Expected publication date is October 9, 2002
Description
Many problems in pure and applied mathematics
boil down to
determining the shape of a surface in space
or constructing
surfaces with prescribed geometric properties.
These problems
range from classical problems in geometry,
elasticity, and
capillarity to problems in computer vision,
medical imaging, and
graphics. There has been a sustained effort
to understand these
questions, but many problems remain open
or only partially solved.
These include determining the shape of a
surface from its metric
and mean curvature (Bonnet's problem), determining
an immersion
from the projectivised Gauss map (Christoffel's
problem) and its
applications to the computer vision problem
on recovering shape
from shading, the construction of surfaces
with prescribed
curvature properties, constructing extremal
surfaces and
interfaces, and representing surface deformations.
This book
studies these questions by presenting a theory
applying to both
global and local questions and emphasizing
conformal immersions
rather than isometric immersions.
The book offers:
A unified and comprehensive presentation
of the quaternionic and
spinor approach to the theory of surface
immersions in three and
four dimensional space.
New geometric invariants of surfaces in space
and new open
problems.
A new perspective and new results on the
classical geometric
problems of surface and surface shape recognition
and surface
representation.
A source of problems to motivate research
and dissertations.
Applications in computer vision and computer
graphics.
Proofs of many results presented by the authors
at colloquia,
conferences, and congresses over the past
two years.
This book describes how to use quaternions
and spinors to study
conformal immersions of Riemann surfaces
into Bbb R^3. The first
part develops the necessary quaternionic
calculus on surfaces,
its application to surface theory and the
study of conformal
immersions and spinor transforms. The integrability
conditions
for spinor transforms lead naturally to Dirac
spinors and their
application to conformal immersions. The
second part presents a
complete spinor calculus on a Riemann surface,
the definition of
a conformal Dirac operator, and a generalized
Weierstrass
representation valid for all surfaces. This
theory is used to
investigate first, to what extent a surface
is determined by its
tangent plane distribution, and second, to
what extent curvature
determines the shape.
The book is geared toward graduate students
and research
mathematicians interested in differential
geometry and geometric
analysis and its applications, computer science,
computer vision,
and computer graphics.
Contents
Conformal immersions via quaternions
Quaternionic calculus and immersions
Applications
Surfaces and Dirac spinors
Spinor algebra
Dirac spinors and conformal immersions
Bibliography
Glossary of symbols
Index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Contemporary Mathematics, Volume:
299
Publication Year: 2002
ISBN: 0-8218-1928-3
Paging: approximately 152 pp.
Binding: Softcover
Description
This book is the first of two proceedings
volumes stemming from
the International Conference and Workshop
on Valuation Theory
held at the University of Saskatchewan (Saskatoon,
SK, Canada).
Valuation theory arose in the early part
of the twentieth century
in connection with number theory and has
many important
applications to geometry and analysis: the
classical application
to the study of algebraic curves and to Dedekind
and Prufer
domains; the close connection to the famous
resolution of the
singularities problem; the study of the absolute
Galois group of
a field; the connection between ordering,
valuations, and
quadratic forms over a formally real field;
the application to
real algebraic geometry; the study of noncommutative
rings; etc.
The special feature of this book is its focus
on current
applications of valuation theory to this
broad range of topics.
Also included is a paper on the history of
valuation theory.
The book is suitable for graduate students
and research
mathematicians working in algebra, algebraic
geometry, number
theory, and mathematical logic.
Contents
S. S. Abhyankar -- Two step descent in modular
Galois theory,
theorems of Burnside and Cayley, and Hilbert's
Thirteenth Problem
M.-E. Alonso and H. Lombardi -- Generalized
Taylor formulae,
computations in real closed valued fields
and quantifier
elimination
S. A. Basarab -- The arithmetic-arboreal
residue structure of a
Prufer domain, I
H.-H. Brungs and G. Torner -- Left valuation
rings, left cones,
and a question of Frege's
V. Cossart, O. Piltant, and A. J. Reguera-Lopez
-- Divisorial
valuations dominating rational surface singularities
T. C. Craven -- Valuations and Hermitian
forms on skew fields
S. D. Cutkosky -- Resolution of morphisms
T. S. Gardener and H. Schoutens -- Rigid
subanalytic sets
J. Grater -- Dubrovin valuation rings and
orders in central
simple algebras
H. G. Grundman and T. L. Smith -- Q-adequate
bicyclic bicubic
fields
D. Harbater, M. van der Put, and R. Guralnick
-- Valued fields
and covers in characteristic p, with an appendix
by R. Guralnick
U. T. Hartl -- Line bundles on rigid analytic
spaces
P. Hitzler and A. K. Seda -- The fixed-point
theorems of Priess-Crampe
and Ribenboim in logic programming
S. K. Khanduja -- The minimum property of
Krasner's constant
H. Lombardi -- About Merckel's lemma
A. Prestel -- Bounds for representations
of polynomials positive
on compact semi-algebraic sets
A. Prestel and N. Schwartz -- Model theory
of real closed rings
P. Roquette -- History of valuation theory--Part
I
J. Schmid -- A density property for PpC-fields
M. van der Put -- Valuation theory in rigid
geometry and curves
over valuation rings
A. R. Wadsworth -- Valuation theory on finite
dimensional
division algebras
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Fields Institute Communications,
Volume: 32
Publication Year: 2002
ISBN: 0-8218-2813-4
Paging: 449 pp.
