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This is the English translation of Professor
Voisin's book
reflecting the discovery of the mirror symmetry phenomenon. The first
chapter is devoted to the geometry of Calabi-Yau manifolds, and the
second describes, as motivation, the ideas from quantum field theory
that led to the discovery of mirror symmetry.
The other chapters deal with more specialized aspects of the
subject: the work of Candelas, de la Ossa, Greene, and Parkes, based
on the fact that under the mirror symmetry hypothesis, the variation
of Hodge structure of a Calabi-Yau threefold determines the
Gromov-Witten invariants of its mirror; Batyrev's construction, which
exhibits the mirror symmetry phenomenon between hypersurfaces of toric
Fano varieties, after a combinatorial classification of the latter;
the mathematical construction of the Gromov-Witten potential, and the
proof of its crucial property (that it satisfies the WDVV equation),
which makes it possible to construct a flat connection underlying a
variation of Hodge structure in the Calabi-Yau case. The book
concludes with the first “naive” Givental computation,
which is a mysterious mathematical justification of the computation of
Candelas, et al.
Titles in this series are co-published with Société Mathématique de France. SMF members are entitled to AMS member discounts.
Graduate students and research mathematicians interested in algebraic geometry; mathematical physicists.
This book … might yet give, even to the non-specialist, some basic orientation in the complicated and rapidly developing world of mirror symmetry.
-- European Mathematical Society Newsletter
Without any doubt, the English version of this panoramic introduction to the phenomenon of mirror symmetry will find a much larger number of interesting readers than the French original could do, and that is what this beautiful text really deserves.
-- Zentralblatt MATH
Partial differential equations (PDEs) and geometric measure
theory (GMT) are branches of analysis whose connections are usually
not emphasized in introductory graduate courses. Yet one cannot
dissociate the notions of mass or electric charge, naturally described
in terms of measures, from the physical potential they
generate. Having such a principle in mind, this book illustrates the
beautiful interplay between tools from PDEs and GMT in a simple and
elegant way by investigating properties such as existence and regularity
of solutions of linear and nonlinear elliptic PDEs.
Inspired by a variety of sources, from the pioneer balayage scheme
of Poincaré to more recent results related to the Thomas–Fermi and
Chern–Simons models, the problems covered in this book follow an
original presentation, intended to emphasize the main ideas in the
proofs. Classical techniques such as regularity theory, maximum
principles and the method of sub- and supersolutions are adapted to
the setting where merely integrability or density assumptions on the
data are available. The distinguished role played by capacities and
precise representatives is also explained.
Other special features are: the remarkable equivalence between
Sobolev capacities and Hausdorff contents in terms of trace
inequalities; the strong approximation of measures in terms of
capacities or densities, normally absent from GMT books; and the rescue of
the strong maximum principle for the Schrödinger operator
involving singular potentials.
This book invites the reader on a trip through modern techniques in
the frontier of elliptic PDEs and GMT and is addressed to graduate
students and researchers with a deep interest in analysis. Most
of the chapters can be read independently, and only a basic knowledge of
measure theory, functional analysis, and Sobolev spaces is
required.
A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Graduate students and research mathematicians interested in analysis.
This is the second of three volumes which will provide a
comprehensive introduction to the modern representation theory of
Frobenius algebras. The first part of the book is devoted to
fundamental results of the representation theory of finite dimensional
hereditary algebras and their tilted algebras, which allow the authors
to describe the representation theory of prominent classes of
Frobenius algebras.
The second part is devoted to basic classical and recent results
concerning the Hochschild extensions of finite dimensional algebras by
duality bimodules and their module categories. Moreover, the shapes of
connected components of the stable Auslander-Reiten quivers of
Frobenius algebras are described.
The only prerequisite for this volume is a basic knowledge of linear
algebra and some results of the first volume. It includes complete
proofs of all results presented and provides a rich supply of examples
and exercises.
A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Graduate students and research mathematicians interested in the representation theory of algebras.
Rolfsen's beautiful book on knots and links can be read by
anyone, from beginner to expert, who wants to learn about knot
theory. Beginners find an inviting introduction to the elements of
topology, emphasizing the tools needed for understanding knots, the
fundamental group and van Kampen's theorem, for example, which are
then applied to concrete problems, such as computing knot groups. For
experts, Rolfsen explains advanced topics, such as the connections
between knot theory and surgery and how they are useful to
understanding three-manifolds.
Besides providing a guide to understanding knot theory, the book offers
“practical” training. After reading it, you will be able to do many things:
compute presentations of knot groups, Alexander polynomials, and other
invariants; perform surgery on three-manifolds; and visualize knots and their
complements. It is characterized by its hands-on approach and emphasis on a
visual, geometric understanding.
Rolfsen offers invaluable insight and strikes a perfect balance
between giving technical details and offering informal
explanations. The illustrations are superb, and a wealth of examples
are included.
Now back in print by the AMS, the book is still a standard reference in knot
theory. It is written in a remarkable style that makes it useful for both
beginners and researchers. Particularly noteworthy is the table of knots and
links at the end. This volume is an excellent introduction to the topic and is suitable
as a textbook for a course in knot theory or 3-manifolds.
Other key books of interest on this topic available from the AMS are The Shoelace Book: A
Mathematical Guide to the Best (and Worst) Ways to Lace your Shoes and
The Knot Book.
Advanced undergraduates, graduate students, and research mathematicians interested in knot theory and its applications to low-dimensional topology.
...a gem and a classic. Every mathematics library should own a copy and every mathematician should read at least some of it. The writing is clear and engaging, while the choice of examples is genius...Rolfsen's book continues to be a beautiful introduction to some beautiful ideas.
