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In the fifteen years since the discovery that Artin's braid groups enjoy a
left-invariant linear ordering, several quite different approaches have
been used to understand this phenomenon. This book is an account of those
approaches, which involve such varied objects and domains as combinatorial
group theory, self-distributive algebra, finite combinatorics, automata,
low-dimensional topology, mapping class groups, and hyperbolic geometry.
The remarkable point is that all these approaches lead to the same
ordering, making the latter rather canonical.
We have attempted to make the ideas in this volume accessible and
interesting to students and seasoned professionals alike. Although the text
touches upon many different areas, we only assume that the reader has some
basic background in group theory and topology, and we include detailed
introductions wherever they may be needed, so as to make the book as
self-contained as possible.
The present volume follows the book, Why are braids orderable?,
written by the same authors and published in 2002 by the Société
Mathématique de France. The current text contains a considerable amount
of new material, including ideas that were unknown in 2002. In addition,
much of the original text has been completely rewritten, with a view to
making it more readable and up-to-date.
Graduate students and research mathematicians interested in braid, group theory, low-dimensional topology.
...this is a timely and very carefully written book describing important, interesting and beautiful results in this new area of research concerning braid groups. It will no doubt create much interest and inspire many more insights into these order structures.
-- Stephen P. Humphries for Mathematical Reviews
Why do solutions of linear analytic PDE
suddenly break down? What is the source of these mysterious
singularities, and how do they propagate? Is there a mean value
property for harmonic functions in ellipsoids similar to that for
balls? Is there a reflection principle for harmonic functions in higher
dimensions similar to the Schwarz reflection principle in the plane?
How far outside of their natural domains can solutions of the
Dirichlet problem be extended? Where do the continued solutions become
singular and why?
This book invites graduate students and young analysts to explore
these and many other intriguing questions that lead to beautiful
results illustrating a nice interplay between parts of modern analysis
and themes in “physical” mathematics of the nineteenth
century. To make the book accessible to a wide audience including
students, the authors do not assume expertise in the theory of
holomorphic PDE, and most of the book is accessible to anyone familiar
with multivariable calculus and some basics in complex analysis and
differential equations.
Graduate students and researchers interested in PDE, especially in holomorphic linear PDE.
In the decade since the discovery that Artin's braid groups enjoy a
left-invariant linear ordering, several quite different approaches have been
applied to understand this phenomenon. This book is an account of those
approaches, involving self-distributive algebra, uniform finite trees,
combinatorial group theory, mapping class groups, laminations, and hyperbolic
geometry.
This volume is suitable for graduate students and research mathematicians
interested in algebra and topology.
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 algebra and topology.
This book, which can be considered as a sequel of the author's famous
book Character Theory of Finite Groups, concerns the character theory
of finite solvable groups and other groups that have an abundance of
normal subgroups.
It is subdivided into three parts: \(\pi\)-theory, character
correspondences, and M-groups. The \(\pi\)-theory
section contains an exposition of D. Gajendragadkar's
\(\pi\)-special characters, and it includes various extensions,
generalizations, and applications of his work. The character
correspondences section proves the McKay character counting conjecture
and the Alperin weight conjecture for solvable groups, and it
constructs a canonical McKay bijection for odd-order groups. In
addition to a review of some basic material on M-groups,
the third section contains an exposition of the use of symplectic
modules for studying M-groups. In particular, an accessible
presentation of E. C. Dade's deep results on monomial characters of
odd prime-power degree is included.
Very little of this material has previously appeared in book form,
and much of it is based on the author's research. By reading a clean
and accessible presentation written by the leading expert in the
field, researchers and graduate students will be inspired to learn and
work in this area that has fascinated the author for decades.
Undergraduate and graduate students and researchers interested in solvable groups, character theory, and finite group theory.
This text is a monograph on algebra, with connections to geometry
and low-dimensional topology. It mainly involves groups, monoids, and
categories, and aims to provide a unified treatment for those
situations in which one can find distinguished decompositions by
iteratively extracting a maximal fragment lying in a prescribed family.
Initiated in 1969 by F. A. Garside in the case of Artin's braid groups,
this approach led to interesting results in a number of
cases, the central notion being what the authors call a Garside family.
The study is far from complete, and the purpose of this
book is to present the current state of the theory and to
invite further research.
The book has two parts: In Part A, the bases of a general theory,
including many easy examples, are developed. In Part B, various more
sophisticated examples are specifically addressed.
To make the content accessible to a wide audience of
nonspecialists, the book's exposition is essentially self-contained and very
few prerequisites are needed. In particular, it should be easy to use
this as a textbook both for Garside theory and for the
more specialized topics investigated in Part B: Artin–Tits
groups, Deligne–Lusztig varieties, groups of algebraic laws,
ordered groups, and structure groups of set-theoretic solutions of the
Yang–Baxter equation. The first part of the book can be used as
the basis for a graduate or advanced undergraduate course.
A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Undergraduate and graduate students interested in Garside theory.
The Moscow Mathematical Olympiad has been
challenging high school students with stimulating, original problems
of different degrees of difficulty for over 75 years. The problems are
nonstandard; solving them takes wit, thinking outside the box, and,
sometimes, hours of contemplation. Some are within the reach of most
mathematically competent high school students, while others are
difficult even for a mathematics professor. Many mathematically
inclined students have found that tackling these problems, or even
just reading their solutions, is a great way to develop mathematical
insight.
