This book develops aspects of category theory
fundamental to the study of algebraic K-theory.
Ring and module theory illustrates category
theory
which provides insight into more advanced
topics in module theory.
Starting with categories in general, the
text then examines categories of K-theory.
This leads to the study of tensor products
and the Morita theory.
The categorical approach to localizations
and completions of modules is formulated
in terms of direct and inverse limits,
prompting a discussion of localization of
categories in general.
Finally, local-global techniques which supply
information about modules from
their localizations and completions and underlie
some interesting applications of
K-theory to number theory and geometry are
considered.
Chapter Contents
1. Basics; 2. Direct sums and their short
exact sequences;
3. Noetherian rings and polynomial rings;
4. Artinian rings and modules; 5. Dedekind
domains;
6. Modules over Dedekind domains.
Dec. 1999 360 pp.
0-521-63276-5
Cambridge
During 1996-97 MSRI held a full academic
year program on
Combinatorics, with special emphasis on the
connections with other branches of
mathematics, such as algebraic geometry,
topology,
commutative algebra, representation theory,
and convex geometry.
The rich combinatorial problems arising from
the study of various
algebraic structures are the subject of this
book, which represents work done
or presented at seminars during the program.
It contains contributions on matroid bundles,
combinatorial representation theory, lattice
points in polyhedra, bilinear forms,
combinatorial differential topology and geometry,
Macdonald polynomials and geometry, enumeration
of matchings,
the generalized Baues problem, and Littlewood-Richardson
semigroups.
These expository articles, written by some
of the most
respected researchers in the field, will
present the state of
the art to graduate students and researchers
in combinatorics
as well as algebra, geometry, and topology.
Chapter Contents
1. Matroid bundles Laura Anderson;
2. Combinatorial representation theory Helene
Barcelo and Arun Ram;
3. An algorithmic theory of lattice points
in polyhedra Alexander
Barvinok and James Pommersheim;
4. Some algebraic properties of the Schechtman-Varchenko
bilinear forms
Graham Denham@and Phil Hanlon;
5. Combinatorial differential topology and
geometry Robin Forman;
6. Macdonald polynomials and geometry Mark
Haiman;
7. Enumeration of matchings: problems and
progress James Propp;
8. The generalized Baues problem Victor Reiner;
9. Littlewood-Richardson semigroups Andrei
Zelevinsky
Sep. 1999 320 pp.
0-521-77087-4
Cambridge
This is an undergraduate textbook
that reveals the intricacies of geometry.
The approach used is that a geometry is a
space together
with a set of transformations of that space
(as argued by Klein in his Erlangen programme).
The authors explore various geometries:
affine, projective, inversive, non-Euclidean
and spherical.
In each case the key results are explained
carefully,
and the relationships between the geometries
are discussed.
This richly illustrated and clearly written
text includes full solutions to
over 200 problems, and is suitable both for
undergraduate courses on geometry
and as a resource for self study.
Chapter Contents
Introduction; 1. Conics; 2. Affine geometry;
3. Projective geometry: lines; 4. Projective
geometry: conics;
5. Inversive geometry; 6. Non-Euclidean geometry;
7. Spherical
geometry; 8. The Kleinian view of geometry;
Appendices
Apr. 1999 497pp
0 521 59193 7/159787-0 (Paper ed.)
Cambridge
2nd International Conference, Xi'an, P.R.
China, 1998
Bifurcation theory consists of two distinct
aspects - static and dynamic.
Static bifurcation theory deals with the
changes that occur in the structure
of the set of zeros of a function as parameters
in the function are varied,
while the dynamic one is concerned with the
changes that occur
in the structure of the limit sets of solutions
of differential equations
as parameters in the vector field are varied.
Its extensive research and numerical analyses
have been conducted in the past years.
This book contains eighteen refereed papers
presented at the conference,
held in Xi'an, China, June 29 - July 3, 1998.
The papers cover recent development of a
wide range of
theoretical and numerical issues of bifurcation
theory.
May 1999. 300 pp.
981-4021-58-X
Springer
Self-Focusing and Wave Collapse
This monograph aims to fill the gap between
the mathematical literature
which significantly contributed during the
last decade to the understanding
of the collapse phenomenon, and applications
to domains like plasma physics
and non-linear optics where this process
provides a fundamental mechanism
for small scale formation and wave dissipation.
This results in a localized heating of the
medium and in the case of
propagation in a dielectric to possible degradation
of the material.
For this purpose, the authors have chosen
to address the problem of wave collapse
by several methods raging from rigorous mathematical
analysis to formal asymptotic
expansions and numerical simulations.
July 1999. 370 pp.
0-387-98611-1
Springer
Statistics of Directional Data, 2nd ed.
