The classical story - of the hypergeometric functions, the configuration space of 4 points on the projective line, elliptic curves, elliptic modular functions and the theta functions - now evolves, in this book, to the story of hypergeometric funktions in 4 variables, the configuration space of 6 points in the projective plane, K3 surfaces, theta functions in 4 variables.
This modern theory has been established by the author and his collaborators in the 1990's; further development to different aspects is expected.
It leads the reader to a fascinating 4-dimensional world.
The author tells the story casually and visually in a plain language, starting form elementary level such as equivalence relations, the exponential function, ... Undergraduate students should be able to enjoy the text.

Discrete and computational geometry are two fields which in recent years have benefitted from the interaction between mathematics and computer science. The results are applicable in areas such as motion planning, robotics, scene analysis, and computer aided design. The book consists of twelve chapters summarizing the most recent results and methods in discrete and computational geometry. All authors are well-known experts in these fields. They give concise and self-contained surveys of the most efficient combinatorical, probabilistic and topological methods that can be used to design effective geometric algorithms for the applications mentioned above. Most of the methods and results discussed in the book have not appeared in any previously published monograph. In particular, this book contains the first systematic treatment of epsilon-nets, geometric tranversal theory, partitions of Euclidean spaces and a general method for the analysis of randomized geometric algorithms. Apart from mathematicians working in discrete and computational geometry this book will also be of great use to computer scientists and engineers, who would like to learn about the most recent results.

Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfangen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen fur die historische wie auch die disziplingeschichtliche Forschung zur Verfugung, die jeweils im historischen Kontext betrachtet werden mussen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

Apollonius's Conics was one of the greatest works of advanced mathematics in antiquity. The work comprised eight books, of which four have come down to us in their original Greek and three in Arabic. By the time the Arabic translations were produced, the eighth book had already been lost. In 1710, Edmond Halley, then Savilian Professor of Geometry at Oxford, produced an edition of the Greek text of the Conics of Books I-IV, a translation into Latin from the Arabic versions of Books V-VII, and a reconstruction of Book VIII. The present work provides the first complete English translation of Halley's reconstruction of Book VIII with supplementary notes on the text. It also contains 1) an introduction discussing aspects of Apollonius's Conics 2) an investigation of Edmond Halley's understanding of the nature of his venture into ancient mathematics, and 3) an appendices giving a brief account of Apollonius's approach to conic sections and his mathematical techniques. This book will be of interest to students and researchers interested in the history of ancient Greek mathematics and mathematics in the early modern period.

Hyperbolic geometry is a classical subject in pure mathematics which has exciting applications in theoretical physics. In this book leading experts introduce hyperbolic geometry and Maass waveforms and discuss applications in quantum chaos and cosmology. The book begins with an introductory chapter detailing the geometry of hyperbolic surfaces and includes numerous worked examples and exercises to give the reader a solid foundation for the rest of the book. In later chapters the classical version of Selberg's trace formula is derived in detail and transfer operators are developed as tools in the spectral theory of Laplace-Beltrami operators on modular surfaces. The computation of Maass waveforms and associated eigenvalues of the hyperbolic Laplacian on hyperbolic manifolds are also presented in a comprehensive way. This book will be valuable to graduate students and young researchers, as well as for those experienced scientists who want a detailed exposition of the subject.

This book offers an introduction to differential geometry for the non-specialist. It includes most of the required material from multivariable calculus, linear algebra, and basic analysis. An intuitive approach and a minimum of prerequisites make it a valuable companion for students of mathematics and physics. The main focus is on manifolds in Euclidean space and the metric properties they inherit from it. Among the topics discussed are curvature and how it affects the shape of space, and the generalization of the fundamental theorem of calculus known as Stokes' theorem.

