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.
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The last few sections of the book present a basic theory of quasisymmetric maps between metric spaces. Much of the material is relatively recent and appears for the first time in book format. There are plenty of exercises. The book is well suited for self-study, or as a text in a graduate course or seminar. The material is relevant to anyone who is interested in analysis and geometry in nonsmooth settings.
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Now quasiconformality is a ubiquitous phenomenon in mathematics. From the work of Dennis Sullivan in the late seventies we know that every space that locally looks like a flat space of Euclid has a quasiconformal structure albeit not a smooth structure in general; and this is true in every dimension but four. Nevertheless, also in dimension four, there are conjecturally an infinity of spaces where one cannot do ordinary, smooth analysis, or calculus, but where one can do quasiconformal, nonsmooth analysis.
The proposal addresses both the need to have a better understanding of this new calculus in nonsmooth spaces and methods how to recognize spaces that are quasiconformally Euclidean. Although the main questions in the proposal are motivated by a basic quest for understanding the structure of space, the consequent study of singularities in transformations has had applications to material sciences, where one wants to know when a deformed body collapses onto itself under a strain.
Lectures on Analysis on Metric Spaces (Universitext)
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Thurston obstructions and Ahlfors regular conformal dimension - ScienceDirect
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View all copies of this ISBN edition:. Synopsis About this title The purpose of this book is to communicate some of the recent advances in this field while preparing the reader for more advanced study.
soilstones.com/wp-content/2020-10-03/41.php Synopsis : Analysis in spaces with no a priori smooth structure has progressed to include concepts from the first order calculus.