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This solutions manual is geared toward instructors for use as a companion
volume to the book,
This is a revised and expanded edition of Kac's original introduction
to algebraic aspects of conformal field theory, which was published by the AMS in
1996. The volume serves as an introduction to algebraic aspects of conformal
field theory, which in the past 15 years revealed a variety of unusual
mathematical notions. Vertex algebra theory provides an effective tool to study
them in a unified way.
In the book, a mathematician encounters new algebraic structures that
originated from Einstein's special relativity postulate and Heisenberg's
uncertainty principle. A physicist will find familiar notions presented in a
more rigorous and systematic way, possibly leading to a better understanding of
foundations of quantum physics.
This revised edition is based on courses given by the author at MIT and at
Rome University in spring 1997. New material is added, including the foundations of
a rapidly growing area of algebraic conformal theory. Also, in some places the
exposition has been significantly simplified.
Graduate students, research mathematicians and physicists working in mathematical aspects of quantum field theory.
Very good introductional book on vertex algebras.
-- Zentralblatt MATH
Essential reading for anyone trying to learn about vertex algebras … well worth buying for experts.
-- Bulletin of the London Mathematical Society
The genesis of these notes was a series of four lectures
given by the first author at the Tata Institute of Fundamental
Research. It evolved into a joint project and contains many
improvements and extensions on the material covered in the original
lectures.
Let \(F\) be a finite extension of \(q\), and
\(E\) an elliptic curve defined over \(F\). The
fundamental idea of the Iwasawa theory of elliptic curves, which grew
out of Iwasawa's basic work on the ideal class groups of cyclotomic
fields, is to study deep arithmetic questions about \(E\) over
\(F\) via the study of coarser questions about the arithmetic
of \(E\) over various infinite extensions of \(F\). At
present, we only know how to formulate this Iwasawa theory when the
infinite extension is a \(p\)-adic Lie extension for a fixed
prime number \(p\). These notes will mainly discuss the
simplest non-trivial example of the Iwasawa theory of \(E\)
over the cyclotomic \(zp\)-extension of \(F\). However,
the authors also make some comments about the Iwasawa theory of
\(E\) over the field obtained by adjoining all
\(p\)-power division points on \(E\) to \(F\).
They discuss in detail a number of numerical examples, which
illustrate the general theory beautifully. In addition, they outline
some of the basic results in Galois cohomology which are used
repeatedly in the study of the relevant Iwasawa modules.
The only changes made to the original notes: The authors take modest
account of the considerable progress which has been made in non-commutative
Iwasawa theory in the intervening years. They also include a short section
on the deep theorems of Kato on the cyclotomic Iwasawa theory of elliptic
curves.
A publication of the Tata Institute of Fundamental Research. Distributed worldwide except in India, Bangladesh, Bhutan, Maldavis, Nepal, Pakistan, and Sri Lanka.
Mathematicians interested in algebraic number theory.
The aim of this book is to explain modern
homotopy theory in a manner accessible to graduate students yet
structured so that experts can skip over numerous linear developments
to quickly reach the topics of their interest. Homotopy theory arises
from choosing a class of maps, called weak equivalences, and then
passing to the homotopy category by localizing with respect to the
weak equivalences, i.e., by creating a new category in which the weak
equivalences are isomorphisms. Quillen defined a model category to be
a category together with a class of weak equivalences and additional
structure useful for describing the homotopy category in terms of the
original category. This allows you to make constructions analogous to
those used to study the homotopy theory of topological spaces.
A model category has a class of maps called weak equivalences plus
two other classes of maps, called cofibrations and fibrations.
Quillen's axioms ensure that the homotopy category exists and that the
cofibrations and fibrations have extension and lifting properties
similar to those of cofibration and fibration maps of topological
spaces. During the past several decades the language of model
categories has become standard in many areas of algebraic topology,
and it is increasingly being used in other fields where homotopy
theoretic ideas are becoming important, including modern algebraic
\(K\)-theory and algebraic geometry.
All these subjects and more are discussed in the book, beginning
with the basic definitions and giving complete arguments in order to
make the motivations and proofs accessible to the novice. The book is
intended for graduate students and research mathematicians working in
homotopy theory and related areas.
Graduate students and research mathematicians.
This book was many years in the writing, and it shows. It is very carefully written, exhaustively (even obsessively) cross-referenced, and precise in all its details. In short, it is an important reference for the subject.
-- Zentralblatt MATH
Operads are powerful tools, and this is the book in which to read about them.
