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.
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.
Download or read book Lectures on Analysis on Metric Spaces written by Juha Heinonen and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 149 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this book is to communicate some of the recent advances in this field while preparing the reader for more advanced study. The material can be roughly divided into three different types: classical, standard but sometimes with a new twist, and recent. The author first studies basic covering theorems and their applications to analysis in metric measure spaces. This is followed by a discussion on Sobolev spaces emphasizing principles that are valid in larger contexts. The last few sections of the book present a basic theory of quasisymmetric maps between metric spaces. Much of the material is recent and appears for the first time in book format.
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: Analysis on metric spaces emerged in the 1990s as an independent research field providing a unified treatment of first-order analysis in diverse and potentially nonsmooth settings. Based on the fundamental concept of upper gradient, the notion of a Sobolev function was formulated in the setting of metric measure spaces supporting a Poincaré inequality. This coherent treatment from first principles is an ideal introduction to the subject for graduate students and a useful reference for experts. It presents the foundations of the theory of such first-order Sobolev spaces, then explores geometric implications of the critical Poincaré inequality, and indicates numerous examples of spaces satisfying this axiom. A distinguishing feature of the book is its focus on vector-valued Sobolev spaces. The final chapters include proofs of several landmark theorems, including Cheeger's stability theorem for Poincaré inequalities under Gromov–Hausdorff convergence, and the Keith–Zhong self-improvement theorem for Poincaré inequalities.
Download or read book Analysis and Geometry of Metric Measure Spaces written by Galia Devora Dafni and published by . This book was released on 2013 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains lecture notes from most of the courses presented at the 50th anniversary edition of the Séminaire de Mathématiques Supérieures in Montréal. 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. In recent decades, metric measure spaces have emerged as a fruitful source of mathematical questions in their own right, and as indispensable tools for addressing classical problems in geometry, topology, dynamical systems, and partial differential eq.
Download or read book Preservation of Bounded Geometry Under Transformations of Metric Spaces written by Xining Li and published by . This book was released on 2015 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the theory of geometric analysis on metric measure spaces, two properties of a metric measure space make the theory richer. These two properties are the doubling property of the measure, and the support of a Poincare ́inequality by the metric measure space. The focus of this dissertation is to show that the doubling property of the measure and the support of a Poincare ́ inequality are preserved by two transformations of the metric measure space: sphericalization (to obtain a bounded space from an unbounded space), and flattening (to obtain an unbounded space from a bounded space). We will show that if the given metric measure space is equipped with an Ahlfors Q-regular measure, then so are the spaces obtained by the sphericalization/flattening transformations. We then show that even if the measure is not Ahlfors regular, if it is doubling, then the transformed measure is still doubling. We then show that if the given metric space satisfies an annular quaisconvexity property and the measure is doubling, and in addition if the metric measure space supports a p-Poincare ́inequality in the sense of Heinonen and Koskela's theory, then so does the transformed metric measure space (under the sphericalization/flattening procedure). Finally, we show that if we relax the annular quasiconvexity condition to an analog of the starlike condition for the metric measure space, then if the metric measure space also satisfies a p-Poincare ́inequality, the transformed space also must satisfy a q-Poincare ́inequality for some p
Download or read book Geometry and Analysis of Metric Spaces via Weighted Partitions written by Jun Kigami and published by Springer Nature. This book was released on 2020-11-16 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of these lecture notes is to propose a systematic framework for geometry and analysis on metric spaces. The central notion is a partition (an iterated decomposition) of a compact metric space. Via a partition, a compact metric space is associated with an infinite graph whose boundary is the original space. Metrics and measures on the space are then studied from an integrated point of view as weights of the partition. In the course of the text: It is shown that a weight corresponds to a metric if and only if the associated weighted graph is Gromov hyperbolic. Various relations between metrics and measures such as bilipschitz equivalence, quasisymmetry, Ahlfors regularity, and the volume doubling property are translated to relations between weights. In particular, it is shown that the volume doubling property between a metric and a measure corresponds to a quasisymmetry between two metrics in the language of weights. The Ahlfors regular conformal dimension of a compact metric space is characterized as the critical index of p-energies associated with the partition and the weight function corresponding to the metric. These notes should interest researchers and PhD students working in conformal geometry, analysis on metric spaces, and related areas.
