Expected publication date is October 3, 2003
Description
The miracle of integral geometry is that
it is often possible to
recover a function on a manifold just from
the knowledge of its
integrals over certain submanifolds. The
founding example is the
Radon transform, introduced at the beginning
of the 20th century.
Since then, many other transforms were found,
and the general
theory was developed. Moreover, many important
practical
applications were discovered. The best known,
but by no means the
only one, being to medical tomography.
This book is a general introduction to integral
geometry, the
first from this point of view for almost
four decades. The
authors, all leading experts in the field,
represent one of the
most influential schools in integral geometry.
The book presents
in detail basic examples of integral geometry
problems, such as
the Radon transform on the plane and in space,
the John
transform, the Minkowski-Funk transform,
integral geometry on the
hyperbolic plane and in the hyperbolic space,
the horospherical
transform and its relation to representations
of $SL(2,\mathbb C)$,
integral geometry on quadrics, etc. The study
of these examples
allows the authors to explain important general
topics of
integral geometry, such as the Cavalieri
conditions, local and
nonlocal inversion formulas, and overdetermined
problems in
integral geometry. Many of the results in
the book were obtained
by the authors in the course of their career-long
work in
integral geometry.
This book is suitable for graduate students
and researchers
working in integral geometry and its applications.
Contents
Radon transform
John transform
Integral geometry and harmonic analysis on
the hyperbolic plane
and in the hyperbolic space
Integral geometry and harmonic analysis on
the group ${G=SL(2,\mathbb
C)}$
Integral geometry on quadrics
Bibliography
Index
Details:
Series: Translations of Mathematical Monographs,
Volume: 220
Publication Year: 2003
ISBN: 0-8218-2932-7
Paging: 170 pp.
Binding: Hardcover
Expected publication date is October 8, 2003
Description
The nonlinear Schrodinger equation has received
a great deal of
attention from mathematicians, particularly
because of its
applications to nonlinear optics. It is also
a good model
dispersive equation, since it is often technically
simpler than
other dispersive equations, such as the wave
or the Korteweg-de
Vries equation. From the mathematical point
of view,
Schrodinger's equation is a delicate problem,
possessing a
mixture of the properties of parabolic and
elliptic equations.
Useful tools in studying the nonlinear Schrodinger
equation are
energy and Strichartz's estimates.
This book presents various mathematical aspects
of the nonlinear
Schrodinger equation. It studies both problems
of local nature (local
existence of solutions, uniqueness, regularity,
smoothing effect)
and problems of global nature (finite-time
blowup, global
existence, asymptotic behavior of solutions).
In principle, the
methods presented apply to a large class
of dispersive semilinear
equations. The first chapter recalls basic
notions of functional
analysis (Fourier transform, Sobolev spaces,
etc.). Otherwise,
the book is mostly self-contained.
It is suitable for graduate students and
research mathematicians
interested in nonlinear partial differential
equations and
applications to mathematical physics.
Contents
Preliminaries
The linear Schrodinger equation
The Cauchy problem in a general domain
The local Cauchy problem
Regularity and the smoothing effect
Global existence and finite-time blowup
Asymptotic behavior in the repulsive case
Stability of bound states in the attractive
case
Further results
Bibliography
Details:
Series: Courant Lecture Notes, Volume: 10
Publication Year: 2003
ISBN: 0-8218-3399-5
Paging: 323 pp.
Binding: Softcover
Expected publication date is October 25,
2003
Description
Over the last four decades, Phillip Griffiths
has been a central
figure in mathematics. During this time,
he made crucial
contributions in several fields, including
complex analysis,
algebraic geometry, and differential systems.
His books and
papers are distinguished by a remarkably
lucid style that invites
the reader to understand not only the subject
at hand, but also
the connections among seemingly unrelated
areas of mathematics.
Even today, many of Griffiths' papers are
used as a standard
source on a subject. Another important feature
of Griffiths'
writings is that they often bring together
classical and modern
mathematics.
