Expected publication date is July 26, 2003
Description
"Lusternik-Schnirelmann category is
like a Picasso painting.
Looking at category from different perspectives
produces
completely different impressions of category's
beauty and
applicability."
--from the Introduction
Lusternik-Schnirelmann category is a subject
with ties to both
algebraic topology and dynamical systems.
The authors take LS-category
as the central theme, and then develop topics
in topology and
dynamics around it. Included are exercises
and many examples. The
book presents the material in a rich, expository
style.
The book provides a unified approach to LS-category,
including
foundational material on homotopy theoretic
aspects, the
Lusternik-Schnirelmann theorem on critical
points, and more
advanced topics such as Hopf invariants,
the construction of
functions with few critical points, connections
with symplectic
geometry, the complexity of algorithms, and
category of $3$-manifolds.
This is the first book to synthesize these
topics. It takes
readers from the very basics of the subject
to the state of the
art. Prerequisites are few: two semesters
of algebraic topology
and, perhaps, differential topology. It is
suitable for graduate
students and researchers interested in algebraic
topology and
dynamical systems.
Contents
Introduction to LS-category
Lower bounds for LS-category
Upper bounds for category
Localization and category
Rational homotopy and category
Hopf invariants
Category and critical points
Category and symplectic topology
Examples, computations and extensions
Topology and analysis
Basic homotopy
Bibliography
Index
Details:
Series: Mathematical Surveys and Monographs,
Volume: 103
Publication Year: 2003
ISBN: 0-8218-3404-5
Paging: 330 pp.
Binding: Hardcover
Expected publication date is August 17, 2003
Description
With graduate school, young mathematicians
take the first step
toward a career in mathematics. During this
period, they make
important decisions that will affect the
rest of their careers.
Here now is a detailed guide to help students
navigate those
years and the years that follow.
In his inimitable and forthright style, Steven
Krantz addresses
the major issues of graduate school, including
deciding where to
go, passing the qualifying exams, finding
an advisor, writing a
thesis, and getting that first job. As with
his earlier guide,
How to Teach Mathematics, Krantz avoids generalities,
giving
clear advice on how to handle real situations.
In addition, he
includes a description of the basic elements
of a mathematical
education, as well as three appendices on
the structure of a
typical department and university and the
standard academic ranks.
Steven G. Krantz is an experienced mathematician
and an award-winning
author. He has worked in many different types
of mathematics
departments, supervised both master's and
doctoral students, and
is currently the Chair of the Mathematics
Department at
Washington University in St. Louis.
Contents
Getting ready for graduate school
Heading off to graduate school
Preliminaries
Essential elements of a graduate education
Pre-thesis work
Thesis work
Sticky wickets
Practical difficulties
Moral difficulties
Post-graduate-school existence
Life after the thesis
Afterthoughts
The elements of mathematics
What mathematics I need to know
Glossary
The administrative structure of a mathematics
department and a
university
The academic ranks
The academic composition of a mathematics
department
A checklist for graduate school
Bibliography
Index
Details:
Publication Year: 2003
ISBN: 0-8218-3455-X
Paging: approximately 240 pp.
Binding: Softcover
Expected publication date is August 31, 2003
Description
Analysis as an independent subject was created
as part of the
scientific revolution in the seventeenth
century. Kepler,
Galileo, Descartes, Fermat, Huygens, Newton,
and Leibniz, to name
but a few, contributed to its genesis. Since
the end of the
seventeenth century, the historical progress
of mathematical
analysis has displayed unique vitality and
momentum. No other
mathematical field has so profoundly influenced
the development
of modern scientific thinking.
Describing this multidimensional historical
development requires
an in-depth discussion which includes a reconstruction
of general
trends and an examination of the specific
problems. This volume
is designed as a collective work of authors
who are proven
experts in the history of mathematics. It
clarifies the
conceptual change that analysis underwent
during its development
while elucidating the influence of specific
applications and
describing the relevance of biographical
and philosophical
backgrounds.
