Expected publication date is November 21,
2003
Description
This book is written by a well-known expert
in classical
algebraic geometry. Tyurin's research was
specifically in
explicit computations to vector bundles on
algebraic varieties.
This is the only available monograph written
from his unique
viewpoint.
Ordinary (abelian) theta functions describe
properties of moduli
spaces of one-dimensional vector bundles
on algebraic curves. Non-abelian
theta functions, which are the main topic
of this book, play a
similar role in the study of higher-dimensional
vector bundles.
The book presents various aspects of the
theory of non-abelian
theta functions and the moduli spaces of
vector bundles,
including their applications to problems
of quantization and to
classical and quantum conformal field theories.
The book is an important source of information
for specialists in
algebraic geometry and its applications to
mathematical aspects
of quantum field theory.
Contents
Quantization procedure
Algebraic curves = Riemann surfaces
Non-abelian theta functions
Symplectic geometry of moduli spaces of vector
bundles
Two versions of CQFT
Three-valent graphs
Analytical aspects of the theory of non-abelian
theta functions
BPU-map
The main weapon
Bibliography
Details:
Series: CRM Monograph Series, Volume: 21
Publication Year: 2003
ISBN: 0-8218-3240-9
Paging: 136 pp.
Binding: Hardcover
Expected publication date is October 24,
2003
Description
In the last twenty years, the theory of holomorphic
dynamical
systems has had a resurgence of activity,
particularly concerning
the fine analysis of Julia sets associated
with polynomials and
rational maps in one complex variable. At
the same time, closely
related theories have had a similar rapid
development, for
example the qualitative theory of differential
equations in the
complex domain.
The meeting, "Etat de la recherche",
held at Ecole
Normale Superieure de Lyon, presented the
current state of the
art in this area, emphasizing the unity linking
the various sub-domains.
This volume contains four survey articles
corresponding to the
talks presented at this meeting.
D. Cerveau describes the structure of polynomial
differential
equations in the complex plane, focusing
on the local analysis in
neighborhoods of singular points. E. Ghys
surveys the theory of
laminations by Riemann surfaces which occur
in many dynamical or
geometrical situations. N. Sibony describes
the present state of
the generalization of the Fatou-Julia theory
for polynomial or
rational maps in two or more complex dimensions.
Lastly, the talk
by J.-C. Yoccoz, written by M. Flexor, considers
polynomials of
degree 2 in one complex variable, and in
particular, with the
hyperbolic properties of these polynomials
centered around the
Jakobson theorem.
This is a general introduction that gives
a basic history of
holomorphic dynamical systems, demonstrating
the numerous and
fruitful interactions among the topics. In
the spirit of the
"Etat de la recherche de la SMF"
meetings, the articles
are written for a broad mathematical audience,
especially
students or mathematicians working in different
fields. This book
is translated from the French edition by
Leslie Kay.
Contents
Holomorphic dynamical systems
Codimension-one holomorphic folations, reduction
of singularities
in low dimensions, and applications
Riemann surface laminations
Dynamics of rational maps on mathbb{P}^k
Dynamics of quadratic polynomials
Details:
Series: SMF/AMS Texts and Monographs, Volume:
10
Publication Year: 2003
ISBN: 0-8218-3228-X
Paging: 197 pp.
Binding: Softcover
Expected publication date is October 23,
2003
Description
This book is an introduction to Cartan's
approach to differential
geometry. Two central methods in Cartan's
geometry are the theory
of exterior differential systems and the
method of moving frames.
This book presents thorough and modern treatments
of both
subjects, including their applications to
both classic and
contemporary problems.
It begins with the classical geometry of
surfaces and basic
Riemannian geometry in the language of moving
frames, along with
an elementary introduction to exterior differential
systems. Key
concepts are developed incrementally with
motivating examples
leading to definitions, theorems, and proofs.
Once the basics of the methods are established,
the authors
develop applications and advanced topics.
One notable application
is to complex algebraic geometry, where they
expand and update
important results from projective differential
geometry.
The book features an introduction to G-structures
and a treatment
of the theory of connections. The Cartan
machinery is also
applied to obtain explicit solutions of PDEs
via Darboux's
method, the method of characteristics, and
Cartan's method of
equivalence.
This text is suitable for a one-year graduate
course in
differential geometry, and parts of it can
be used for a one-semester
course. It has numerous exercises and examples
throughout. It
will also be useful to experts in areas such
as PDEs and
algebraic geometry who want to learn how
moving frames and
exterior differential systems apply to their
fields.
Contents
Moving frames and exterior differential systems
Euclidean geometry and Riemannian geometry
Projective geometry
Cartan-Kahler I: Linear algebra and constant-coefficient
homogeneous systems
Cartan-Kahler II: The Cartan algorithm for
linear Pfaffian
systems
Applications to PDE
Cartan-Kahler III: The general case
Geometric structures and connections
Linear algebra and representation theory
Differential forms
Complex structures and complex manifolds
Initial value problems
Hints and answers to selected exercises
Bibliography
Index
Details:
Series: Graduate Studies in Mathematics,
Volume: 61
Publication Year: 2003
ISBN: 0-8218-3375-8
Paging: 378 pp.
