Description
An n-dimensional mu-component boundary link
is a codimension 2
embedding of spheres L=sqcup_{mu}S^n subset
S^{n+2} such that
there exist mu disjoint oriented embedded
(n+1)-manifolds which
span the components of L. An F_mu-link is
a boundary link
together with a cobordism class of such spanning
manifolds.
The F_mu-link cobordism group C_n(F_mu) is
known to be trivial
when n is even but not finitely generated
when n is odd. Our main
result is an algorithm to decide whether
two odd-dimensional F_mu-links
represent the same cobordism class in C_{2q-1}(F_mu)
assuming
q>1. We proceed to compute the isomorphism
class of C_{2q-1}(F_mu),
generalizing Levine's computation of the
knot cobordism group C_{2q-1}(F_1).
Our starting point is the algebraic formulation
of Levine, Ko and
Mio who identify C_{2q-1}(F_mu) with a surgery
obstruction group,
the Witt group G^{(-1)^q,mu}(Z) of mu-component
Seifert matrices.
We obtain a complete set of torsion-free
invariants by passing
from integer coefficients to complex coefficients
and by applying
the algebraic machinery of Quebbemann, Scharlau
and Schulte.
Signatures correspond to `algebraically integral'
simple self-dual
representations of a certain quiver (directed
graph with loops).
These representations, in turn, correspond
to algebraic integers
on an infinite disjoint union of real affine
varieties.
To distinguish torsion classes, we consider
rational coefficients
in place of complex coefficients, expressing
G^{(-1)^q,mu}(mathbb{Q})
as an infinite direct sum of Witt groups
of finite-dimensional
division mathbb{Q}-algebras with involution.
The Witt group of
every such algebra appears as a summand infinitely
often.
The theory of symmetric and hermitian forms
over these division
algebras is well-developed. There are five
classes of algebras to
be considered; complete Witt invariants are
available for four
classes, those for which the local-global
principle applies. An
algebra in the fifth class, namely a quaternion
algebra with non-standard
involution, requires an additional Witt invariant
which is
defined if all the local invariants vanish.
Contents
Introduction
Main results
Preliminaries
Morita Equivalence
Devissage
Varieties of representations
Generalizing Pfister's theorem
Characters
Detecting rationality and integrality
Representation varieties: Two examples
Number theory invariants
All division algebras occur
Appendix I. Primitive element theorems
Appendix II. Hermitian categories
Bibliography
Index
Details:
Series: Memoirs of the American Mathematical
Society, Volume: 165
Publication Year: 2003
ISBN: 0-8218-3340-5
Paging: 110 pp.
Binding: Softcover
Description
This paper is devoted to the connective K
homology and cohomology
of finite groups G. We attempt to give a
systematic account from
several points of view.
In Chapter 1, following Quillen [50, 51],
we use the methods of
algebraic geometry to study the ring ku^*(BG)
where ku denotes
connective complex K-theory. We describe
the variety in terms of
the category of abelian p-subgroups of G
for primes p dividing
the group order. As may be expected, the
variety is obtained by
splicing that of periodic complex K-theory
and that of integral
ordinary homology, however the way these
parts fit together is of
interest in itself. The main technical obstacle
is that the
Kunneth spectral sequence does not collapse,
so we have to show
that it collapses up to isomorphism of varieties.
In Chapter 2 we give several families of
new complete and
explicit calculations of the ring ku^*(BG).
This illustrates the
general results of Chapter 1 and their limitations.
In Chapter 3 we consider the associated homology
ku_*(BG). We
identify this as a module over ku^*(BG) by
using the local
cohomology spectral sequence. This gives
new specific
calculations, but also illuminating structural
information,
including remarkable duality properties.
Finally, in Chapter 4 we make a particular
study of elementary
abelian groups V. Despite the group-theoretic
simplicity of V,
the detailed calculation of ku^*(BV) and
ku_*(BV) exposes a very
intricate structure, and gives a striking
illustration of our
methods. Unlike earlier work, our description
is natural for the
action of GL(V).
