Expected publication date is September 21, 2005
Description
Self-similar groups (groups generated by automata) initially
appeared as examples of groups that are easy to define but have
exotic properties like nontrivial torsion, intermediate growth,
etc. This book studies the self-similarity phenomenon in group
theory and shows its intimate relationship with dynamical systems
and more classical self-similar structures, such as fractals,
Julia sets, and self-affine tilings. This connection is
established through the central topics of the book, which are the
notions of the iterated monodromy group and limit space.
A wide variety of examples and different applications of self-similar
groups to dynamical systems and vice versa are discussed. In
particular, it is shown that Julia sets can be reconstructed from
the respective iterated monodromy groups and that groups with
exotic properties can appear not just as isolated examples, but
as naturally defined iterated monodromy groups of rational
functions.
The book offers important, new mathematics that will open new
avenues of research in group theory and dynamical systems. It is
intended to be accessible to a wide readership of professional
mathematicians.
Contents
Basic definitions and examples
Algebraic theory
Limit spaces
Orbispaces
Iterated monodromy groups
Examples and applications
Bibliography
Index
Details:
Series: Mathematical Surveys and Monographs, Volume 117
Publication Year: 2005
ISBN: 0-8218-3831-8
Paging: 231 pp.
Binding: Hardcover
Expected publication date is September 28, 2005
Description
This monograph contains exciting original mathematics that will
inspire new directions of research in algebraic geometry.
Developed here is an arithmetic analog of the theory of ordinary
differential equations, where functions are replaced by integer
numbers, the derivative operator is replaced by a "Fermat
quotient operator", and differential equations (viewed as
functions on jet spaces) are replaced by "arithmetic
differential equations". The main application of this theory
concerns the construction and study of quotients of algebraic
curves by correspondence with infinite orbits. Any such quotient
usually reduces to a point in algebraic geometry. But many of the
above quotients cease to be trivial (and become quite interesting)
if one enlarges algebraic geometry by using arithmetic
differential equations in place of algebraic equations.
This book, in part, follows a series of papers written by the
author. However, a substantial amount of the material has never
been published before. For most of the book, the only
prerequisites are the basic facts of algebraic geometry and
algebraic number theory. It is suitable for graduate students and
researchers interested in algebraic geometry.
Contents
Main concepts and results
Preliminaries from algebraic geometry
Outline of delta-geometry
General theory
Global theory
Local theory
Birational theory
Applications
Spherical correspondences
Flat correspondences
Hyperbolic correspondences
List of results
Bibliography
Index
Details:
Series: Mathematical Surveys and Monographs, Volume 118
Publication Year: 2005
ISBN: 0-8218-3862-8
Paging: approximately 360 pp.
Binding: Hardcover
Expected publication date is September 3, 2005
"From a review of the first edition:
""Beautifully written and well organized ...
indispensable for those interested in certain areas of
mathematical physics for the expert and beginner alike. The
author deserves to be congratulated both for his work in unifying
a subject and for showing workers in the field new directions for
future development.""
-- Zentralblatt MATH
Description
This is a second edition of a well-known book stemming from the
author's lectures on the theory of trace ideals in the algebra of
operators in a Hilbert space. Because of the theory's many
different applications, the book was widely used and much in
demand.
For this edition, the author has added four chapters on the
closely related theory of rank one perturbations of self-adjoint
operators. He has also included a comprehensive index and an
addendum describing some developments since the original notes
were published.
This book continues to be a vital source of information for those
interested in the theory of trace ideals and in its applications
to various areas of mathematical physics.
Contents
Preliminaries
Calkin's theory of operator ideals and symmetrically normed
ideals; convergence theorems for mathcal J_P
Trace, determinant, and Lidskii's theorem
f(x)g(-inabla)
Fredholm theory
Scattering with a trace condition
Bound state problems
Lots of inequalities
Regularized determinants and renormalization in quantum field
theory
An introduction to the theory on a Banach space
Borel transforms, the Krein spectral shift, and all that
Spectral theory of rank one perturbations
Localization in the Anderson model following Aizenman-Molchanov
The Xi function
Addenda
Bibliography
Index
Details:
Series: Mathematical Surveys and Monographs, Volume 120
Publication Year: 2005
ISBN: 0-8218-3581-5
Paging: 150 pp.