—Bulletin of the London Mathematical
Society
Operads are mathematical devices that describe algebraic structures
of many varieties and in various categories. Operads are particularly
important in categories with a good notion of “homotopy”,
where they play a key role in organizing hierarchies of higher
homotopies. Significant examples from algebraic topology first
appeared in the sixties, although the formal definition and
appropriate generality were not forged until the seventies. In the
nineties, a renaissance and further development of the theory were
inspired by the discovery of new relationships with graph cohomology,
representation theory, algebraic geometry, derived categories, Morse
theory, symplectic and contact geometry, combinatorics, knot theory,
moduli spaces, cyclic cohomology, and, last but not least, theoretical
physics, especially string field theory and deformation
quantization.
The book contains a detailed and comprehensive historical
introduction describing the development of operad theory from the
initial period when it was a rather specialized tool in homotopy
theory to the present when operads have a wide range of applications
in algebra, topology, and mathematical physics. Many results and
applications currently scattered in the literature are brought
together here along with new results and insights. The basic
definitions and constructions are carefully explained and include many
details not found in any of the standard literature.
Graduate students, research mathematicians, and mathematical physicists interested in homotopy theory, gauge theory, and string theory.
Operads are powerful tools, and this is the book in which to read about them.
-- Bulletin of the London Mathematical Society
The first book whose main goal is the theory of operads per se … a book such as this one has been long awaited by a wide scientific readership, including mathematicians and theoretical physicists … Written in a way to stimulate thought and abundant in references, spanning from 1898 through 2001, the book under review is guaranteed to contribute to the constant quest of mathematics for novel ideas and effective applications … a great piece of mathematical literature and will be helpful to anyone who needs to use operads, from graduate students to mature mathematicians and physicists.
-- Mathematical Reviews, Featured Review
The questions that have been at the center of invariant theory since the 19th
century have revolved around the following themes: finiteness, computation, and
special classes of invariants. This book begins with a survey of many concrete
examples chosen from these themes in the algebraic, homological, and
combinatorial context. In further chapters, the authors pick one or the other
of these questions as a departure point and present the known answers, open
problems, and methods and tools needed to obtain these answers. Chapter 2
deals with algebraic finiteness. Chapter 3 deals with combinatorial finiteness.
Chapter 4 presents Noetherian finiteness. Chapter 5 addresses homological
finiteness. Chapter 6 presents special classes of invariants, which deal with
modular invariant theory and its particular problems and features. Chapter 7
collects results for special classes of invariants and coinvariants such as
(pseudo) reflection groups and representations of low degree. If the ground
field is finite, additional problems appear and are compensated for in part by
the emergence of new tools. One of these is the Steenrod algebra, which the
authors introduce in Chapter 8 to solve the inverse invariant theory problem,
around which the authors have organized the last three chapters.
The book contains numerous examples to illustrate the theory, often of more
than passing interest, and an appendix on commutative graded algebra, which
provides some of the required basic background. There is an extensive reference
list to provide the reader with orientation to the vast literature.
Graduate students and research mathematicians interested in commutative rings, algebras, and algebraic topology.
[The book] covers a lot of information and various instructive examples.
-- Zentralblatt MATH
Both the material and the treatment would be ideal for a postgraduate course, or for inclusion in … [a] postgraduate ‘crash course’ dealing with topics in modern algebra … In addition to recommending this book to all who want to learn about invariant theory, I also recommend it to those in search of a scholium on typography (to be found on pages 357 and 358), which introduces the reader to such esoterica as Zapfian italics!
-- Bulletin of the LMS
Together with the companion volume by the same author, Operators,
Functions, and Systems: An Easy Reading. Volume 1: Hardy, Hankel, and
Toeplitz, Mathematical Surveys and Monographs, Vol. 92, AMS, 2002, this
unique work combines four major topics of modern analysis and its
applications:
A. Hardy classes of holomorphic functions,
B. Spectral theory of Hankel and Toeplitz operators,
C. Function models for linear operators and free
interpolations, and
D. Infinite-dimensional system theory and signal
processing.
This volume contains Parts C and D.
Function models for linear operators and free interpolations:
This is a universal topic and, indeed, is the most influential operator
theory technique in the post-spectral-theorem era. In this book, its
capacity is tested by solving generalized Carleson-type interpolation
problems.
Infinite-dimensional system theory and signal processing: This
topic is the touchstone of the three previously developed techniques. The
presence of this applied topic in a pure mathematics environment reflects
important changes in the mathematical landscape of the last 20 years, in
that the role of the main consumer and customer of harmonic, complex, and
operator analysis has more and more passed from differential equations,
scattering theory, and probability to control theory and signal
processing.
This and the companion volume are geared toward a wide audience of
readers, from graduate students to professional mathematicians. They
develop an elementary approach to the subject while retaining an expert
level that can be applied in advanced analysis and selected applications.