Contents
Introduction
General properties of the ku-cohomology of
finite groups
Examples of ku-cohomology of finite groups
The ku-homology of finite groups
The ku-homology and ku-cohomology of elementary
abelian groups
Appendix A. Conventions
Appendix B. Indices
Appendix. Bibliography
Details:
Series: Memoirs of the American Mathematical
Society, Volume: 165
Publication Year: 2003
ISBN: 0-8218-3366-9
Paging: 127 pp.
Binding: Softcover
Description
In this paper, bifurcations of stationary
and time-periodic
solutions to reaction-diffusion systems are
studied. We develop a
center-manifold and normal form theory for
radial dynamics which
allows for a complete description of radially
symmetric patterns.
In particular, we show the existence of localized
pulses near
saddle-nodes, critical Gibbs kernels in the
cusp, focus patterns
in Turing instabilities, and active or passive
target patterns in
oscillatory instabilities.
Contents
Introduction
Instabilities in one space dimension
Stationary radially symmetric patterns
Time-periodic radially symmetric patterns
Discussion
Bibliography
Details:
Series: Memoirs of the American Mathematical
Society, Volume: 165
Publication Year: 2003
ISBN: 0-8218-3373-1
Paging: 86 pp.
Binding: Softcover
Description
Interpolation of Weighted Banach Lattices
It is known that for many, but not all, compatible
couples of
Banach spaces (A_{0},A_{1}) it is possible
to characterize all
interpolation spaces with respect to the
couple via a simple
monotonicity condition in terms of the Peetre
K-functional. Such
couples may be termed Calderon-Mityagin couples.
The main results
of the present paper provide necessary and
sufficient conditions
on a couple of Banach lattices of measurable
functions (X_{0},X_{1})
which ensure that, for all weight functions
w_{0} and w_{1}, the
couple of weighted lattices (X_{0,w_{0}},X_{1,w_{1}})
is a
Calderon-Mityagin couple. Similarly, necessary
and sufficient
conditions are given for two couples of Banach
lattices (X_{0},X_{1})
and (Y_{0},Y_{1}) to have the property that,
for all choices of
weight functions w_{0}, w_{1}, v_{0} and
v_{1}, all relative
interpolation spaces with respect to the
weighted couples (X_{0,w_{0}},X_{1,w_{1}})
and (Y_{0,v_{0}},Y_{1,v_{1}}) may be described
via an obvious
analogue of the above-mentioned K-functional
monotonicity
condition.
A number of auxiliary results developed in
the course of this
work can also be expected to be useful in
other contexts. These
include a formula for the K-functional for
an arbitrary couple of
lattices which offers some of the features
of Holmstedt's formula
for K(t,f;L^{p},L^{q}), and also the following
uniqueness theorem
for Calderon's spaces X^{1-theta }_{0}X^{theta
}_{1}: Suppose
that the lattices X_0, X_1, Y_0 and Y_1 are
all saturated and
have the Fatou property. If X^{1-theta }_{0}X^{theta
}_{1} = Y^{1-theta
}_{0}Y^{theta }_{1} for two distinct values
of theta in (0,1),
then X_{0} = Y_{0} and X_{1} = Y_{1}. Yet
another such auxiliary
result is a generalized version of Lozanovskii's
formula left(
X_{0}^{1-theta }X_{1}^{theta }right) ^{prime
}=left (X_{0}^{prime
}right) ^{1-theta }left( X_{1}^{prime }right)
^{theta } for the
associate space of X^{1-theta }_{0}X^{theta
}_{1}.
A Characterization of Relatively Decomposable
Banach Lattices
Two Banach lattices of measurable functions
X and Y are said to
be relatively decomposable if there exists
a constant D such that
whenever two functions f and g can be expressed
as sums of
sequences of disjointly supported elements
of X and Y
respectively, f = sum^{infty }_{n=1} f_{n}
and g = sum^{infty
}_{n=1} g_{n}, such that | g_{n}| _{Y} le
| f_{n}| _{X} for all n
= 1, 2, ldots, and it is given that f in
X, then it follows that
g in Y and | g| _{Y} le D| f| _{X}.
Relatively decomposable lattices appear naturally
in the theory
of interpolation of weighted Banach lattices.
It is shown that X and Y are relatively decomposable
if and only
if, for some r in [1,infty ], X satisfies
a lower r-estimate and
Y satisfies an upper r-estimate. This is
also equivalent to the
condition that X and ell ^{r} are relatively
decomposable and
also ell ^{r} and Y are relatively decomposable.
