ideal introduction for graduate students and a useful reference for experts.
Author : Luigi Ambrosio
Publisher : Springer Science & Business Media
ISBN 13 : 376438722X
Total Pages : 333 pages
Book Rating : 4.7/5 (643 download)
Download or read book Gradient Flows written by Luigi Ambrosio and published by Springer Science & Business Media. This book was released on 2008-10-29 with total page 333 pages. 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.
Author : Nicola Gigli
Publisher : Springer Nature
ISBN 13 : 3030386139
Total Pages : 212 pages
Book Rating : 4.0/5 (33 download)
Download or read book Lectures on Nonsmooth Differential Geometry written by Nicola Gigli and published by Springer Nature. This book was released on 2020-02-10 with total page 212 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Author : Fabrice Baudoin
Publisher : Springer Nature
ISBN 13 : 3030841413
Total Pages : 312 pages
Book Rating : 4.0/5 (38 download)
Download or read book New Trends on Analysis and Geometry in Metric Spaces written by Fabrice Baudoin and published by Springer Nature. This book was released on 2022-02-04 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book includes four courses on geometric measure theory, the calculus of variations, partial differential equations, and differential geometry. Authored by leading experts in their fields, the lectures present different approaches to research topics with the common background of a relevant underlying, usually non-Riemannian, geometric structure. In particular, the topics covered concern differentiation and functions of bounded variation in metric spaces, Sobolev spaces, and differential geometry in the so-called Carnot–Carathéodory spaces. The text is based on lectures presented at the 10th School on "Analysis and Geometry in Metric Spaces" held in Levico Terme (TN), Italy, in collaboration with the University of Trento, Fondazione Bruno Kessler and CIME, Italy. The book is addressed to both graduate students and researchers.
Author : Mikhail Gromov
Publisher : Springer Science & Business Media
ISBN 13 : 0817645837
Total Pages : 594 pages
Book Rating : 4.8/5 (176 download)
Download or read book Metric Structures for Riemannian and Non-Riemannian Spaces written by Mikhail Gromov and published by Springer Science & Business Media. This book was released on 2007-06-25 with total page 594 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an English translation of the famous "Green Book" by Lafontaine and Pansu (1979). It has been enriched and expanded with new material to reflect recent progress. Additionally, four appendices, by Gromov on Levy's inequality, by Pansu on "quasiconvex" domains, by Katz on systoles of Riemannian manifolds, and by Semmes overviewing analysis on metric spaces with measures, as well as an extensive bibliography and index round out this unique and beautiful book.
Author : James J Yeh
Publisher : World Scientific
ISBN 13 : 9813200421
Total Pages : 308 pages
Book Rating : 4.8/5 (132 download)
Download or read book Metric In Measure Spaces written by James J Yeh and published by World Scientific. This book was released on 2019-11-18 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: Measure and metric are two fundamental concepts in measuring the size of a mathematical object. Yet there has been no systematic investigation of this relation. The book closes this gap.
Author : Gerard Walschap
Publisher : Springer Science & Business Media
ISBN 13 : 0387218262
Total Pages : 235 pages
Book Rating : 4.3/5 (872 download)
Download or read book Metric Structures in Differential Geometry written by Gerard Walschap and published by Springer Science & Business Media. This book was released on 2012-08-23 with total page 235 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers an introduction to the theory of differentiable manifolds and fiber bundles. It examines bundles from the point of view of metric differential geometry: Euclidean bundles, Riemannian connections, curvature, and Chern-Weil theory are discussed, including the Pontrjagin, Euler, and Chern characteristic classes of a vector bundle. These concepts are illustrated in detail for bundles over spheres.