The four parts of Selected Works--Analytic
Geometry, Algebraic
Geometry, Variations of Hodge Structures,
and Differential
Systems--are organized according to the subject
matter and are
supplemented by Griffiths' brief, but extremely
illuminating,
personal reflections on the mathematical
content and the times in
which they were produced.
Griffiths' Selected Works provide the reader
with a panoramic
view of important and exciting mathematics
during the second half
of the 20th century.
Contents
Part I. Analytic Geometry
P. Griffiths -- Commentary on Vector bundles
Vector Bundles
Ph. A. Griffiths -- The extension problem
for compact
submanifolds of complex manifolds I
P. A. Griffiths -- The extension problem
in complex analysis II;
embeddings with positive normal bundle
P. A. Griffiths -- Hermitian differential
geometry, Chern
classes, and positive vector bundles
P. A. Griffiths -- Period spaces and Hodge
theory
P. A. Griffiths and W. Schmid -- Locally
homogeneous complex
manifolds
P. Griffiths and W. Schmid -- Recent developments
in Hodge theory:
A discussion of techniques and results
P. R. Deligne, P. Griffiths, J. Morgan, and
D. Sullivan -- Real
homotopy theory of Kahler manifolds
Analytic geometry
P. A. Griffiths -- Holomorphic mapping into
canonical algebraic
varieties
J. Carlson and P. Griffiths -- A defect relation
for
equidimensional holomorphic mappings between
algebraic varieties
P. Griffiths -- Complex differentiable and
integral geometry and
curvature integrals associated to singularities
of complex
analytic varieties
M. Cornalba and P. Griffiths -- Analytic
cycles and vector
bundles on non-compact algebraic varieties
M. Green and P. Griffiths -- Two applications
of algebraic
geometry to entire holomorphic mappings
Acknowledgments
Selected titles
Part 2. Algebric Geometry
P. A. Griffiths -- Introductory comments
to Part 2
Cycles and deformation theory
P. A. Griffiths -- Some results on algebraic
cycles on algebraic
manifolds
P. A. Griffiths -- Complex-analytic properties
of certain Zariski
open sets on algebraic varieties
C. H. Clemens and P. A. Griffiths -- The
intermediate Jacobian of
the cubic threefold
Abel's theorem
P. A. Griffiths -- Variations on a theorem
of Abel
P. Griffiths and J. Harris -- A Poncelet
theorem in space
P. Griffiths and J. Harris -- Residues and
zero-cycles on
algebraic varieties
S. S. Chern and P. Griffiths -- Abel's theorem
and webs
Algebraic and differential geometry
P. A. Griffiths -- Complex analysis and algebraic
geometry
P. Griffiths and J. Harris -- Algebraic geometry
and local
differential geometry
Loci of divisors
P. Griffiths and J. Harris -- The variety
of special linear
systems on a general algebraic curve
P. Griffiths and J. Harris -- On the Noether-Lefschetz
theorem
and some remarks on codimension-two cycles
E. Arbarello, M. Cornalba, P. Griffiths,
and J. Harris -- Special
divisors on algebraic curves
Acknowledgments
Selected Titles
Part 3. Variations of Hodge Structures
P. A. Griffiths -- Introductory comments
to part 3
Periods of integrals
P. A. Griffiths -- Periods of integrals on
algebraic naifolds, I
(Construction and properties of the modular
varieties)
P. A. Griffiths -- Periods of integrals on
algebraic manifolds,
II (Local study of the period mapping)
Ph. A. Griffiths -- Periods of integrals
on algebraic manifolds
III (some global differential-geometric properties
of the period
mapping)
P. A. Griffiths -- On the periods of certain
rational integrals I
P. A. Griffiths -- On the periods of certain
rational integrals:
II
P. A. Griffiths -- Periods of integrals on
algebraic manifolds:
