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One of the most eminent of contemporary mathematicians, Shing-Tung
Yau has received numerous honors, including the 1982 Fields Medal,
considered the highest honor in mathematics, for his work in
differential geometry. He is known also for his work in algebraic
and Kähler geometry, general relativity, and string theory. His
influence in the development and establishment of these areas of
research has been great. These five volumes reproduce a
comprehensive selection of his published mathematical papers of the
years 1971 to 1991—a period of groundbreaking accomplishments in
numerous disciplines including geometric analysis, Kähler
geometry, and general relativity. The editors have organized the
contents of this collection by subject area—metric geometry and
minimal submanifolds; metric geometry and harmonic functions;
eigenvalues and general relativity; and Kähler geometry. Also
presented are expert commentaries on the subject matter, and
personal reminiscences that shed light on the development of the
ideas which appear in these papers.
Comprising volumes 28 and 29 of the ALM series, this outstanding
collection presents all the survey papers of Shing-Tung Yau
published to date (through 2013), each with Yau's own commentary.
Among these are several papers not otherwise easily accessible.
Also presented are several commentaries on Yau's work written by
outstanding scholars from around the world especially for
publication here. Shing-Tung Yau's work is mainly in differential
geometry, and he is one of the originators of the broad subject of
geometric analysis - in which he remains one of the most active
participants. His contributions have had an influence on both
physics and mathematics, and he has long been active at the
interface between geometry and theoretical physics. His proof of
the positive energy theorem in general relativity demonstrated -
sixty years after its discovery - that Einstein's theory is
consistent and stable. His proof of the Calabi conjecture allowed
physicists - using Calabi-Yau compactification - to show that
string theory is a viable candidate for a unified theory of nature.
Calabi-Yau manifolds are part of the standard toolkit of string
theorists today. He was awarded the Fields Medal in 1982 and the
Wolf Prize in 2010, and has received many other honors. The
Selected Expository Works of Shing-Tung Yau with Commentary
provides the reader with systematic commentary on all aspects of
mathematics by a contemporary master. The reader can thereby see
the world of mathematics through his particular perspective, and
gain understanding of the motivation and evolution of mathematical
ideas.
In this volume are collected notes of lectures delivered at the
First In ternational Research Institute of the Mathematical Society
of Japan. This conference, held at Tohoku University in July 1993,
was devoted to geometry and global analysis. Subsequent to the
conference, in answer to popular de mand from the participants, it
was decided to publish the notes of the survey lectures. Written by
the lecturers themselves, all experts in their respective fields,
these notes are here presented in a single volume. It is hoped that
they will provide a vivid account of the current research, from the
introduc tory level up to and including the most recent results,
and will indicate the direction to be taken by future researeh.
This compilation begins with Jean-Pierre Bourguignon's notes
entitled "An Introduction to Geometric Variational Problems,"
illustrating the gen eral framework of the field with many examples
and providing the reader with a broad view of the current research.
Following this, Kenji Fukaya's notes on "Geometry of Gauge Fields"
are concerned with gauge theory and its applications to
low-dimensional topology, without delving too deeply into technical
detail. Special emphasis is placed on explaining the ideas of infi
nite dimensional geometry that, in the literature, are often hidden
behind rigorous formulations or technical arguments."
One of the most eminent of contemporary mathematicians, Shing-Tung
Yau has received numerous honors, including the 1982 Fields Medal,
considered the highest honor in mathematics, for his work in
differential geometry. He is known also for his work in algebraic
and Kähler geometry, general relativity, and string theory. His
influence in the development and establishment of these areas of
research has been great. These five volumes reproduce a
comprehensive selection of his published mathematical papers of the
years 1971 to 1991—a period of groundbreaking accomplishments in
numerous disciplines including geometric analysis, Kähler
geometry, and general relativity. The editors have organized the
contents of this collection by subject area—metric geometry and
minimal submanifolds; metric geometry and harmonic functions;
eigenvalues and general relativity; and Kähler geometry. Also
presented are expert commentaries on the subject matter, and
personal reminiscences that shed light on the development of the
ideas which appear in these papers.
One of the most eminent of contemporary mathematicians, Shing-Tung
Yau has received numerous honors, including the 1982 Fields Medal,
considered the highest honor in mathematics, for his work in
differential geometry. He is known also for his work in algebraic
and Kähler geometry, general relativity, and string theory. His
influence in the development and establishment of these areas of
research has been great. These five volumes reproduce a
comprehensive selection of his published mathematical papers of the
years 1971 to 1991—a period of groundbreaking accomplishments in
numerous disciplines including geometric analysis, Kähler
geometry, and general relativity. The editors have organized the
contents of this collection by subject area—metric geometry and
minimal submanifolds; metric geometry and harmonic functions;
eigenvalues and general relativity; and Kähler geometry. Also
presented are expert commentaries on the subject matter, and
personal reminiscences that shed light on the development of the
ideas which appear in these papers.
This volume of Surveys in Differential Geometry is dedicated to the
three most eminent contributors to the subject of regularity and
existence of nonlinear partial differential equations, which has
played such an important role in geometry. These are Richard
Hamilton, Leon Simon, and Karen Uhlenbeck. Presented topics
include: analysis related to minimal submanifolds, Yang-Mills
theory, Kähler metrics, Monge-Ampère equations, curve flows, and
general relativity.
Geometric Analysis combines differential equations with
differential geometry. An important aspect of geometric analysis is
to approach geometric problems by studying differential equations.
Besides some known linear differential operators such as the
Laplace operator, many differential equations arising from
differential geometry are nonlinear. A particularly important
example is the Monge-Ampere equation. Applications to geometric
problems have also motivated new methods and techniques in
differential equations. The field of geometric analysis is broad
and has had many striking applications. This handbook of geometric
analysis presents introductions and survey papers treating
important topics in geometric analysis, with their applications to
related fields. It can be used as a reference by graduate students
and by researchers in related areas.
In the Spring of 1984, the two authors gave a series of lectures in
the Institute for Advanced Studies in Princeton. These lectures,
which continued throughout the 1984-1985 academic year, are
published in this volume. This greatly anticipated volume is an
essential reference tool for Differential Geometry. It describes
the major achievements in Differential Geometry, which progressed
rapidly in the 20th century.
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