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This book gives a unified, complete, and
self-contained exposition of the main algebraic theorems of invariant
theory for matrices in a characteristic free approach. More precisely,
it contains the description of polynomial functions in several
variables on the set of \(m\times m\) matrices with coefficients in an
infinite field or even the ring of integers, invariant under
simultaneous conjugation.
Following Hermann Weyl's classical approach, the ring of invariants
is described by formulating and proving
(1) the first fundamental theorem that describes a set of
generators in the ring of invariants, and
(2) the second fundamental theorem that describes relations
between these generators.
The authors study both the case of matrices over a field of
characteristic 0 and the case of matrices over a field of positive
characteristic. While the case of characteristic 0 can be treated
following a classical approach, the case of positive characteristic
(developed by Donkin and Zubkov) is much harder. A presentation of
this case requires the development of a collection of tools. These
tools and their application to the study of invariants are exlained in
an elementary, self-contained way in the book.
Undergraduate and graduate students and researchers interested in linear algebra, representation theory, and invariant theory.
Following Faltings and Vojta's work proving the Mordell-Lang
conjecture for abelian varieties and Raynaud's work proving the
Manin-Mumford conjecture, many new diophantine questions appeared,
often described as problems of unlikely intersections. The arithmetic
of moduli spaces of abelian varieties and, more generally, Shimura
varieties has been parallel-developed around the central
André-Oort conjecture. These two themes can be placed in a
common frame—the Zilber-Pink conjecture.
This volume is an introduction to these problems and to the various
techniques used: geometry, height theory, reductive groups and Hodge
theory, Shimura varieties, and model theory via the notion of
o-minimal structure. The volume contains texts corresponding to
courses presented at CIRM in May 2011 by Philipp Habegger, Gaël
Rémond, Thomas Scanlon, Emmanuel Ullmo, and Andrei Yafaev and
an ample introduction by E. Ullmo centered on the notion of
bi-algebraicity aimed at a presentation of the general
setting.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
Graduate students and researchers interested in the Zilber-Pink conjecture.
This volume presents an overview of the research in real algebraic geometry. The volume contains an introduction and five survey articles. The topics are real rational surfaces, o-minimal geometry, analytic arcs and real analytic singularities, algorithms in real algebraic geometry, positive polynomials, and sums of squares.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
Anyone interested in real algebraic geometry.
This is the first of a three volume collection
devoted to the geometry, topology, and curvature of 2-dimensional
spaces. The collection provides a guided tour through a wide range of
topics by one of the twentieth century's masters of geometric
topology. The books are accessible to college and graduate students
and provide perspective and insight to mathematicians at all levels
who are interested in geometry and topology.
The first volume begins with length measurement as dominated by the
Pythagorean Theorem (three proofs) with application to number theory;
areas measured by slicing and scaling, where Archimedes uses the
physical weights and balances to calculate spherical volume and is led
to the invention of calculus; areas by cut and paste, leading to the
Bolyai-Gerwien theorem on squaring polygons; areas by counting,
leading to the theory of continued fractions, the efficient rational
approximation of real numbers, and Minkowski's theorem on convex
bodies; straight-edge and compass constructions, giving complete
proofs, including the transcendence of \(e\) and
\(\pi\), of the impossibility of squaring the circle,
duplicating the cube, and trisecting the angle; and finally to a
construction of the Hausdorff-Banach-Tarski paradox that shows some
spherical sets are too complicated and cloudy to admit a well-defined
notion of area.
Undergraduate and graduate students and researchers interested in topology.
This book contains four essays on expository writing of books and papers at the research level and at the level of graduate texts. The authors were the four members of the AMS Committee on Expository Writing.
Real algebraic varieties are ubiquitous. They are the first
objects students encounter while learning about coordinates. But the
systematic study of these objects, however elementary they may be, is
formidable.
This book is will provide the basics of this rich theory
and for more advanced readers, will provide many fundamental
results often missing from the available literature, the
“folklore”. In particular, the introduction of topological
methods of the theory to non-specialists is one of the original
features of the book.
The first three chapters introduce the basis and classical methods
of real and complex algebraic geometry. The last three chapters each
focus on one more specific aspect of real algebraic varieties. The
book offers a panorama of classical knowledge and also addresses the
major developments of the last twenty years in terms of the topology
and geometry of varieties of dimension two and three, without
forgetting the curves, the central subject of Hilbert's famous
sixteenth problem. Various level exercises are given, and the
solutions to many of them are provided at the end of each
chapter.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
Graduate students and research mathematicians interested in real algebraic varieties.
The 7th Seasonal Institute of the Mathematical Society of
Japan on Hyperbolic Geometry and Geometric Group Theory was held from
July 30–August 5, 2014, at the University of Tokyo. This
volume, the proceedings of the meeting, collects survey and research
articles by international specialists in this fast-growing field.
