Series: Progress in Mathematics, Vol. 227
316 pages , 6 1/8 x 9 , hardcover
ISBN: 3-7643-3408-8, published 2004
expected release date: 12/18/2003
ABOUT THIS BOOK
The deep and relatively new field of vertex
operator algebras is
intimately related to a variety of areas
in mathematics and
physics: for example, the concepts of "monstrous
moonshine,"
infinite-dimensional Lie theory, string theory,
and conformal
field theory. This book introduces the reader
to the fundamental
theory of vertex operator algebras and its
basic techniques and
examples. Beginning with a detailed presentation
of the
theoretical foundations and proceeding to
a range of
applications, the text includes a number
of new, original results
and also highlights and brings fresh perspective
to important
works of many researchers.
After introducing the elementary "formal
calculus''
underlying the subject, the book provides
an axiomatic
development of vertex operator algebras and
their modules,
expanding on the early contributions of R.
Borcherds, I. Frenkel,
J. Lepowsky, A. Meurman, Y.-Z. Huang, C.
Dong, Y. Zhu and others.
The concept of a "representation'' of
a vertex (operator)
algebra is treated in detail, following and
extending the work of
H. Li; this approach is used to construct
important families of
vertex (operator) algebras and their modules.
Requiring only a familiarity with basic algebra,
"Introduction
to Vertex Operator Algebras and Their Representations"
will
be useful for graduate students and researchers
in mathematics
and physics. The book's self-contained presentation
of the core
topics will equip readers to embark on many
active research
directions related to vertex operator algebras,
group theory,
representation theory, and string theory.
TABLE OF CONTENTS
* Preface
* Introduction
* Formal Calculus
* Vertex Operator Algebras: The Axiomatic
Basics
* Modules
* Vertex Algebras
* Families of Vertex Operator Algebras
* References
* Index
370 pages , 6 1/8 x 9 , 10 illus., hardcover
ISBN: 3-7643-3521-1, published 2004
ABOUT THIS BOOK
This monograph is devoted to the study of
Koethe-Bochner function
spaces, an area of research at the intersection
of Banach space
theory, harmonic analysis, probability, and
operator theory. A
number of significant results---many scattered
throughout the
literature---are distilled and presented
here, giving readers a
comprehensive view of Koethe-Bochner function
spaces from the
subject's origins in functional analysis
to its connections to
other disciplines. Key features and topics:*
Considerable
background material provided, including a
compilation of
important theorems and concepts in classical
functional analysis,
as well as a discussion of the Dunford-Pettis
Property, tensor
products of Banach spaces, relevant geometry,
and the basic
theory of conditional expectations and martingales
* Rigorous
treatment of Koethe-Bochner spaces, encompassing
convexity,
measurability, stability properties, Dunford-Pettis
operators,
and Talagrand spaces, with a particular emphasis
on open problems
* Detailed examination of Talagrand's Theorem,
Bourgain's
Theorem, and the Diaz-Kalton Theorem, the
latter extended to
arbitrary measure spaces.
Series: Progress in Mathematics, Vol. 226
288 pages , 6 1/8 x 9 , 10 illus., hardcover
ISBN: 3-7643-3259-X, published 2004
ABOUT THIS BOOK
One of the great successes of twentieth century
mathematics has
been the remarkable qualitative understanding
of rational and
integral points on curves, gleaned in part
through the theorems
of Mordell, Weil, Siegel, and Faltings. It
has become clear that
the study of rational and integral points
has deep connections to
other branches of mathematics: complex algebraic
geometry, Galois
and ,tale cohomology, transcendence theory
and diophantine
approximation, harmonic analysis, automorphic
forms, and analytic
number theory. This text, which focuses on
higher dimensional
varieties, provides precisely such an interdisciplinary
view of
the subject. It is a digest of research and
survey papers by
leading specialists; the book documents current
knowledge in
higher-dimesional arithmetic and gives indications
for future
research. It will be valuable to not only
to practitioners in the
field, but to a wide audience of mathematicians
and graduate
students with an interest in arithmetic geometry.