Binding: Hardcover
Expected publication date is October 2, 2005
Description
This book focuses on finiteness conjectures and results in
ordinary differential equations (ODEs) and Diophantine geometry.
During the past twenty-five years, much progress has been
achieved on finiteness conjectures, which are the offspring of
the second part of Hilbert's 16th problem. Even in its simplest
case, this is one of the very few problems on Hilbert's list
which remains unsolved. These results are about existence and
estimation of finite bounds for the number of limit cycles
occurring in certain families of ODEs. The book describes this
progress, the methods used (bifurcation theory, asymptotic
expansions, methods of differential algebra, or geometry) and the
specific results obtained. The finiteness conjectures on limit
cycles are part of a larger picture that also includes finiteness
problems in other areas of mathematics, in particular those in
Diophantine geometry where remarkable results were proved during
the same period of time. There is a chapter devoted to finiteness
results in Diophantine geometry obtained by using methods of
differential algebra, which is a connecting element between these
parallel developments in the book.
The volume can be used as an independent study text for advanced
undergraduates and graduate students studying ODEs or
applications of differential algebra to differential equations
and Diophantine geometry. It is also a good entry point for
researchers interested these areas, in particular, in limit
cycles of ODEs, and in finiteness problems.
Contributors to the volume include Andrei A. Bolibrukh and
Alexandru Buium. Available from the AMS by A. Buium is Arithmetic
Differential Equations, as Volume 118 in the Mathematical Surveys
and Monographs series.
Contents
Finiteness problems in differential equations and Diophantine
geometry
Linear differential equations, Fuchsian inequalities and
multiplicities of zeros
Quantitative theory of ordinary differential equations and
tangential Hilbert 16th problem
Around Hilbert-Arnol'd problem
Finiteness results in differential algebraic geometry and
Diophantine geometry
o-minimal structures, real analytic geometry, and transseries
List of lectures
List of participants
List of participants II
Details:
Series: CRM Monograph Series, Volume 24
Publication Year: 2005
ISBN: 0-8218-2805-3
Paging: approximately 224 pp.
Binding: Hardcover
Expected publication date is October 20, 2005
Description
This introductory textbook puts forth a clear and focused point
of view on the differential geometry of curves and surfaces,
emphasizing the global aspects. The excellent collection of
examples and exercises (with hints) will help students in
learning the material. Advanced undergraduates and graduate
students will find this a nice entry point to differential
geometry.
In order to study the global properties of curves and surfaces,
it is necessary to have more sophisticated tools than are usually
found in textbooks on the topic. In particular, students must
have a firm grasp on certain topological theories. Indeed, this
monograph treats the Gauss-Bonnet Theorem and discusses the Euler
characteristic. The authors also cover Alexandrov's theorem on
embedded compact surfaces in mathbb{R}^3 with constant mean
curvature. The last chapter addresses the global geometry of
curves, including periodic space curves and the four vertices
theorem for plane curves that are not necessarily convex.
This volume is suitable for advanced undergraduates, graduate
students, and researchers interested in the differential geometry
of curves and surfaces. It can also be used as an introduction to
a more general study of differential geometry.
This book is jointly published by the AMS and the Real Sociedad
Matematica Espanola (RSME).
Contents
Plane and space curves
Surfaces in Euclidean space
The second fundamental form
Separation and orientability
Integration on surfaces
Global extrinsic geometry
Intrinsic geometry of surfaces
The Gauss-Bonnet theorem
Global geometry of curves
Bibliography
Index
Details:
Series: Graduate Studies in Mathematics, Volume: 69
Publication Year: 2005
ISBN: 0-8218-3815-6
Paging: approximately 384 pp.
Binding: Hardcover