In 2006 the Moscow Center for Continuous Mathematical Education
began publishing a collection of problems from the Moscow Mathematical
Olympiads, providing for each an answer (and sometimes a hint) as well
as one or more detailed solutions. This volume represents the years
2000–2005.
The problems and the accompanying material are well suited for math
circles. They are also appropriate for problem-solving classes and
practice for regional and national mathematics
competitions.
In the interest of fostering a greater awareness and appreciation of
mathematics and its connections to other disciplines and everyday life, MSRI
and the AMS are publishing books in the Mathematical Circles Library series as
a service to young people, their parents and teachers, and the mathematics
profession.
Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).
High school and undergraduate students interested in problem solving and mathematical circles.
The Moscow Mathematical Olympiad has been challenging high school
students with stimulating, original problems of different degrees of
difficulty for over 75 years. The problems are nonstandard; solving them
takes wit, thinking outside the box, and, sometimes, hours of
contemplation. Some are within the reach of most mathematically
competent high school students, while others are difficult even for a
mathematics professor. Many mathematically inclined students have found
that tackling these problems, or even just reading their solutions, is a
great way to develop mathematical insight.
In 2006 the Moscow Center for Continuous Mathematical Education began
publishing a collection of problems from the Moscow Mathematical
Olympiads, providing for each an answer (and sometimes a hint) as well as
one or more detailed solutions. This volume represents the years
1993–1999.
The problems and the accompanying material are well suited for math
circles. They are also appropriate for problem-solving classes and
practice for regional and national mathematics competitions.
In the interest of fostering a greater awareness and appreciation of
mathematics and its connections to other disciplines and everyday life, MSRI
and the AMS are publishing books in the Mathematical Circles Library series as
a service to young people, their parents and teachers, and the mathematics
profession.
Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).
High school and undergraduate students interested in problem solving; mathematical circles.
New Horizons in Geometry represents the fruits of 15 years of work in geometry by a remarkable team of prize-winning authors Tom Apostol and Mamikon Mnatsakanian. It serves as a capstone to an amazing collaboration. Apostol and Mamikon provide fresh and powerful insights into geometry that requires only a modest background in mathematics. Using new and intuitively rich methods, they give beautifully illustrated proofs of results, the majority of which are new, and frequently develop extensions of familiar theorems that are often surprising and sometimes astounding. It is mathematical exposition of the highest order. The hundreds of full color illustrations by Mamikon are visually enticing and provide great motivation to read further and savor the wonderful results. Lengths, areas, and volumes of curves, surfaces, and solids are explored from a visually captivating perspective. It is an understatement to say that Apostol and Mamikon have breathed new life into geometry.
In a remarkable display of mathematical versatility and imagination, the authors present us with a wealth of geometrical gems. These beautiful and often surprising results deal with a multitude of geometric forms, their interrelationships, and in many cases, their connection with patterns underlying the laws of nature.
-- Don Chakerian
… The authors provide fresh and powerful insights into geometry that requires only a modest background in mathematics. Using new and intuitively rich methods, they give beautifully illustrated proofs of results and extensions of familiar theorems. Lengths, areas and volumes of curves, surfaces and solids are explored from a visually captivating perspective. … The hundreds of full color illustrations are visually enticing and provide great motivation to read further and savor the wonderful results. This book is a must have for any geometer.
-- Dirk Keppen, Zentrallblatt MATH
Readers of New Horizons in Geometry are in for a great ride in the spirit of Archimedes through a beautiful geometrical landscape that will give you considerable pleasure and a heightened appreciation for a wonderful subject.
-- Don Albers, former Director of MAA Publications
Within the subject of topological dynamics,
there has been considerable recent interest in systems where the
underlying topological space is a Cantor set. Such systems have an
inherently combinatorial nature, and seminal ideas of Anatoly Vershik
allowed for a combinatorial model, called the Bratteli-Vershik model,
for such systems with no non-trivial closed invariant subsets. This
model led to a construction of an ordered abelian group which is an
algebraic invariant of the system providing a complete classification
of such systems up to orbit equivalence.
The goal of this book is to give a statement of this classification
result and to develop ideas and techniques leading to it. Rather than
being a comprehensive treatment of the area, this book is aimed at
students and researchers trying to learn about some surprising
connections between dynamics and algebra. The only background material
needed is a basic course in group theory and a basic course in general
topology.
Undergraduate and graduate students and researchers interested in dynamical systems.
A well-written, inviting textbook designed for
a one-semester, junior-level course in elementary number theory. The
intended audience will have had exposure to proof writing, but not
necessarily to abstract algebra. That audience will be well prepared
by this text for a second-semester course focusing on algebraic number
theory. The approach throughout is geometric and intuitive; there are
over 400 carefully designed exercises, which include a balance of
calculations, conjectures, and proofs. There are also nine
substantial student projects on topics not usually covered in a
first-semester course, including Bernoulli numbers and polynomials,
geometric approaches to number theory, the \(p\)-adic numbers,
quadratic extensions of the integers, and arithmetic generating
functions.
An instructor's manual for this title is available
electronically. Please send email to textbooks@ams.org for more
information.