-- Scott A. Taylor, MAA Reviews
Looking at a sequence of zeros and ones, we
often feel that it is not random, that is, it is not plausible as an
outcome of fair coin tossing. Why? The answer is provided by
algorithmic information theory: because the sequence is compressible,
that is, it has small complexity or, equivalently, can be produced by a short
program. This idea, going back to Solomonoff, Kolmogorov, Chaitin,
Levin, and others, is now the starting point of algorithmic
information theory.
The first part of this book is a textbook-style exposition of the
basic notions of complexity and randomness; the second part covers
some recent work done by participants of the “Kolmogorov
seminar” in Moscow (started by Kolmogorov himself in the 1980s)
and their colleagues.
This book contains numerous exercises (embedded in the text) that
will help readers to grasp the material.
Graduate students and researchers interested in topics related to an algorithmic approach to complexity and randomness.
This is a very well written book. Through explicit examples and (at times elaborate) calculations, the authors are able to provide answers to some important questions in the theory of elliptic equations. It is a remarkable feat that the seemingly different worlds of nonassociative algebras and that of nonlinear elliptic equations can be combined so effectively in a self-contained book of this size.
-- Florin Catrina, Zentralblatt Math
The ultimate goal of this book is to explain
that the Grothendieck–Teichmüller group, as defined by Drinfeld in
quantum group theory, has a topological interpretation as a group of
homotopy automorphisms associated to the little 2-disc operad. To
establish this result, the applications of methods of algebraic
topology to operads must be developed. This volume is devoted
primarily to this subject, with the main objective of developing a
rational homotopy theory for operads.
The book starts with a comprehensive review of the general theory
of model categories and of general methods of homotopy theory. The
definition of the Sullivan model for the rational homotopy of spaces
is revisited, and the definition of models for the rational homotopy
of operads is then explained. The applications of spectral sequence
methods to compute homotopy automorphism spaces associated to operads
are also explained. This approach is used to get a topological
interpretation of the Grothendieck–Teichmüller group in the
case of the little 2-disc operad.
This volume is intended for graduate students and researchers
interested in the applications of homotopy theory methods in operad
theory. It is accessible to readers with a minimal background in
classical algebraic topology and operad theory.
Graduate students and researchers interested in algebraic topology and algebraic geometry.
Destined to remain the classic treatment of the subject … for many years to come … Although an inspection of the table of contents reveals a coverage so extensive that the work of more than 600 pages might lead one at first to regard this book as an encyclopedia on the subject, a reading of any chapter of the text will impress the reader as a friendly lecture revealing an unusual appreciation of both rigor and the computing technique so important to the statistician. In a word, Jordan's work is a most readable and detailed record of lectures on the Calculus of Finite Differences which will certainly appeal tremendously to the statistician and which could have been written only by one possessing a deep appreciation of mathematical statistics.
-- Harry C. Carver, founder and former Editor of the Annals of Mathematical Statistics
This book aims to give students a chance to begin exploring
some introductory to intermediate topics in combinatorics, a
fascinating and accessible branch of mathematics centered around
(among other things) counting various objects and sets.
The book includes chapters featuring tools for solving counting
problems, proof techniques, and more to give students a broad
foundation to build on. The only prerequisites are a solid background
in arithmetic, some basic algebra, and a love for learning
mathematics.
A publication of XYZ Press. Distributed in North America by the American Mathematical Society.
Middle and high school students interested in mathematics competition preparation.
The only prerequisites to study the book are a background in arithmetic and some basic algebra. I believe that this will be a useful source for a wide range of introductory to intermediate audiences. Instructors preparing students for mathematical competitions will find the book very suitable for that purpose.
-- Mehdi Hassani, MAA Reviews
This is a book of problems in abstract algebra for strong undergraduates or beginning graduate students. It can be used as a supplement to a course or for self-study. The book provides more variety and more challenging problems than are found in most algebra textbooks. It is intended for students wanting to enrich their learning of mathematics by tackling problems that take some thought and effort to solve. The book contains problems on groups (including the Sylow Theorems, solvable groups, presentation of groups by generators and relations, and structure and duality for finite abelian groups); rings (including basic ideal theory and factorization in integral domains and Gauss's Theorem); linear algebra (emphasizing linear transformations, including canonical forms); and fields (including Galois theory). Hints to many problems are also included.
Undergraduate and graduate students interested in teaching and learning undergraduate algebra.
I'll certainly be using some of the problems the next time I teach algebra. I'm even tempted to make it the only textbook for the course.
-- Fernando Q. Gouvêa, MAA Reviews
This book offers a gentle introduction to the mathematics of both sides of game theory: combinatorial and classical. The combination allows for a dynamic and rich tour of the subject united by a common theme of strategic reasoning. Designed as a textbook for an undergraduate mathematics class and with ample material and limited dependencies between the chapters, the book is adaptable to a variety of situations and a range of audiences. Instructors, students, and independent readers alike will appreciate the flexibility in content choices as well as the generous sets of exercises at various levels.
Undergraduate students, graduate students, and researchers interested in game theory.
The topics covered here are chosen for a broad and versatile look at the subject, the writing style is clear and enjoyable, examples are plentiful, and there is a good selection of exercises, both computational and proof-oriented...In addition to clear and engaging writing, and a good selection of exercises, this book also boasts an excellent bibliography...I have no hesitation whatsoever recommending it as a text for an introductory undergraduate course.
-- Mark Hunacek, MAA Reviews