Contributors
include: P. Swinnerton-Dyer * B. Hassett
* Yu. Tschinkel * J.
Shalika * R. Takloo-Bighash * J.-L. Colliot-Th,lSne
* A. de Jong
* Ph. Gille * D. Harari * J. Harris * B.
Mazur * W. Raskind * J.
Starr * T. Wooley
TABLE OF CONTENTS
Preface * P. Swinnerton-Dyer: Diophantine
Equations: Progress and
Problems * P. Swinnerton-Dyer: Weak Approximation
on Del Pezzo
Surfaces of Degree 4 * B. Hassett and Yu.
Tschinkel: Weak
Approximation Over Function Fields * B. Hassett
and Yu. Tschinkel:
Universal Torsors and Cox Rings * J. Shalika,
R. Takloo-Bighash,
Yu. Tschinkel: Rational Points on Compactifications
of Semi-Simple
Groups * J.-L. Colliot-Th,lSne* A. de Jong
* Ph. Gille * D.
Harari * J. Harris * B. Mazur * W. Raskind
* J. Starr * T. Wooley
Series: Progress in Mathematics, Vol. 228
328 pages , 6 1/8 x 9 , hardcover
ISBN: 3-7643-3373-1, published 2004
ABOUT THIS BOOK
Semisimple Lie groups, and their algebraic
analogues over fields
other than the reals, are of fundamental
importance in geometry,
number theory, and mathematical physics.
Three independent, self-contained
volumes, under the general title 'Lie Theory,'
feature survey
work and original results by well-established
researchers in key
areas of semisimple Lie theory. A wide spectrum
of topics is
treated, with emphasis on the interplay between
representation
theory and the geometry of adjoint orbits
for Lie algebras over
fields of possibly finite characteristic,
as well as for infinite-dimensional
Lie algebras. Also covered is unitary representation
theory and
branching laws for reductive subgroups, an
active part of modern
representation theory. Finally, there is
a thorough discussion of
compactifications of symmetric spaces, number
theory via
Selberg's trace formula, and harmonic analysis
through a far-reaching
generalization of Harish-Chandra's Plancherel
formula for
semisimple Lie groups. Ideal for graduate
students and
researchers, 'Lie Theory' provides a broad,
clearly focused
examination of semisimple Lie groups and
their integral
importance to research in many branches of
mathematics. 'Lie
Theory: Lie Algebras and Representations'
contains J. C.
Jantzen's Nilpotent Orbits in Representation
Theory,' and K.-H.
Neeb's 'Infinite Dimensional Groups and their
Representations.'
Both are comprehensive treatments of the
relevant geometry of
orbits in Lie algebras, or their duals, and
the correspondence to
representations. 'Lie Theory: Unitary Representations,
Number
Theory, and Compactifications' contains work
by A. Borel and L.
Ji, T. Kobayashi and J.-P. Labesse. 'Lie
Theory: Harmonic
Analysis on Symmetric Spaces' features work
by E. van den Ban, P.
Delorme, and H. Schlichtkrull.
TABLE OF CONTENTS
Preface * J.C. Jantzen, 'Nilpotent Orbits
in Representation
Theory': * Introduction * Nilpotent Orbits
for Classical Groups *
Some General Results * Centralisers in the
Classical Cases * Bala-Carter
Theory * Centralisers * The Nilpotent Cone
I * The Nilpotent Cone
II * Functions on Orbits and Orbit Closures
* Associated
Varieties * Springer's Fibres and Steinberg's
Triples * Paving
Springer's Fibres * l-adic and Perverse Stuff
* Springer's
Representations * References * K.-H. Neeb,
'Infinite Dimensional
Groups and their Representations': * Introduction
* The Finite-Dimensional
Case * Split Lie Algebras * Unitary Highest
Weight Modules *
Banach-Lie Groups * Holomorphic Representations
of Classical
Banach-Lie Groups * Geometry of Coadjoint
Orbits of Banach-Lie
Groups * Coadjoint Orbits and Complex Line
Bundles for U2(H) *
Appendix: The Topology of Classical Banach-Lie
Groups *
References