Binding: Hardcover
Expected publication date is December 6,
2003
Description
This volume presents the proceedings of a
workshop on Inverse
Problems and Applications and a special session
on Inverse
Boundary Problems and Applications.
Inverse problems arise in practical situations,
such as medical
imaging, exploration geophysics, and non-destructive
evaluation
where measurements made in the exterior of
a body are used to
deduce properties of the hidden interior.
A large class of
inverse problems arise from a physical situation
modeled by
partial differential equations. The inverse
problem is to
determine some coefficients of the equation
given some
information about solutions. Analysis of
such problems is a
fertile area for interaction between pure
and applied mathematics.
This interplay is well represented in this
volume where several
theoretical and applied aspects of inverse
problems are
considered.
The book includes articles on a broad range
of inverse problems
including the inverse conductivity problem,
inverse problems for
Maxwell's equations, time reversal mirrors,
ultrasound using
elastic pressure waves, inverse problems
arising in the
environment, inverse scattering for the three-body
problem, and
optical tomography. Also included are several
articles on unique
continuation and on the study of propagation
of singularities for
hyperbolic equations in anisotropic media.
This volume is suitable for graduate students
and research
mathematicians interested in inverse problems
and applications.
Contents
G. Alessandrini, A. Morassi, and E. Rosset
-- Size estimates
V. Bacchelli, C. D. Pagani, and F. Saleri
-- Uniqueness in the
inverse conductivity problem for thin imperfections
weakly or
strongly conducting
E. Beretta and E. Francini -- Asymptotic
formulas for
perturbations in the electromagnetic fields
due to the presence
of thin inhomogeneities
L. Borcea, G. Papanicolaou, and C. Tsogka
-- A resolution study
for imaging and time reversal in random media
L. Escauriaza and S. Vessella -- Optimal
three cylinder
inequalities for solutions to parabolic equations
with Lipschitz
leading coefficients
M. Giudici -- Some problems for the application
of inverse
techniques to environmental modeling
V. Isakov, G. Nakamura, and J.-N. Wang --
Uniqueness and
stability in the Cauchy problem for the elasticity
system with
residual stress
L. Ji and J. McLaughlin -- Using a Hankel
function expansion to
identify stiffness for the boundary impulse
input experiment
C. E. Kenig, G. Ponce, and L. Vega -- On
the uniqueness of
solutions of higher order nonlinear dispersive
equations
Y. V. Kurylev, M. Lassas, and E. Somersalo
-- Reconstruction of a
manifold from electromagnetic boundary measurements
A. Lorenzi and E. Paparoni -- Direct and
inverse problems for
second-order integro-differential operator
equations in an
unbounded time interval
C. J. Nolan and G. Uhlmann -- Geometrical
optics for generic
anisotropic materials
M. Piana and M. Bertero -- Linear approaches
in microwave
tomography
A. Tamasan -- Optical tomography in weakly
anisotropic scattering
media
G. Uhlmann and A. Vasy -- Inverse problems
in three-body
scattering
Details:
Series: Contemporary Mathematics,Volume:
333
Publication Year: 2003
ISBN: 0-8218-3367-7
Paging: approximately 224 pp.
Binding: Softcover
Expected publication date is November 21,
2003
Description
It is remarkable that so much about Lie groups
could be packed
into this small book. But after reading it,
students will be well-prepared
to continue with more advanced, graduate-level
topics in
differential geometry or the theory of Lie
groups.
The theory of Lie groups involves many areas
of mathematics:
algebra, differential geometry, algebraic
geometry, analysis, and
differential equations. In this book, Arvanitoyeorgos
outlines
enough of the prerequisites to get the reader
started. He then
chooses a path through this rich and diverse
theory that aims for
an understanding of the geometry of Lie groups
and homogeneous
spaces. In this way, he avoids the extra
detail needed for a
thorough discussion of representation theory.
Lie groups and homogeneous spaces are especially
useful to study
in geometry, as they provide excellent examples
where quantities
(such as curvature) are easier to compute.
A good understanding
of them provides lasting intuition, especially
in differential
geometry.
The author provides several examples and
computations. Topics
discussed include the classification of compact
and connected Lie
groups, Lie algebras, geometrical aspects
of compact Lie groups
and reductive homogeneous spaces, and important
classes of
homogeneous spaces, such as symmetric spaces
and flag manifolds.
Applications to more advanced topics are
also included, such as
homogeneous Einstein metrics, Hamiltonian
systems, and
homogeneous geodesics in homogeneous spaces.
The book is suitable for advanced undergraduates,
graduate
students, and research mathematicians interested
in differential
geometry and neighboring fields, such as
topology, harmonic
analysis, and mathematical physics.
Contents
Lie groups
Maximal tori and the classification theorem
The geometry of a compact Lie group
Homogeneous spaces
The geometry of a reductive homogeneous space
Symmetric spaces
Generalized flag manifolds
Advanced topics
Bibliography
Index
Details:
Series: Student Mathematical Library, Volume:
22
Publication Year: 2003
ISBN: 0-8218-2778-2
Paging: 141 pp.
Binding: Softcover