Jordan theory has developed rapidly in the last three decades, but very few books describe its diverse applications. Here, the author discusses some recent advances of Jordan theory in differential geometry, complex and functional analysis, with the aid of numerous examples and concise historical notes. These include: the connection between Jordan and Lie theory via the Tits-Kantor-Koecher construction of Lie algebras; a Jordan algebraic approach to infinite dimensional symmetric manifolds including Riemannian symmetric spaces; the one-to-one correspondence between bounded symmetric domains and JB*-triples; and applications of Jordan methods in complex function theory. The basic structures and some functional analytic properties of JB*-triples are also discussed. The book is a convenient reference for experts in complex geometry or functional analysis, as well as an introduction to these areas for beginning researchers. The recent applications of Jordan theory discussed in the book should also appeal to algebraists.

In recent years, number theory and arithmetic geometry have been enriched by new techniques from noncommutative geometry, operator algebras, dynamical systems, and K-Theory. This volume collects and presents up-to-date research topics in arithmetic and noncommutative geometry and ideas from physics that point to possible new connections between the fields of number theory, algebraic geometry and noncommutative geometry. The articles collected in this volume present new noncommutative geometry perspectives on classical topics of number theory and arithmetic such as modular forms, class field theory, the theory of reductive p-adic groups, Shimura varieties, the local L-factors of arithmetic varieties. They also show how arithmetic appears naturally in noncommutative geometry and in physics, in the residues of Feynman graphs, in the properties of noncommutative tori, and in the quantum Hall effect.

This is a revised edition of the ?rst printing which appeared in 2002. The book is based on lectures at the University of Bergen, Norway.Over the years these lectures have covered many different aspects and facets of the wonderful ?eld of geometry. Consequently it has never been possible to give a full and ?nal account of geometry as such, at an undergraduatelevel: A carefully consideredselection has always been necessary. The present book constitutes the main central themes of these selections. One of the groups I am aiming at, is future teachers of mathematics. All too oftenthe textsdealingwith geometrywhichgo intothe syllabusforteacher-students present the material in ways which appear pedantic and formalistic, suppressing the very powerful and dynamic character of this old ?eld, which at the same time so young. Geometry is a ?eld of mathematical insight, research, history and source of artistic inspiration. And not least important, an integral part of our common cultural heritage.

Exploring theories and applications developed during the last 30 years, Digital Geometry in Image Processing presents a mathematical treatment of the properties of digital metric spaces and their relevance in analyzing shapes in two and three dimensions. Unlike similar books, this one connects the two areas of image processing and digital geometry, highlighting important results of digital geometry that are currently used in image analysis and processing. The book discusses different digital geometries in multi-dimensional integral coordinate spaces. It also describes interesting properties of the geometries, including metric and topological properties, shapes of circles and spheres, proximity to Euclidean norms, and number theoretic representations of geometric objects such as straight lines and circles. The authors-all active researchers in image processing and digital geometry-demonstrate how these concepts and properties are useful in various techniques for image processing and analysis. In particular, the book covers applications in object representation and shape analysis. With many figures (some in color) and end-of-chapter exercises, this book provides an in-depth, unified account of digital metrics, the characterization of digital curves and straight lines, and their uses in shape analysis. It gives you insight on the latest two- and three-dimensional image processing applications.

In the first half of the 19th century geometry changed radically, and withina century it helped to revolutionize both mathematics and physics. It also put the epistemology and the philosophy of science on a new footing. In this volume a sound overview of this development is given by leading mathematicians, physicists, philosophers, and historians of science. This interdisciplinary approach gives this collection a unique character. It can be used by scientists and students, but it also addresses a general readership.

Intuitively, a foliation corresponds to a decomposition of a manifold into a union of connected, disjoint submanifolds of the same dimension, called leaves, which pile up locally like pages of a book. The theory of foliations, as it is known, began with the work of C. Ehresmann and G. Reeb, in the 1940's; however, as Reeb has himself observed, already in the last century P. Painleve saw the necessity of creating a geometric theory (of foliations) in order to better understand the problems in the study of solutions of holomorphic differential equations in the complex field. The development of the theory of foliations was however provoked by the following question about the topology of manifolds proposed by H. Hopf in the 3 1930's: "Does there exist on the Euclidean sphere S a completely integrable vector field, that is, a field X such that X* curl X * 0?" By Frobenius' theorem, this question is equivalent to the following: "Does there exist on the 3 sphere S a two-dimensional foliation?" This question was answered affirmatively by Reeb in his thesis, where he 3 presents an example of a foliation of S with the following characteristics: There exists one compact leaf homeomorphic to the two-dimensional torus, while the other leaves are homeomorphic to two-dimensional planes which accu mulate asymptotically on the compact leaf. Further, the foliation is C"".