Graduate students and research mathematicians interested in operator theory, functions of a complex variable, systems theory, and control.
Together with the companion volume by the same author,
A. Hardy classes of holomorphic functions,
B. Spectral theory of Hankel and Toeplitz operators,
C. Function models for linear operators and free interpolations, and
D. Infinite-dimensional system theory and signal processing.
This volume contains Parts A and B.
Hardy classes of holomorphic functions is known to be the
most powerful tool in complex analysis for a variety of applications,
starting with Fourier series, through the Riemann
\(\zeta\)-function, all the way to Wiener's theory of signal
processing.
Spectral theory of Hankel and Toeplitz operators becomes the
supporting pillar for a large part of harmonic and complex analysis and
for many of their applications. In this book, moment problems,
Nevanlinna-Pick and Carathéodory interpolation, and the best rational
approximations are considered to illustrate the power of Hankel and
Toeplitz operators.
The book is geared toward a wide audience of readers, from graduate
students to professional mathematicians, interested in operator theory and
functions of a complex variable. The two volumes develop an elementary
approach while retaining an expert level that can be applied in advanced
analysis and selected applications.
Graduate students and research mathematicians interested in analysis.
Subriemannian geometries, also known as Carnot-Carathéodory
geometries, can be
viewed as limits of Riemannian geometries. They also arise in physical
phenomenon involving “geometric phases” or holonomy. Very roughly
speaking, a subriemannian geometry consists of a manifold endowed with a
distribution (meaning a \(k\)-plane field, or subbundle of the tangent
bundle), called horizontal together with an inner product on that
distribution. If \(k=n\), the dimension of the manifold, we get the
usual Riemannian geometry. Given a subriemannian geometry, we can define the
distance between two points just as in the Riemannian case, except we are only
allowed to travel along the horizontal lines between two points.
The book is devoted to the study of subriemannian geometries, their
geodesics, and their applications. It starts with the simplest
nontrivial example of a subriemannian geometry: the two-dimensional
isoperimetric problem reformulated as a problem of finding
subriemannian geodesics. Among topics discussed in other chapters of
the first part of the book the author mentions an elementary
exposition of Gromov's surprising idea to use subriemannian geometry
for proving a theorem in discrete group theory and Cartan's method of
equivalence applied to the problem of understanding invariants
(diffeomorphism types) of distributions. There is also a chapter
devoted to open problems.
The second part of the book is devoted to applications of subriemannian
geometry. In particular, the author describes in detail the following four
physical problems: Berry's phase in quantum mechanics, the problem of a falling
cat righting herself, that of a microorganism swimming, and a phase problem
arising in the \(N\)-body problem. He shows that all these problems can
be studied using the same underlying type of subriemannian geometry: that of a
principal bundle endowed with \(G\)-invariant metrics.
Reading the book requires introductory knowledge of differential geometry,
and it can serve as a good introduction to this new, exciting area of
mathematics.
Graduate students and research mathematicians interested in geometry and topology.
Very comprehensive and elegantly written book.
-- Mathematical Reviews
It was undoubtedly a necessary task to collect all the results on the concentration of measure during the past years in a monograph. The author did this very successfully and the book is an important contribution to the topic. It will surely influence further research in this area considerably. The book is very well written, and it was a great pleasure for the reviewer to read it.
—Mathematical Reviews
The observation of the concentration of measure
phenomenon is inspired by isoperimetric inequalities. A familiar
example is the way the uniform measure on the standard sphere
\(S^n\) becomes concentrated around the equator as the
dimension gets large. This property may be interpreted in terms of
functions on the sphere with small oscillations, an idea going back to
Lévy. The phenomenon also occurs in probability, as a version
of the law of large numbers, due to Emile Borel. This book offers the
basic techniques and examples of the concentration of measure
phenomenon. The concentration of measure phenomenon was put forward in
the early seventies by V. Milman in the asymptotic geometry of Banach
spaces. It is of powerful interest in applications in various areas,
such as geometry, functional analysis and infinite-dimensional
integration, discrete mathematics and complexity theory, and
probability theory. Particular emphasis is on geometric, functional,
and probabilistic tools to reach and describe measure concentration in
a number of settings.
The book presents concentration functions and inequalities, isoperimetric and
functional examples, spectrum and topological applications, product measures,
entropic and transportation methods, as well as aspects of M. Talagrand's deep
investigation of concentration in product spaces and its application in
discrete mathematics and probability theory, supremum of Gaussian and empirical
processes, spin glass, random matrices, etc. Prerequisites are a basic
background in measure theory, functional analysis, and probability theory.
Graduate students and research mathematicians interested in measure and integration, functional analysis, convex and discrete geometry, and probability theory and stochastic processes.