EMS Textbooks in Mathematics Vol. 1
ISBN 3-03719-001-9
January 2004, 192 pages, hardcover, 16.5
cm x 23.5 cm.
This book is written as a text for a course
aimed at 3rd or 4th
year students. Only some familiarity with
elementary linear
algebra and probability is directly assumed,
but some maturity is
required. The students may specialize in
discrete mathematics,
computer science, or communication engineering.
The book is also
a suitable introduction to coding theory
for researchers from
related fields or for professionals who want
to supplement their
theoretical basis. The book gives the coding
basics for working
on projects in any of the above areas, but
material specific to
one of these fields has not been included.
The chapters cover the
codes and decoding methods that are currently
of most interest in
research, development, and application. They
give a relatively
brief presentation of the essential results,
emphasizing the
interrelations between different methods
and proofs of all
important results. A sequence of problems
at the end of each
chapter serves to review the results and
give the student an
appreciation of the concepts. In addition,
some problems and
suggestions for projects indicate direction
for further work. The
presentation encourages the use of programming
tools for studying
codes, implementing decoding methods, and
simulating performance.
Specific examples of programming exercises
are provided on the
book's home page.
EMS Monographs in Mathematics Vol. 1
ISBN 3-03719-000-0
December 2003, 352 pages, hardcover, 16.5
cm x 23.5 cm.
The elements of many classical combinatorial
structures can be
naturally decomposed into components. Permutations
can be
decomposed into cycles, polynomials over
a finite field into
irreducible factors, mappings into connected
components. In all
of these examples, and in many more, there
are strong
similarities between the numbers of components
of different sizes
that are found in the decompositions of `typical'
elements of
large size. For instance, the total number
of components grows
logarithmically with the size of the element,
and the size of the
largest component is an appreciable fraction
of the whole. This
book explains the similarities in asymptotic
behaviour as the
result of two basic properties shared by
the structures: the
conditioning relation and the logarithmic
condition. The
discussion is conducted in the language of
probability, enabling
the theory to be developed under rather general
and explicit
conditions; for the finer conclusions, Stein's
method emerges as
the key ingredient. The book is thus of particular
interest to
graduate students and researchers in both
combinatorics and
probability theory.
ESI Lectures in Mathematics and Physics Vol.
1
ISBN 3-03719-002-7
December 2003, 100 pages, softcover, 17.0
cm x 24.0 cm.
In 1914, E. Cartan posed the problem to find
all irreducible real
linear Lie algebras. An updated exposition
of his work was given
by Iwahori (1959). This theory reduces the
classification of
irreducible real representations of a real
Lie algebra to a
description of the so-called self-conjugate
irreducible complex
representations of this algebra and to the
calculation of an
invariant of such a representation (with
values +1 or -1) which
is called the index. Moreover, these two
problems were reduced to
the case when the Lie algebra is simple and
the highest weight of
its irreducible complex representation is
fundamental. A complete
case-by-case classification for all simple
real Lie algebras was
given (without proof) in the tables of Tits
(1967). But actually
a general solution of these problems is contained
in a paper of
Karpelevich (1955) (written in Russian and
not widely known),
where inclusions between real forms induced
by a complex
representation were studied. WE begin with
a simplified (and
somewhat extended and corrected) exposition
of the main part of
this paper and relate it to the theory of
Cartan - Iwahori. We
conclude with some tables, where an involution
of the Dynkin
diagram which allows us to find self-conjugate
representations is
described and explicit formulas for the index
are given. In a
short addendum, written by J. v. Silhan,
this involution is
interpreted in terms of the Satake diagram.
The book is aimed at
students in Lie groups, Lie algebras and
their representations,
as well as researchers in any field where
these theories are used.
The reader is supposed to know the classical
theory of complex
semisimple Lie algebras and their finite
dimensional
representation; the main facts are presented
without proofs in
Section 1. In the remaining sections the
exposition is made with
detailed proofs, including the correspondence
between real forms
and involutive automorphisms, the Cartan
decompositions and the
conjugacy of maximal compact subgroups of
the automorphism group.
Zurich Lectures in Advanced Mathematics Vol.
1
ISBN 3-03719-003-5
January 2004, 144 pages, softcover, 17.0
cm x 24.0 cm.
This book is an introduction to the modern
theory of partial
hyperbolicity with applications to stable
ergodicity theory of
smooth dynamical systems. It provides a systematic
treatment of
the theory and describes all the basic concepts
and major results
that have been obtained in the area since
its creation around the
early 1970s. It can be used as a textbook
for a graduate student
course and is also of interest to professional
mathematicians
working in the field of dynamical systems
and their applications.
EMS Series of Lectures in Mathematics Vol.
1
ISBN 3-03719-004-3
January 2004, 250 pages, softcover, 17.0
cm x 24.0 cm.
This book gives a detailed account of the
analytic foundations of
gauge theory -- Uhlenbeck's compactness theorems
for general
connections and for Yang-Mills connections.
It intends to guide
graduate students into the analysis of Yang-Mills
theory as well
as to serve as a reference for researchers
in the field. The book
is largely self-contained. It contains a
number of appendices (e.g.
on Sobolev spaces of maps between manifolds)
and an introductory
part covering the $L^p$-regularity theory
for the inhomogenous
Neumann problem. The two main parts contain
the full proofs of
Uhlenbeck's weak and strong compactness theorems
on closed
manifolds as well as their generalizations
to manifolds with
boundary and noncompact manifolds. These
parts include a number
of useful analytic tools such as general
patching constructions
and local slice theorems.