A book about numbers sounds rather dull. This one is not. Instead it is a lively story about one thread of mathematics-the concept of "number" told by eight authors and organized into a historical narrative that leads the reader from ancient Egypt to the late twentieth century. It is a story that begins with some of the simplest ideas of mathematics and ends with some of the most complex. It is a story that mathematicians, both amateur and professional, ought to know. Why write about numbers? Mathematicians have always found it diffi cult to develop broad perspective about their subject....

Five papers by distinguished American and European mathematicians describe some current trends in mathematics in the perspective of the recent past and in terms of expectations for the future. Among the subjects discussed are algebraic groups, quadratic forms, topological aspects of global analysis, variants of the index theorem, and partial differential equations.

An up-to-date report on the current status of important research topics in algebraic geometry and its applications, such as computational algebra and geometry, singularity theory algorithms, numerical solutions of polynomial systems, coding theory, communication networks, and computer vision. Contributions on more fundamental aspects of algebraic geometry include expositions related to counting points on varieties over finite fields, Mori theory, linear systems, Abelian varieties, vector bundles on singular curves, degenerations of surfaces, and mirror symmetry of Calabi-Yau manifolds.

An up-to-date report on the current status of important research topics in algebraic geometry and its applications, such as computational algebra and geometry, singularity theory algorithms, numerical solutions of polynomial systems, coding theory, communication networks, and computer vision. Contributions on more fundamental aspects of algebraic geometry include expositions related to counting points on varieties over finite fields, Mori theory, linear systems, Abelian varieties, vector bundles on singular curves, degenerations of surfaces, and mirror symmetry of Calabi-Yau manifolds.

Die Schwierigkeit Mathematik zu lernen und zu lehren ist jedem bekannt, der einmal mit diesem Fach in Beruhrung gekommen ist. Begriffe wie "reelle oder komplexe Zahlen, Pi" sind zwar jedem gelaufig, aber nur wenige wissen, was sich wirklich dahinter verbirgt. Die Autoren dieses Bandes geben jedem, der mehr wissen will als nur die Hulle der Begriffe, eine meisterhafte Einfuhrung in die Magie der Mathematik und schlagen einzigartige Brucken fur Studenten.

Die Rezensenten der ersten beiden Auflagen uberschlugen sich.

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In recent years new topological methods, especially the theory of sheaves founded by J. LERAY, have been applied successfully to algebraic geometry and to the theory of functions of several complex variables. H. CARTAN and J. -P. SERRE have shown how fundamental theorems on holomorphically complete manifolds (STEIN manifolds) can be for mulated in terms of sheaf theory. These theorems imply many facts of function theory because the domains of holomorphy are holomorphically complete. They can also be applied to algebraic geometry because the complement of a hyperplane section of an algebraic...

Im Mittelpunkt des Buches steht eine Konstruktion mit Hilfe von Geradenkonfigurationen in der komplex-projektiven Ebene, die uberraschende Beziehungen zur elementaren Geometrie aufzeigt: Aus der beruhmten Miyaoka-Yau-Ungleichung fur die Chernschen Zahlen einer algebraischen Flache folgen Aussagen uber Geraden- und Punktkonfigurationen, fur die kein direkter Beweis bekannt ist. Der Grenzfall der Ungleichung ist eine Proportionalitatsbeziehung, die genau die Flachen charakterisiert, deren universelle Uberlagerung die Vollkugel im komplex-zweidimensionalen Raum ist. Die Methoden gestatten die...

This book consists almost entirely of papers delivered at the Seminar on partial differential equations held at Max-Planck-Institut in the spring of 1984. They give an insight into important recent research activities. Some further developments are also included.