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 manifold is holo morphically complete. J. -P. SERRE has obtained important results on algebraic manifolds by these and other methods. Recently many of his results have been proved for algebraic varieties defined over a field of arbitrary characteristic. K. KODAIRA and D. C. SPENCER have also applied sheaf theory to algebraic geometry with great success. Their methods differ from those of SERRE in that they use techniques from differential geometry (harmonic integrals etc. ) but do not make any use of the theory of STEIN manifolds. M. F. ATIYAH and W. V. D. HODGE have dealt successfully with problems on integrals of the second kind on algebraic manifolds with the help of sheaf theory. I was able to work together with K. KODAIRA and D. C. SPENCER during a stay at the Institute for Advanced Study at Princeton from 1952 to 1954.
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.
During the winter term 1987/88 I gave a course at the University of Bonn under the title "Manifolds and Modular Forms". I wanted to develop the theory of "Elliptic Genera" and to learn it myself on this occasion. This theory due to Ochanine, Landweber, Stong and others was relatively new at the time. The word "genus" is meant in the sense of my book "Neue Topologische Methoden in der Algebraischen Geometrie" published in 1956: A genus is a homomorphism of the Thorn cobordism ring of oriented compact manifolds into the complex numbers. Fundamental examples are the signature and the A-genus. The A-genus equals the arithmetic genus of an algebraic manifold, provided the first Chern class of the manifold vanishes. According to Atiyah and Singer it is the index of the Dirac operator on a compact Riemannian manifold with spin structure. The elliptic genera depend on a parameter. For special values of the parameter one obtains the signature and the A-genus. Indeed, the universal elliptic genus can be regarded as a modular form with respect to the subgroup r (2) of the modular group; the two cusps 0 giving the signature and the A-genus. Witten and other physicists have given motivations for the elliptic genus by theoretical physics using the free loop space of a manifold.
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.
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. While we each view our specialty as having roots in the past, and sometimes having connec tions to other specialties in the present, we seldom see the panorama of mathematical development over thousands of years. Numbers attempts to give that broad perspective, from hieroglyphs to K-theory, from Dedekind cuts to nonstandard analysis.
This book presents the proceedings from the conference on algebraic geometry in honor of Professor Friedrich Hirzebruch's 70th Birthday. The event was held at the Stefan Banach International Mathematical Center in Warsaw (Poland). The topics covered in the book include intersection theory, singularities, low-dimensional manifolds, moduli spaces, number theory, and interactions between mathematical physics and geometry. Also included are articles from notes of two special lectures. The first, by Professor M. Atiyah, describes the important contributions to the field of geometry by Professor Hirzebruch. The second article contains notes from the talk delivered at the conference by Professor Hirzebruch. Contributors to the volume are leading researchers in the field.
Mathematics has a certain mystique, for it is pure and ex- act, yet demands remarkable creativity. This reputation is reinforced by its characteristic abstraction and its own in- dividual language, which often disguise its origins in and connections with the physical world. Publishing mathematics, therefore, requires special effort and talent. Heinz G|tze,who has dedicated his life to scientific pu- blishing, took up this challenge with his typical enthusi- asm. This Festschrift celebrates his invaluable contribu- tions to the mathematical community, many of whose leading members he counts among his personal friends. The articles, written by mathematicians from around the world and coming from diverse fields, portray the important role of mathematics in our culture. Here, the reflections of important mathematicians, often focused on the history of mathematics, are collected, in recognition of Heinz G|tze's life-longsupport of mathematics.
Im Mittelpunkt des Buches steht eine Konstruktion mit Hilfe von Geradenkonfigurationen in der komplex-projektiven Ebene, die überraschende Beziehungen zur elementaren Geometrie aufzeigt: Aus der berühmten Miyaoka-Yau-Ungleichung für die Chernschen Zahlen einer algebraischen Fläche folgen Aussagen über Geraden- und Punktkonfigurationen, für die kein direkter Beweis bekannt ist. Der Grenzfall der Ungleichung ist eine Proportionalitätsbeziehung, die genau die Flächen charakterisiert, deren universelle Überlagerung die Vollkugel im komplex-zweidimensionalen Raum ist. Die Methoden gestatten die Konstruktion von Flächen aus dieser besonders interessanten Klasse, für die bislang wenig explizite Beispiele bekannt waren.