Summary of main results and discussion of
open problems
Variations of Hodge structures
J. Carlson, M. Green, P. Griffiths, and J.
Harris --
Infinitesimal variations of Hodge structure
(I)
P. Griffiths and J. Harris -- Infinitesimal
variations of Hodge
structure (II): An infinitesimal invariant
of Hodge classes
P. A. Griffiths -- Infinitesimal variations
of Hodge structure (III):
Determinantal varieties and the infinitesimal
invariant of normal
functions
Acknowledgments
Selected Titles
Part 4. Differential Systems
P. A. Griffiths -- Introductory comments
to part 4
Moving frames and differential geometry
P. Griffiths -- On Cartan's method of Lie
groups and moving
frames as applied to uniqueness and existence
questions in
differential geometry
Differential systems and Hodge structure
P. A. Griffiths -- Poincare and algebraic
geometry
R. L. Bryant and P. A. Griffiths -- Some
observations on the
infinitesimal period relations for regular
threefolds with
trivial canonical bundle
R. L. Bryant and P. Griffiths -- Reduction
for constrained
variational problems and $\int\frac{1}{2}
\kappa^2 ds$
Integrability
P. A. Griffiths -- Linearizing flows and
a cohomological
interpretation of Lax equations
The characteristic variety and its geometry
P. A. Griffiths -- Some aspects of exterior
differential systems
R. L. Bryant and P. A. Griffiths -- Characteristic
cohomology of
differential systems (I): General theory
R. L. Bryant and P. A. Griffiths -- Characteristic
cohomology of
differential systems II: conservation laws
for a class of
parabolic equations
R. Bryant, P. Griffiths, and L. Hsu -- Hyperbolic
exterior
differential systems and their conservation
laws, Part I
R. Bryant, P. Griffiths, and L. Hsu -- Hyperbolic
exterior
differential systems and their conservation
laws, Part II
Acknowledgments
Selected Titles
Details:
Series: Collected Works, Volume: 18
Publication Year: 2003
ISBN: 0-8218-1066-9
Paging: 2598 pp.
Binding: Hardcover
Expected publication date is October 5, 2003
Description
This book contains contributions by an impressive
list of leading
mathematicians. The articles include high-level
survey and
research papers exploring contemporary issues
in geometric
analysis, differential geometry, and several
complex variables.
Many of the articles will provide graduate
students with a good
entry point into important areas of modern
research.
The material is intended for researchers
and graduate students
interested in several complex variables and
complex geometry.
Contents
B. Berndtsson -- Bergman kernels related
to Hermitian line
bundles over compact complex manifolds
J. P. D'Angelo -- A gentle introduction to
points of finite type
on real hypersurfaces
P. Eberlein -- The moduli space of 2-step
nilpotent Lie algebras
of type $(p,q)$
F. Forstneric -- The homotopy principle in
complex analysis: A
survey
K. Grove -- Finiteness theorems in riemannian
geometry
Y. Itokawa, Y. Machigashira, and K. Shiohama
-- Generalized
Toponogov's theorem for manifolds with radial
curvature bounded
below
H. Jacobowitz -- The global isometric embedding
problem
K.-T. Kim and S. G. Krantz -- The Bergman
metric invariants and
their boundary behavior
S. Kobayashi -- Natural connections in almost
complex manifolds
S. Kumar and J. Millson -- The generalized
triangle inequalities
for rank 3 symmetric spaces of noncompact
type
J. D. McNeal -- Subelliptic estimates and
scaling in the
$\bar\partial$-Neumann problem
N. Mok -- Negativity of curvature on spaces
parametrizing Hodge
decompositions of reduced first cohomology
groups
T. Ohsawa -- On the extension of $L^2$ holomorphic
functions VI--A
limiting case
P. Petersen -- Variations on a theme of Synge
L. P. Rothschild -- Mappings between real
submanifolds in complex
space
B. Shiffman, T. Tate, and S. Zelditch --
Harmonic analysis on
toric varieties
B. Wong -- On complex manifolds with noncompact
automorphism
groups
H.-H. Wu and F. Zheng -- Kahler manifolds
with slightly positive
bisectional curvature
Details:
Series: Contemporary Mathematics, Volume:
332
Publication Year: 2003
ISBN: 0-8218-3273-5
Paging: approximately 320 pp.
Binding: Softcover