This volume is recommended for researchers and graduate
students interested in hyperbolic geometry, geometric group theory,
and low-dimensional topology.
Published for the Mathematical Society of Japan by Kinokuniya, Tokyo, and distributed worldwide, except in Japan, by the AMS.
Volumes in this series are freely available electronically 5 years post-publication.
Graduate students and researchers interested in hyperbolic geometry, geometric group theory, and low-dimensional topology.
Now available in Second Edition:
MBK/107
This book is an introduction to the modern approach to the
theory of Markov chains. The main goal of this approach is to
determine the rate of convergence of a Markov chain to the stationary
distribution as a function of the size and geometry of the state
space. The authors develop the key tools for estimating convergence
times, including coupling, strong stationary times, and spectral
methods. Whenever possible, probabilistic methods are emphasized. The
book includes many examples and provides brief introductions to some
central models of statistical mechanics. Also provided are accounts
of random walks on networks, including hitting and cover times, and
analyses of several methods of shuffling cards. As a prerequisite,
the authors assume a modest understanding of probability theory and
linear algebra at an undergraduate level. Markov Chains and Mixing
Times is meant to bring the excitement of this active area of
research to a wide audience.
Markov Chains and Mixing Times is a magical book, managing to be both friendly and deep. It gently introduces probabilistic techniques so that an outsider can follow. At the same time, it is the first book covering the geometric theory of Markov chains and has much that will be new to experts. It is certainly THE book that I will use to teach from. I recommend it to all comers, an amazing achievement.
-- Persi Diaconis, Mary V. Sunseri Professor of Statistics and Mathematics, Stanford University
A superb introduction to Markov chains which treats riffle shuffling and stationary times...
-- Sami Assaf, University of Southern California, Persi Diaconis, Stanford University, and Kannan Soundararajan, Stanford University, in their paper "Riffle Shuffles with Biased Cuts"
In this book, [the authors] rapidly take a well-prepared undergraduate to the frontiers of research. Short, focused chapters with clear logical dependencies allow readers to use the book in multiple ways.
-- CHOICE Magazine
This book is a beautiful introduction to Markov chains and the analysis of their convergence towards a stationary distribution. Personally, I enjoyed very much the lucid and clear writing style of the exposition in combination with full mathematical rigor and the fascinating relations of the theory of Markov chains to several other mathematical areas.
-- Zentralblatt MATH
Throughout the book, the authors generously provide concrete examples that motivate theory and illustrate ideas. I expect this superb book to be widely used in graduate courses around the world, and to become a standard reference.
-- Mathematical Reviews
This book is about differentiation of
functions. It is divided into two parts, which can be used as
different textbooks, one for an advanced undergraduate course in
functions of one variable and one for a graduate course on Sobolev
functions. The first part develops the theory of monotone, absolutely
continuous, and bounded variation functions of one variable and their
relationship with Lebesgue–Stieltjes measures and Sobolev
functions. It also studies decreasing rearrangement and curves. The
second edition includes a chapter on functions mapping time into
Banach spaces.
The second part of the book studies functions of several
variables. It begins with an overview of classical results such as
Rademacher's and Stepanoff's differentiability theorems, Whitney's
extension theorem, Brouwer's fixed point theorem, and the divergence
theorem for Lipschitz domains. It then moves to distributions, Fourier
transforms and tempered distributions.
The remaining chapters are a treatise on Sobolev functions. The
second edition focuses more on higher order derivatives and it
includes the interpolation theorems of Gagliardo and Nirenberg. It
studies embedding theorems, extension domains, chain rule,
superposition, Poincaré's inequalities and traces.
A major change compared to the first edition is the chapter on
Besov spaces, which are now treated using interpolation theory.
Graduate students and researchers interested in Sobolev spaces, particularly their applications to PDEs.
This book offers a modern, up-to-date
introduction to quasiconformal mappings from an explicitly geometric
perspective, emphasizing both the extensive developments in mapping
theory during the past few decades and the remarkable applications of
geometric function theory to other fields, including dynamical
systems, Kleinian groups, geometric topology, differential geometry,
and geometric group theory. It is a careful and detailed introduction
to the higher-dimensional theory of quasiconformal mappings from the
geometric viewpoint, based primarily on the technique of the conformal
modulus of a curve family. Notably, the final chapter describes the
application of quasiconformal mapping theory to Mostow's celebrated
rigidity theorem in its original context with all the necessary
background.
This book will be suitable as a textbook for
graduate students and researchers interested in beginning to work on
mapping theory problems or learning the basics of the geometric
approach to quasiconformal mappings. Only a basic background in
multidimensional real analysis is assumed.
Graduate students and researchers interested in mapping theory.
[T]he book takes a wider approach to the modern theory of quasiconformal mappings and its applications than what is usual in more specialized books.
-- Olli Martio, Zentralblatt MATH