Contents
Interpolation of weighted Banach lattices,
by Michael Cwikel and
Per G. Nilsson
Introduction
Definitions, terminology and preliminary
results
The main results
A uniqueness theorem
Two properties of the K-functional for a
couple of Banach
lattices
Characterizations of couples which are uniformly
Calderon-Mityagin
for all weights
Some uniform boundedness principles for interpolation
of Banach
lattices
Appendix: Lozanovskii's formula for general
Banach lattices of
measurable functions
References
A characterization of relatively decomposable
Banach lattices, by
Michael Cwikel, Per G. Nilsson and Gideon
Schechtman
Introduction
Equal norm upper and lower p-estimates and
some other preliminary
results
Completion of the proof of the main theorem
Application to the problem of characterizing
interpolation spaces
References
Details:
Series: Memoirs of the American Mathematical
Society,Volume: 165
Publication Year: 2003
ISBN: 0-8218-3382-0
Paging: 127 pp.
Binding: Softcover
Expected publication date is October 16,
2003
Description
"There were lots of young analysts who
flocked to Chicago in
those years, but virtually from the start
it was clear that Tom
had a special brilliance ... Eventually,
the mathematical door
would open a crack as Tom discovered a new
technique, usually of
astonishing originality. The end would now
be in sight, as [he]
unleashed his tremendous technical abilities
... Time after time,
[Wolff] would pick a central problem in an
area and solve it.
After a few more results, the field would
be changed forever ...
In the mathematical community, the common
and rapid response to
these breakthroughs was that they were seen,
not just as
watershed events, but as lightning strikes
that permanently
altered the landscape."
--Peter W. Jones, Yale University
"Tom Wolff was not only a deep thinker
in mathematics but
also a technical master."
--Barry Simon, California Institute of Technology
Thomas H. Wolff was a leading analyst and
winner of the Salem and
Bocher Prizes. He made significant contributions
to several areas
of harmonic analysis, in particular to geometrical
and measure-theoretic
questions related to the Kakeya needle problem.
Wolff attacked
the problem with awesome power and originality,
using both
geometric and combinatorial ideas. This book
provides an inside
look at the techniques used and developed
by Wolff. It is based
on a graduate course on Fourier analysis
he taught at Caltech.
The selection of the material is somewhat
unconventional in that
it leads the reader, in Wolff's unique and
straightforward way,
through the basics directly to current research
topics. The book
demonstrates how harmonic analysis can provide
penetrating
insights into deep aspects of modern analysis.
It is an
introduction to the subject as a whole and
an overview of those
branches of harmonic analysis that are relevant
to the Kakeya
conjecture.
The first few chapters cover the usual background
material: the
Fourier transform, convolution, the inversion
theorem, the
uncertainty principle, and the method of
stationary phase.
However, the choice of topics is highly selective,
with emphasis
on those frequently used in research inspired
by the problems
discussed in later chapters. These include
questions related to
the restriction conjecture and the Kakeya
conjecture, distance
sets, and Fourier transforms of singular
measures. These problems
are diverse, but often interconnected; they
all combine
sophisticated Fourier analysis with intriguing
links to other
areas of mathematics, and they continue to
stimulate first-rate
work.
The book focuses on laying out a solid foundation
for further
reading and research. Technicalities are
kept to a minimum, and
simpler but more basic methods are often
favored over the most
recent methods. The clear style of the exposition
and the quick
progression from fundamentals to advanced
topics ensure that both
graduate students and research mathematicians
will benefit from
the book.
Contents
The L^1 Fourier transform
The Schwartz space
Fourier inversion and the Plancherel theorem
Some specifics, and L^p for p<2
The uncertainty principle
The stationary phase method
The restriction problem
Hausdorff measures
Sets with maximal Fourier dimension and distance
sets
The Kakeya problem
Recent work connected with the Kakeya problem
Bibliography for Chapter 11
Historical notes
Bibliography
Details:
Series: University Lecture Series, Volume:
29
Publication Year: 2003
ISBN: 0-8218-3449-5
Paging: 137 pp.
Binding: Softcover