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In August 1995 an international symposium on "Quasiconformal
Mappings and Analysis" was held in Ann Arbor on the occasion of
Professor Fred- erick W. Gehring's 70th birthday and his impending
retirement from the Mathematics Department at the University of
Michigan. The concept of the symposium was to feature broad survey
talks on a wide array of topics related to Gehring's basic research
contributions in the field of quasicon- formal mappings,
emphasizing their relations to other parts of analysis. Principal
speakers were Kari Astala, Albert Baernstein, Clifford Earle, Pe-
ter Jones, Irwin Kra, OUi Lehto, Gaven Martin, Dennis Sullivan, and
Jussi Vaisala. Financial support was provided by the National
Science Founda- tion, with additional grants from the University of
Michigan and from the Institute for Mathematics and its
Applications. The symposium was a great success. The speakers rose
to the occasion and presented excellent survey lectures. The
present volume was conceived as a means for disseminating those
expositions to a wider audience. Ad- ditional mathematicians, some
of whom had not been able to attend the symposium, were invited to
contribute similar articles. The result is a fit- ting tribute to
Fred Gehring's pre-eminent role in developing the theory of
quasiconformal mappings, through his own research and writings and
lec- tures, and through his supervision of graduate students. The
volume begins with descriptions of Gehring's mathematical career
and an overview of his research achievements.
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.
Analysis in spaces with no a priori smooth structure has progressed to include concepts from the first order calculus. In particular, there have been important advances in understanding the infinitesimal versus global behavior of Lipschitz functions and quasiconformal mappings in rather general settings; abstract Sobolev space theories have been instrumental in this development. The purpose of this book is to communicate some of the recent work in the area while preparing the reader to study more substantial, related articles. 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 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.
In August 1995 an international symposium on "Quasiconformal
Mappings and Analysis" was held in Ann Arbor on the occasion of
Professor Fred- erick W. Gehring's 70th birthday and his impending
retirement from the Mathematics Department at the University of
Michigan. The concept of the symposium was to feature broad survey
talks on a wide array of topics related to Gehring's basic research
contributions in the field of quasicon- formal mappings,
emphasizing their relations to other parts of analysis. Principal
speakers were Kari Astala, Albert Baernstein, Clifford Earle, Pe-
ter Jones, Irwin Kra, OUi Lehto, Gaven Martin, Dennis Sullivan, and
Jussi Vaisala. Financial support was provided by the National
Science Founda- tion, with additional grants from the University of
Michigan and from the Institute for Mathematics and its
Applications. The symposium was a great success. The speakers rose
to the occasion and presented excellent survey lectures. The
present volume was conceived as a means for disseminating those
expositions to a wider audience. Ad- ditional mathematicians, some
of whom had not been able to attend the symposium, were invited to
contribute similar articles. The result is a fit- ting tribute to
Fred Gehring's pre-eminent role in developing the theory of
quasiconformal mappings, through his own research and writings and
lec- tures, and through his supervision of graduate students. The
volume begins with descriptions of Gehring's mathematical career
and an overview of his research achievements.
The Ahlfors-Bers Colloquia commemorate the mathematical legacy of
Lars Ahlfors and Lipman Bers. The core of this legacy lies in the
fields of geometric function theory, Teichmuller theory, hyperbolic
manifolds, and partial differential equations. However, the work of
Ahlfors and Bers has impacted and created interactions with many
other fields, such as algebraic geometry, mathematical physics,
dynamics, geometric group theory, number theory, and topology. The
triannual Ahlford-Bers colloquia serve as a venue to disseminate
the relevant work to the wider mathematical community and bring the
key participants together to ponder future directions in the field.
The present volume includes a wide range of articles in the fields
central to this legacy. The majority of articles present new
results, but there are expository articles as well.
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 Poincare 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 Poincare
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 Poincare inequalities under Gromov-Hausdorff convergence, and
the Keith-Zhong self-improvement theorem for Poincare inequalities.
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