Author : Guy David
Publisher :
ISBN 13 :
Total Pages : 109 pages
Book Rating : 4.:/5 (893 download)
Download or read book Lipschitz Maps in Metric Spaces written by Guy David and published by . This book was released on 2014 with total page 109 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this dissertation we study Lipschitz and bi-Lipschitz mappings on abstract, non-smooth metric measure spaces. The dissertation consists of two separate parts. The first part considers a well-known class of questions that ask the following: If X and Y are metric measure spaces and f is a Lipschitz mapping from X to Y whose image has positive measure, then must f have large pieces on which it is bi-Lipschitz? Building on methods of David (who is not the present author) and Semmes, we answer this question in the affirmative for Lipschitz mappings between certain types of Ahlfors regular topological manifolds. In general, these manifolds need not admit bi-Lipschitz embeddings into any Euclidean space. To prove the result, we use some facts on the Gromov-Hausdorff convergence of manifolds and a topological theorem of Bonk and Kleiner. This also yields a new proof of the uniform rectifiability of some metric manifolds. In the second part, we study the class of ``Lipschitz differentiability spaces'' introduced by Cheeger. These are spaces on which an appropriate version of Rademacher's theorem holds. We show that if an Ahlfors regular Lipschitz differentiability space has a differentiable structure of maximal dimension, then at almost every point all its tangents are uniformly rectifiable. In particular, it admits Euclidean tangents at almost every point. Conversely, we show that if the dimension of the differentiable structure is not extremal, then the space is strongly unrectifiable, in the sense of Ambrosio-Kirchheim. In proving these results, we generalize some results of Cheeger from the setting of doubling spaces with Poincare inequalities (PI spaces) to general doubling Lipschitz differentiability spaces. The starting point is a result of Bate on the local structure of these spaces.
Author : Vladimir Igorevich Bogachev
Publisher : American Mathematical Soc.
ISBN 13 : 082184993X
Total Pages : 506 pages
Book Rating : 4.8/5 (218 download)
Download or read book Differentiable Measures and the Malliavin Calculus written by Vladimir Igorevich Bogachev and published by American Mathematical Soc.. This book was released on 2010-07-21 with total page 506 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides the reader with the principal concepts and results related to differential properties of measures on infinite dimensional spaces. In the finite dimensional case such properties are described in terms of densities of measures with respect to Lebesgue measure. In the infinite dimensional case new phenomena arise. For the first time a detailed account is given of the theory of differentiable measures, initiated by S. V. Fomin in the 1960s; since then the method has found many various important applications. Differentiable properties are described for diverse concrete classes of measures arising in applications, for example, Gaussian, convex, stable, Gibbsian, and for distributions of random processes. Sobolev classes for measures on finite and infinite dimensional spaces are discussed in detail. Finally, we present the main ideas and results of the Malliavin calculus--a powerful method to study smoothness properties of the distributions of nonlinear functionals on infinite dimensional spaces with measures. The target readership includes mathematicians and physicists whose research is related to measures on infinite dimensional spaces, distributions of random processes, and differential equations in infinite dimensional spaces. The book includes an extensive bibliography on the subject.
Author : Nicola Gigli
Publisher : American Mathematical Soc.
ISBN 13 : 1470427656
Total Pages : 174 pages
Book Rating : 4.4/5 (74 download)
Download or read book Nonsmooth Differential Geometry-An Approach Tailored for Spaces with Ricci Curvature Bounded from Below written by Nicola Gigli and published by American Mathematical Soc.. This book was released on 2018-02-23 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author discusses in which sense general metric measure spaces possess a first order differential structure. Building on this, spaces with Ricci curvature bounded from below a second order calculus can be developed, permitting the author to define Hessian, covariant/exterior derivatives and Ricci curvature.
Author : Ştefan Cobzaş
Publisher : Springer
ISBN 13 : 3030164896
Total Pages : 605 pages
Book Rating : 4.0/5 (31 download)
Download or read book Lipschitz Functions written by Ştefan Cobzaş and published by Springer. This book was released on 2019-05-23 with total page 605 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this book is to present various facets of the theory and applications of Lipschitz functions, starting with classical and culminating with some recent results. Among the included topics we mention: characterizations of Lipschitz functions and relations with other classes of functions, extension results for Lipschitz functions and Lipschitz partitions of unity, Lipschitz free Banach spaces and their applications, compactness properties of Lipschitz operators, Bishop-Phelps type results for Lipschitz functionals, applications to best approximation in metric and in metric linear spaces, Kantorovich-Rubinstein norm and applications to duality in the optimal transport problem, Lipschitz mappings on geodesic spaces. The prerequisites are basic results in real analysis, functional analysis, measure theory (including vector measures) and topology, which, for reader's convenience, are surveyed in the first chapter of the book.