This innovative textbook introduces a new pattern-based approach to learning proof methods in the mathematical sciences. Readers will discover techniques that will enable them to learn new proofs across different areas of pure mathematics with ease. The patterns in proofs from diverse fields such as algebra, analysis, topology and number theory are explored. Specific topics examined include game theory, combinatorics and Euclidean geometry, enabling a broad familiarity. The author, an experienced lecturer and researcher renowned for his innovative view and intuitive style, illuminates a wide range of techniques and examples from duplicating the cube to triangulating polygons to the infinitude of primes to the fundamental theorem of algebra. Intended as a companion for undergraduate students, this text is an essential addition to every aspiring mathematician's toolkit.

This introduction treats the classical isoperimetric inequality in Euclidean space and contrasting rough inequalities in noncompact Riemannian manifolds. In Euclidean space the emphasis is on a most general form of the inequality sufficiently precise to characterize the case of equality, and in Riemannian manifolds the emphasis is on those qualitiative features of the inequality that provide insight into the coarse geometry at infinity of Riemannian manifolds. The treatment in Euclidean space features a number of proofs of the classical inequality in increasing generality, providing in the process a transition from the methods of classical differential geometry to those of modern geometric measure theory; and the treatment in Riemannian manifolds features discretization techniques, and applications to upper bounds of large time heat diffusion in Riemannian manifolds. The result is an introduction to the rich tapestry of ideas and techniques of isoperimetric inequalities, a subject that has its beginnings in classical antiquity and which continues to inspire fresh ideas in geometry and analysis to this very day--and beyond

Convex geometry is at once simple and amazingly rich. While the classical results go back many decades, during that previous to this book's publication in 1999, the integral geometry of convex bodies had undergone a dramatic revitalization, brought about by the introduction of methods, results and, most importantly, new viewpoints, from probability theory, harmonic analysis and the geometry of finite-dimensional normed spaces. This book is a collection of research and expository articles on convex geometry and probability, suitable for researchers and graduate students in several branches of mathematics coming under the broad heading of 'Geometric Functional Analysis'. It continues the Israel GAFA Seminar series, which is widely recognized as the most useful research source in the area. The collection reflects the work done at the program in Convex Geometry and Geometric Analysis that took place at MSRI in 1996.

Mathematicians and non-mathematicians alike have long been fascinated by geometrical problems, particularly those that are intuitive in the sense of being easy to state, perhaps with the aid of a simple diagram. Each section in the book describes a problem or a group of related problems. Usually the problems are capable of generalization of variation in many directions. The book can be appreciated at many levels and is intended for everyone from amateurs to research mathematicians.

During the past few decades, the gradual merger of Discrete Geometry and the newer discipline of Computational Geometry has provided enormous impetus to mathematicians and computer scientists interested in geometric problems. This 2005 volume, which contains 32 papers on a broad range of topics of interest in the field, is an outgrowth of that synergism. It includes surveys and research articles exploring geometric arrangements, polytopes, packing, covering, discrete convexity, geometric algorithms and their complexity, and the combinatorial complexity of geometric objects, particularly in low dimension. There are points of contact with many applied areas such as mathematical programming, visibility problems, kinetic data structures, and biochemistry, as well as with algebraic topology, geometric probability, real algebraic geometry, and combinatorics.

