Gert-Martin Greuel – författare

This book constitutes the proceedings of the 5th International Conference on Mathematical Software, ICMS 2015, held in Berlin, Germany, in July 2016.

This volume brings together recent, original research and survey articles by leading experts in several fields that include singularity theory, algebraic geometry and commutative algebra. The motivation for this collection comes from the wide-ranging research of the distinguished mathematician, Antonio Campillo, in these and related fields. Besides his influence in the mathematical community stemming from his research, Campillo has also endeavored to promote mathematics and mathematicians'' networking everywhere, especially in Spain, Latin America and Europe. Because of his impressive achievements throughout his career, we dedicate this book to Campillo in honor of his 65th birthday. 

Researchers and students from the world-wide, and in particular Latin American and European, communities in singularities, algebraic geometry, commutative algebra, coding theory, and other fields covered in the volume, will have interest in this book.

Singularity theory is a field of intensive study in modern mathematics with fascinating relations to algebraic geometry, complex analysis, commutative algebra, representation theory, theory of Lie groups, topology, dynamical systems, and many more, and with numerous applications in the natural and technical sciences.

This book presents the basic singularity theory of analytic spaces, including local deformation theory, and the theory of plane curve singularities. Plane curve singularities are a classical object of study, rich of ideas and applications, which still is in the center of current research and as such provides an ideal introduction to the general theory. Deformation theory is an important technique in many branches of contemporary algebraic geometry and complex analysis. This introductory text provides the general framework of the theory while still remaining concrete.

In the first part of the book the authors develop the relevant techniques, including the Weierstraß preparation theorem, the finite coherence theorem etc., and then treat isolated hypersurface singularities, notably the finite determinacy, classification of simple singularities and topological and analytic invariants. In local deformation theory, emphasis is laid on the issues of versality, obstructions, and equisingular deformations. The book moreover contains a new treatment of equisingular deformations of plane curve singularities including a proof for the smoothness of the mu-constant stratum which is based on deformations of the parameterization. Computational aspects of the theory are discussed as well. Three appendices, including basic facts from sheaf theory, commutative algebra, and formal deformation theory, make the reading self-contained.

The material, which can be found partly in other books and partly in research articles, is presented from a unified point of view for the first time. It is given with complete proofs, new in many cases. The book thuscan serve as source for special courses in singularity theory and local algebraic and analytic geometry.

From the reviews of the first edition:

"It is certainly no exaggeration to say that … A Singular Introduction to Commutative Algebra aims to lead a further stage in the computational revolution in commutative algebra … . Among the great strengths and most distinctive features … is a new, completely unified treatment of the global and local theories. … making it one of the most flexible and most efficient systems of its type....another strength of Greuel and Pfister''s book is its breadth of coverage of theoretical topics in the portions of commutative algebra closest to algebraic geometry, with algorithmic treatments of almost every topic....Greuel and Pfister have written a distinctive and highly useful book that should be in the library of every commutative algebraist and algebraic geometer, expert and novice alike."

J.B. Little, MAA, March 2004

The second edition is substantially enlarged by a chapter on Groebner bases in non-commtative rings, a chapter on characteristic and triangular sets with applications to primary decomposition and polynomial solving and an appendix on polynomial factorization including factorization over algebraic field extensions and absolute factorization, in the uni- and multivariate case.

In unserer technisierten Welt stoßen wir überall auf Mathematik. Mathematik ist eine Basiswissenschaft und der Schlüssel für bahnbrechende Innovationen. Sie macht viele Produkte und Dienstleistungen überhaupt erst möglich und ist damit ein wichtiger Produktions- und Wettbewerbsfaktor.

Im vorliegenden Buch berichten 19 große internationale Unternehmen sowie die Bundesagentur für Arbeit wie unverzichtbar Mathematik für ihren Erfolg heute geworden ist. Ein spannender und lehrreicher Einblick in die Mathematik, der mit oft zitierten und negativen Vorurteilen gründlich aufräumt.