Author : Xianzhe Dai
Publisher : Springer Science & Business Media
ISBN 13 : 3034802579
Total Pages : 401 pages
Book Rating : 4.0/5 (348 download)
Download or read book Metric and Differential Geometry written by Xianzhe Dai and published by Springer Science & Business Media. This book was released on 2012-06-01 with total page 401 pages. Available in PDF, EPUB and Kindle. Book excerpt: Metric and Differential Geometry grew out of a similarly named conference held at Chern Institute of Mathematics, Tianjin and Capital Normal University, Beijing. The various contributions to this volume cover a broad range of topics in metric and differential geometry, including metric spaces, Ricci flow, Einstein manifolds, Kähler geometry, index theory, hypoelliptic Laplacian and analytic torsion. It offers the most recent advances as well as surveys the new developments. Contributors: M.T. Anderson J.-M. Bismut X. Chen X. Dai R. Harvey P. Koskela B. Lawson X. Ma R. Melrose W. Müller A. Naor J. Simons C. Sormani D. Sullivan S. Sun G. Tian K. Wildrick W. Zhang
Author : Yann Ollivier
Publisher : Cambridge University Press
ISBN 13 : 1139993623
Total Pages : 317 pages
Book Rating : 4.1/5 (399 download)
Download or read book Optimal Transport written by Yann Ollivier and published by Cambridge University Press. This book was released on 2014-08-07 with total page 317 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of optimal transportation has its origins in the eighteenth century when the problem of transporting resources at a minimal cost was first formalised. Through subsequent developments, particularly in recent decades, it has become a powerful modern theory. This book contains the proceedings of the summer school 'Optimal Transportation: Theory and Applications' held at the Fourier Institute in Grenoble. The event brought together mathematicians from pure and applied mathematics, astrophysics, economics and computer science. Part I of this book is devoted to introductory lecture notes accessible to graduate students, while Part II contains research papers. Together, they represent a valuable resource on both fundamental and advanced aspects of optimal transportation, its applications, and its interactions with analysis, geometry, PDE and probability, urban planning and economics. Topics covered include Ricci flow, the Euler equations, functional inequalities, curvature-dimension conditions, and traffic congestion.
Author : U. Meierfrankenfeld
Publisher : American Mathematical Soc.
ISBN 13 : 1470418770
Total Pages : 356 pages
Book Rating : 4.4/5 (74 download)
Download or read book The Local Structure Theorem for Finite Groups With a Large $p$-Subgroup written by U. Meierfrankenfeld and published by American Mathematical Soc.. This book was released on 2016-06-21 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt: Let p be a prime, G a finite Kp-group S a Sylow p-subgroup of G and Q a large subgroup of G in S (i.e., CG(Q)≤Q and NG(U)≤NG(Q) for 1≠U≤CG(Q)). Let L be any subgroup of G with S≤L, Op(L)≠1 and Q⋬L. In this paper the authors determine the action of L on the largest elementary abelian normal p-reduced p-subgroup YL of L.
Author : Galia Devora Dafni
Publisher : American Mathematical Soc.
ISBN 13 : 0821894188
Total Pages : 241 pages
Book Rating : 4.8/5 (218 download)
Download or read book Analysis and Geometry of Metric Measure Spaces written by Galia Devora Dafni and published by American Mathematical Soc.. This book was released on 2013 with total page 241 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contains lecture notes from most of the courses presented at the 50th anniversary edition of the Seminaire de Mathematiques Superieure in Montreal. This 2011 summer school was devoted to the analysis and geometry of metric measure spaces, and featured much interplay between this subject and the emergent topic of optimal transportation.