Beginning with Jackson networks and ending
with spatial queuing
systems, this book describes several basic
stochastic network processes,
with the focus on network processes that
have tractable expressions for
the equilibrium probability distribution
of the numbers of units at the stations.
Intended for graduate students and researchers
in engineering,
science and mathematics interested in the
basics of stochastic networks
that have been developed over the last twenty
years,
the text assumes a graduate course in stochastic
processes without measure theory,
emphasizing multi-dimensional Markov processes.
Alongside self-contained material on point
processes involving real
analysis, the book also contains complete
introductions to reversible
Markov processes, Palm probabilities for
stationary systems,
Little laws for queuing systems and space-time
Poisson processes.
July 1999 330 pp.
0-387-98773-8
Springer
This invaluable book is based on E Witten's
lectures on
topological quantum field theory
which were presented in the spring of 1989
at Fine Hall, Princeton.
At that time Witten unified several important
mathematical works in terms of
quantum field theory, most notably the Donaldson
polynomial,
Gromov/Floer homology and Jones polynomials.
In the book Witten explains his three-dimensional
construction of
Jones polynomials via ChernSimons gauge
theory.
The classical moduli space consists of unitary
flat connections.
To quantize it, he constructs a Hilbert space
consisting of holomorphic sections
of the natural determinant bundle over the
moduli space,
or non-Abelian theta functions.
He explains that the quantization is independent
of complex structures
so that the theory becomes topological.
He then constructs morphisms in terms of
the Feynman path integral.
He provides the relevant background,
such as the Feynman-Kac formula and Feynman
diagrams,
thus making the book more accessible to mathematicians.
His construction leads to many beautiful
applications,
such as the skein relation, the surgery formula
and a proof of Verlinde's formula.
Sen Hu has included several appendices providing
details left out of
Witten's lectures, and has added two more
chapters to update some
developments. In particular, he has included
the combinatorial
approach to computing the partition function
and the method of
localization invented by E Witten.
Nov. 1999 200 pp.
981-02-3908-4/3909-2(pbk)
World Sci.
Albert Einstein was one of the principal
founders
of the quantum and relativity theories.
Until 1925, when the BoseEinstein statistics
was discovered,
he made great contributions to the foundations
of quantum theory.
However, after the discovery of quantum mechanics
by Heisenberg and
wave mechanics by Schrdinger, with the consequent
development of the principles of uncertainty
and complementarity,
it would seem that Einstein's views completely
changed.
In his theory of the Brownian motion, Einstein
had invoked the theory
of probability to establish the reality of
atoms and molecules;
but, in 1916-17, when he wished to predict
the exact instant
when an atom would radiate - and developed
his theory of the A and B
coefficients - he wondered whether the "quantum
absorption and
emission of light could ever be understood
in the sense of
the complete causality requirement, or would
a statistical residue remain?
I must admit that there I lack the courage
of my convictions.
But I would be very unhappy to
renounce complete causality", as he
wrote to his friend Max Born.
However, he wrote later to Born that quantum
mechanics "is certainly imposing",
but "an inner voice tells me that it
is not the real thing ...
It does not bring us closer to the secret
of the 'Old One'.
I, at any rate, am convinced that He is not
playing at dice".
At the 1927 and 1930 Solvay Conferences on
Physics in Brussels,
Einstein engaged in profound discussions
with Niels Bohr and others
about his conviction regarding classical
determinism versus
the statistical causality of quantum mechanics.
Aug. 1999 150pp.
981-02-3913-0
World Sci.
What has been said about
The Quantum Theory of Fields,Vol. 1 and 2;
"The insight and depth of treatment
which singles this book out from others
in this field can be largely attributed to
Weinberg's authority as an originator
of many of the ideas in the book ...
Experienced researchers and beginn-ing graduate
students alike will delight
in the gems of wisdom to be found in these
pages.
This book combines exposition of technical
detail with physical insight
in a unique manner that confirms the promise
of Volume I
and I have no doubt that these two volumes
will rapidly constitute
the classic treatment of this important subject'.
Michael B. Green, CERN Courier
"Weinberg" Modern Applications
goes to the boundaries of
our present understanding of field theory.
It is unmatched by any other book on quantum
field theory for its depth,
generality and definitive character, and
it will be an essential reference
for serious students and researchers in elementary
particle physics.
O. W. Greenberg, Physics Today
In this third volume , Nobel Laureate Steven
Weinberg
continues his masterly exposition of quantum
field theory.
This volume presents a self-conteined, uo-to-date
and comprehensive
introduction to supersymmetry,
an area of theoretical physics likely to
be at
the physics of elementary particles and gravitation.
Much material in this book is not found in
other books
on supersymmetry.
Sep. 1999 500 pp.
0-521-66000-9
Vol. 1: Foundations; 55001-7 / Vol. 2: Modern
Applications; 5502-5
Cambridge