Lectures on Analysis on Metric Spaces

Lectures on Analysis on Metric Spaces
Author: Juha Heinonen
Publisher: Springer Science & Business Media
Total Pages: 158
Release: 2001
Genre: Mathematics
ISBN: 9780387951041

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.

Lectures on Analysis on Metric Spaces

Lectures on Analysis on Metric Spaces
Author: Juha Heinonen
Publisher: Springer Science & Business Media
Total Pages: 149
Release: 2012-12-06
Genre: Mathematics
ISBN: 1461301319

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.

Topics on Analysis in Metric Spaces

Topics on Analysis in Metric Spaces
Author: Luigi Ambrosio
Publisher: Oxford University Press, USA
Total Pages: 148
Release: 2004
Genre: Mathematics
ISBN: 9780198529385

This book presents the main mathematical prerequisites for analysis in metric spaces. It covers abstract measure theory, Hausdorff measures, Lipschitz functions, covering theorums, lower semicontinuity of the one-dimensional Hausdorff measure, Sobolev spaces of maps between metric spaces, and Gromov-Hausdorff theory, all developed ina general metric setting. The existence of geodesics (and more generally of minimal Steiner connections) is discussed on general metric spaces and as an application of the Gromov-Hausdorff theory, even in some cases when the ambient space is not locally compact. A brief and very general description of the theory of integration with respect to non-decreasing set functions is presented following the Di Giorgi method of using the 'cavalieri' formula as the definition of the integral. Based on lecture notes from Scuola Normale, this book presents the main mathematical prerequisites for analysis in metric spaces. Supplemented with exercises of varying difficulty it is ideal for a graduate-level short course for applied mathematicians and engineers.

New Trends on Analysis and Geometry in Metric Spaces

New Trends on Analysis and Geometry in Metric Spaces
Author: Fabrice Baudoin
Publisher: Springer Nature
Total Pages: 312
Release: 2022-02-04
Genre: Mathematics
ISBN: 3030841413

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.

Topology of Metric Spaces

Topology of Metric Spaces
Author: S. Kumaresan
Publisher: Alpha Science Int'l Ltd.
Total Pages: 172
Release: 2005
Genre: Computers
ISBN: 9781842652503

"Topology of Metric Spaces gives a very streamlined development of a course in metric space topology emphasizing only the most useful concepts, concrete spaces and geometric ideas to encourage geometric thinking, to treat this as a preparatory ground for a general topology course, to use this course as a surrogate for real analysis and to help the students gain some perspective of modern analysis." "Eminently suitable for self-study, this book may also be used as a supplementary text for courses in general (or point-set) topology so that students will acquire a lot of concrete examples of spaces and maps."--BOOK JACKET.

Lectures on Real Analysis

Lectures on Real Analysis
Author: Finnur Lárusson
Publisher: Cambridge University Press
Total Pages: 128
Release: 2012-06-07
Genre: Mathematics
ISBN: 1139511041

This is a rigorous introduction to real analysis for undergraduate students, starting from the axioms for a complete ordered field and a little set theory. The book avoids any preconceptions about the real numbers and takes them to be nothing but the elements of a complete ordered field. All of the standard topics are included, as well as a proper treatment of the trigonometric functions, which many authors take for granted. The final chapters of the book provide a gentle, example-based introduction to metric spaces with an application to differential equations on the real line. The author's exposition is concise and to the point, helping students focus on the essentials. Over 200 exercises of varying difficulty are included, many of them adding to the theory in the text. The book is perfect for second-year undergraduates and for more advanced students who need a foundation in real analysis.

Gradient Flows

Gradient Flows
Author: Luigi Ambrosio
Publisher: Springer Science & Business Media
Total Pages: 333
Release: 2008-10-29
Genre: Mathematics
ISBN: 376438722X

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.

Real Analysis

Real Analysis
Author: N. L. Carothers
Publisher: Cambridge University Press
Total Pages: 420
Release: 2000-08-15
Genre: Mathematics
ISBN: 9780521497565

A text for a first graduate course in real analysis for students in pure and applied mathematics, statistics, education, engineering, and economics.

Analysis and Geometry of Metric Measure Spaces

Analysis and Geometry of Metric Measure Spaces
Author: Galia Devora Dafni
Publisher: American Mathematical Soc.
Total Pages: 241
Release: 2013
Genre: Mathematics
ISBN: 0821894188

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.