This book describes the global properties of simply-connected spaces
that are non-positively curved in the sense of groups which act on such spaces by asymmetries.
The theory of these objects is developed in a manner accessible to anyone familiar
with the rudiments of topology and group theory:
non-trivial theorems are proved by concatenating elementary geometric arguments,
and many examples are given.
Part I is an introduction to the geometry of geodesic spaces.
In Part II the basic theory of spaces with upper curvature bounds is developed.
More specialized topics, such as complexes of groups, are covered in Part III.
The book is divided into three parts, each part is divided
into chapters and the chapters have various subheadings.
1999 575 pp.
3-540-64324-9 16,220.
Inverse Galois Theory is concerned with the question
which finite groups occur as Galois Groups over a given field.
In particular, this includes the question on the structure and the representations of
the absolute Galois group of K and also the question about its finite epimorphic images,
the so- called inverse problem of Galois theory.
In all these areas important progress was made in the last few years.
The aim of the book is to give a consistent and reasonably complete survey of these results,
with the main emphasis on the rigidity method and its applications.
This monograph presents the state-of-the-art
concerning methods and results in the inverse Galois problem,
it contains a new cohomological proof for the famous theorem of
Shafarevich on the realization of solvable groups as Galois groups over global fields,
and, among others, it gives an introduction to the
very recent results on fundamental groups in positive characteristics.
1999 430 pp.
3-540-62890-8 9,400.
Grobner Deformations of
Hypergeometric Differential Equations
Perturbation techniques have a long tradition in analysis;
Grobner deformations of left ideals in the Weyl algebra
are the algebraic analogue to classical perturbation techniques.
The algorithmic methods introduced here are particularly useful for Studying the systems
of multidimensional hypergeometric PDEs introduced by Gelfand, Kapranov and Zelevinsky.
The Grobner deformation of these GKZ hypergeometric systems
reduces problems concerning hypergeometric functions to questions about commutative monomial ideals,
and leads to an unexpected interplay between analysis and combinatorics.
1999 350 pp.
3-540-66065-8 6,620.