Download or read book Geometric Analysis and Function Spaces written by Steven George Krantz and published by American Mathematical Soc.. This book was released on 1993-01-01 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book brings into focus the synergistic interaction between analysis and geometry by examining a variety of topics in function theory, real analysis, harmonic analysis, several complex variables, and group actions. Krantz's approach is motivated by examples, both classical and modern, which highlight the symbiotic relationship between analysis and geometry. Creating a synthesis among a host of different topics, this book is useful to researchers in geometry and analysis and may be of interest to physicists, astronomers, and engineers in certain areas. The book is based on lectures presented at an NSF-CBMS Regional Conference held in May 1992.
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.
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.
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.
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.
Download or read book An Invitation to Alexandrov Geometry written by Stephanie Alexander and published by Springer. This book was released on 2019-05-08 with total page 88 pages. Available in PDF, EPUB and Kindle. Book excerpt: Aimed toward graduate students and research mathematicians, with minimal prerequisites this book provides a fresh take on Alexandrov geometry and explains the importance of CAT(0) geometry in geometric group theory. Beginning with an overview of fundamentals, definitions, and conventions, this book quickly moves forward to discuss the Reshetnyak gluing theorem and applies it to the billiards problems. The Hadamard–Cartan globalization theorem is explored and applied to construct exotic aspherical manifolds.
Download or read book Pure Metric Geometry written by Anton Petrunin and published by Springer. This book was released on 2023-11-19 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book serves as an introductory asset for learning metric geometry by delivering an in-depth examination of key constructions and providing an analysis of universal spaces, injective spaces, the Gromov-Hausdorff convergence, and ultralimits. This book illustrates basic examples of domestic affairs of metric spaces, this includes Alexandrov geometry, geometric group theory, metric-measure spaces and optimal transport. Researchers in metric geometry will find this book appealing and helpful, in addition to graduate students in mathematics, and advanced undergraduate students in need of an introduction to metric geometry. Any previous knowledge of classical geometry, differential geometry, topology, and real analysis will be useful in understanding the presented topics.
Download or read book Analysis on Metric measure Spaces written by Alireza Ranjbar-Motlagh and published by . This book was released on 1998 with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Lipschitz Algebras written by Nik Weaver and published by World Scientific. This book was released on 1999 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Lipschitz algebras Lp(M), for M a complete metric space, are quite analogous to the spaces C(omega) and Linfinity(X), for omega a compact Hausdorff space and X a sigma-finite measure space. Although the Lipschitz algebras have not been studied as thoroughly as these better-known cousins, it is becoming increasingly clear that they play a fundamental role in functional analysis, and are also useful in many applications, especially in the direction of metric geometry. This book gives a comprehensive treatment of (what is currently known about) the beautiful theory of these algebras.
Download or read book Measure Topology and Fractal Geometry written by Gerald A. Edgar and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the reviews: "In the world of mathematics, the 1980's might well be described as the "decade of the fractal". Starting with Benoit Mandelbrot's remarkable text The Fractal Geometry of Nature, there has been a deluge of books, articles and television programmes about the beautiful mathematical objects, drawn by computers using recursive or iterative algorithms, which Mandelbrot christened fractals. Gerald Edgar's book is a significant addition to this deluge. Based on a course given to talented high- school students at Ohio University in 1988, it is, in fact, an advanced undergraduate textbook about the mathematics of fractal geometry, treating such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology. However, the book also contains many good illustrations of fractals (including 16 color plates), together with Logo programs which were used to generate them. ... Here then, at last, is an answer to the question on the lips of so many: 'What exactly is a fractal?' I do not expect many of this book's readers to achieve a mature understanding of this answer to the question, but anyone interested in finding out about the mathematics of fractal geometry could not choose a better place to start looking." #Mathematics Teaching#1