The first ten chapters of the book outline
chronological
development and the last three chapters survey
the history of
differential equations, the calculus of variations,
and
functional analysis.
Special features are a separate chapter on
the development of the
theory of complex functions in the nineteenth
century and two
chapters on the influence of physics on analysis.
One is about
the origins of analytical mechanics, and
one treats the
development of boundary-value problems of
mathematical physics (especially
potential theory) in the nineteenth century.
The book presents an accurate and very readable
account of the
history of analysis. Each chapter provides
a comprehensive
bibliography. Mathematical examples have
been carefully chosen so
that readers with a modest background in
mathematics can follow
them. It is suitable for mathematical historians
and a general
mathematical audience.
Contents
R. Thiele -- Antiquity
J. van Maanen -- Precursors of differentiation
and integration
N. Guicciardini -- Newton's method and Leibniz's
calculus
H. N. Jahnke -- Algebraic analysis in the
18th century
M. Panza -- The origins of analytic mechanics
in the 18th century
J. Lutzen -- The foundation of analysis in
the 19th century
T. Archibald -- Analysis and physics in the
nineteenth century:
The case of boundary-value problems
U. Bottazzini -- Complex function theory,
1780-1900
T. Hochkirchen -- Theory of measure and integration
from Riemann
to Lebesgue
M. Epple -- The end of the science of quantity:
Foundations of
analysis, 1860-1910
T. Archibald -- Differential equations: A
historical overview to
circa 1900
C. Fraser -- The calculus of variations:
A historical survey
R. Siegmund-Schultze -- The origins of functional
analysis
Index of names
Subject index
Details:
Series: History of Mathematics, Volume: 24
Publication Year: 2003
ISBN: 0-8218-2623-9
Paging: approximately 432 pp.
Binding: Hardcover
Expected publication date is July 20, 2003
Description
This volume addresses algebraic invariants
that occur in the
confluence of several important areas of
mathematics, including
number theory, algebra, and arithmetic algebraic
geometry. The
invariants are analogues for Galois cohomology
of the
characteristic classes of topology, which
have been extremely
useful tools in both topology and geometry.
It is hoped that
these new invariants will prove similarly
useful. Early versions
of the invariants arose in the attempt to
classify the quadratic
forms over a given field.
The authors are well-known experts in the
field. Serre, in
particular, is recognized as both a superb
mathematician and a
master author. His book on Galois cohomology
from the 1960s was
fundamental to the development of the theory.
Merkurjev, also an
expert mathematician and author, co-wrote
The Book of Involutions
(Volume 44 in the AMS Colloquium Publications
series), an
important work that contains preliminary
descriptions of some of
the main results on invariants described
here.
The book also includes letters between Serre
and some of the
principal developers of the theory. It will
be of interest to
graduate students and research mathematicians
interested in
number theory and Galois cohomology.
Contents
Cohomological invariants, Witt invariants,
and trace forms
Contents
Introduction
The notion of "invariant"
Cohomological preliminaries: The local case
Cohomological preliminaries: The function
field case
Specialization properties of cohomological
invariants
Restriction and corestriction of invariants
Cohomological invariants of $O_n,SO_n,\ldots$
Cohomological invariants of etale algebras
Witt invariants
The trace form in dimension $\le 7$
A letter from M. Rost to J-P. Serre
A letter from J-P. Serre to R. S. Garibaldi
A letter from B. Totaro to J-P. Serre
Rost invariants of simply connected algebraic
groups
Contents
Rost invariants of simply connected algebraic
groups
The groups $H^{d+1}(F,\mathbb{Q}/\mathbb{Z}(d))$
Tables of Dynkin indices
Bibliography
Index of notation
Index of terms
Details:
Series: University Lecture Series,Volume:
28
Publication Year: 2003
ISBN: 0-8218-3287-5
Paging: approximately 176 pp.
Binding: Softcover