Binding: Hardcover
Description
This book is a collection of survey articles
on several topics
related to the general notion of integrability.
It stems from a
workshop on "Mathematical Methods of
Regular Dynamics"
dedicated to Sophie Kowalevski. Leading experts
introduce
corresponding areas in depth. The book provides
a broad overview
of research, from the pioneering work of
the nineteenth century
to the developments of the 1970s through
the present.
The book begins with two historical papers
by R. L. Cooke on
Kowalevski's life and work. Following are
15 research surveys on
integrability issues in differential and
algebraic geometry,
classical complex analysis, discrete mathematics,
spinning tops,
Painleve equations, global analysis on manifolds,
special
functions, etc. It concludes with Kowalevski's
famous paper
published in Acta Mathematica in 1889, "Sur
le probleme de
la rotation d'un corps solide autour d'un
point fixe".
The book is suitable for graduate students
in pure and applied
mathematics, the general mathematical audience
studying
integrability, and research mathematicians
interested in
differential and algebraic geometry, analysis,
and special
functions.
Contents
R. L. Cooke -- The life of S. V. Kovalevskaya
R. L. Cooke -- Kovalevskaya's mathematical
work
B. Enriquez -- The KZB connection: Parametrizations,
flat
sections and q-deformation
L. Gavrilov -- Jacobians of singularized
spectral curves and
completely integrable systems
L. Haine -- The q-hypergeometric equation,
Askey-Wilson type
solitons and rational curves with singularities
K. Hikami -- Quantum discrete soliton equations
E. I. Horozov -- Dual algebras of differential
operators
J.-S. Hu and M. Yan -- A link between two
fundamental
contributions of Kowalevski
A. A. Kapaev -- Monodromy deformation approach
to the scaling
limit of the Painleve first equation
V. B. Kuznetsov -- Kowalevski top revisited
D. Markushevich -- Some algebro-geometric
integrable systems
versus classical ones
M. Mazzocco -- Painleve sixth equation as
isomonodromic
deformations equation of an irregular system
A. Nakayashiki and F. A. Smirnov -- Euler
characteristics of
theta divisors of Jacobians for spectral
curves
E. Previato -- Reduction theory, elliptic
solitons and integrable
systems
T. Sasaki and M. Yoshida -- Schwarzian derivatives
and
uniformization
A. O. Smirnov -- Elliptic solitons and Heun's
equation
V. V. Sokolov -- Generalized Kowalevski top:
New integrable cases
on e(3) and so(4)
Reprint of the Original Paper
S. Kowalevski -- Sur le probleme de la rotation
d'un corps solide
autour d'un point fixe
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: CRM Proceedings & Lecture Notes,
Volume: 32
Publication Year: 2002
ISBN: 0-8218-2885-1
Paging: 372 pp.
Binding: Softcover
Expected publication date is August 25, 2002
Reviews of the Previous Volumes:
"This lively presentation of an amazingly wide spectrum of
happenings in mathematics is impressive ... [this book] should be
presented to a wide audience even outside mathematics, which
could be fascinated by the ideas, concepts and beauty of the
mathematical topics."
-- European Mathematical Society Newsletter
"The articles are very well written, and usually include
quotes from the mathematicians who were involved in the work in
question, giving the whole thing a more "human" feel.
This book offers professionals a way to keep abreast of what's
going on in the field and also gives us a way to share with our
students and colleagues some of the excitement of doing
mathematics. Don't miss it."
-- MAA Online
Description
Mathematicians like to point out that mathematics is universal.
In spite of this, most people continue to view it as either
mundane (balancing a checkbook) or mysterious (cryptography).
This fifth volume of the What's Happening series contradicts that
view by showing that mathematics is indeed found everywhere--in
science, art, history, and our everyday lives.
Here is some of what you'll find in this volume:
Mathematics and Science
Mathematical biology: Mathematics was key to cracking the genetic
code. Now, new mathematics is needed to understand the three-dimensional
structure of the proteins produced from that code.
Celestial mechanics and cosmology: New methods have revealed a
multitude of solutions to the three-body problem. And other new
work may answer one of cosmology's most fundamental questions:
What is the size and shape of the universe?
Mathematics and Everyday Life
Traffic jams: New models are helping researchers understand where
traffic jams come from-and maybe what to do about them!
Small worlds: Researchers have found a short distance from theory
to applications in the study of small world networks.
Elegance in Mathematics
Beyond Fermat's Last Theorem: Number theorists are reaching
higher ground after Wiles' astounding 1994 proof: new
developments in the elegant world of elliptic curves and modular
functions.
The Millennium Prize Problems: The Clay Mathematics Institute has
offered a million dollars for solutions to seven important and
difficult unsolved problems.
These are just some of the topics of current interest that are
covered in this latest volume of What's Happening in the
Mathematical Sciences. The book has broad appeal for a wide
spectrum of mathematicians and scientists, from high school
students through advanced-level graduates and researchers.
Contents
Introduction
New heights for number theory
Nothing to sphere but sphere itself
A mathematical twist to protein folding
Finite math
The mathematics of traffic jams
Rewriting history
It's a small, big, small, big world
A celestial Pas de Trois
Think and grow rich
Ising on the cake
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: What's Happening in the Mathematical Sciences, Volume: 5
Publication Year: 2002
ISBN: 0-8218-2904-1
Paging: 95 pp.
Binding: Softcover