When we studied complex variables in the late 1960s, modem geometry on the complex fie1d and complex function theory were identified in teaching and research as several complex variables. A beginner in the field at that time would have the experience of jumping from the sheaf-theoretical methods employed in the theory of analytic spaces to the P.D.E. methods of the a- problem, with the c1ear understanding that the phenomena lying behind such different methods and problems were the same. A few years later, new important discoveries made c1ear that complex differential geometry was also in the same company. Looking at the historical development of the subject in the first half of the twentieth century shows this was not astonishing. The origin of the theory of functions of several complex variables was tardier than the familiar of analytic functions of one complex variable. The first comprehensive theory textbook by Behnke and Thullen, in the 1930s, expounded the foundations ofthe general theory as set up by Weierstrass, Cousin, Hartogs, and Poincare and c1early put in evidence that the difficulties were all but solved. In aseries of papers from 1936 to 1953, Oka introduced a brilliant collection of new ideas and systematically eliminated aU difficulties. Oka's work had in itse1f a fruitful seed and contained the premises for the opening of wider horizons.

Convexity is important in theoretical aspects of mathematics and also for economists and physicists. In this monograph the author provides a comprehensive insight into convex sets and functions including the infinite-dimensional case and emphasizing the analytic point of view. Chapter one introduces the reader to the basic definitions and ideas that play central roles throughout the book. The rest of the book is divided into four parts: convexity and topology on infinite-dimensional spaces; Loewner's theorem; extreme points of convex sets and related issues, including the Krein-Milman theorem and Choquet theory; and a discussion of convexity and inequalities. The connections between disparate topics are clearly explained, giving the reader a thorough understanding of how convexity is useful as an analytic tool. A final chapter overviews the subject's history and explores further some of the themes mentioned earlier. This is an excellent resource for anyone interested in this central topic.

This 1998 book describes the progress that had been made towards the development of a comprehensive understanding of the formation of complex, disorderly patterns under conditions far from equilibrium. The application of fractal geometry and scaling concepts to the quantitative description and understanding of structure formed under non-equilibrium conditions is described. Self-similar fractals, multi-fractals and scaling methods are discussed, with examples, to facilitate applications in the physical sciences. Computer simulations and experimental studies are emphasised, but the author also includes discussion of theoretical advances in the subject. Much of the book deals with diffusion-limited growth processes and the evolution of rough surfaces, although a broad range of other applications is also included. The techniques and topics will be relevant to graduate students and researchers in physics, chemistry, materials science, engineering and the earth sciences, interested in applying the ideas of fractals and scaling.

Elliptic cohomology is an extremely beautiful theory with both geometric and arithmetic aspects. The former is explained by the fact that the theory is a quotient of oriented cobordism localised away from 2, the latter by the fact that the coefficients coincide with a ring of modular forms. The aim of the book is to construct this cohomology theory, and evaluate it on classifying spaces BG of finite groups G. This class of spaces is important, since (using ideas borrowed from 'Monstrous Moonshine') it is possible to give a bundle-theoretic definition of EU-(BG). Concluding chapters also discuss variants, generalisations and potential applications.

The three main themes of this book, probability theory, differential geometry, and the theory of integrable systems, reflect the broad range of mathematical interests of Henry McKean, to whom it is dedicated. Written by experts in probability, geometry, integrable systems, turbulence, and percolation, the seventeen papers included here demonstrate a wide variety of techniques that have been developed to solve various mathematical problems in these areas. The topics are often combined in an unusual and interesting fashion to give solutions outside of the standard methods. The papers contain some exciting results and offer a guide to the contemporary literature on these subjects.

The purpose of this 1982 book is to present an introduction to developments which had taken place in finite group theory related to finite geometries. This book is practically self-contained and readers are assumed to have only an elementary knowledge of linear algebra. Among other things, complete descriptions of the following theorems are given in this book; the nilpotency of Frobneius kernels, Galois and Burnside theorems on permutation groups of prime degree, the Omstrom Wagner theorem on projective planes, and the O'Nan and Ito theorems on characterizations of projective special linear groups. Graduate students and professionals in pure mathematics will continue to find this account of value.

This book is aimed at theoretical and mathematical physicists and mathematicians interested in modern gravitational physics. I have thus tried to use language familiar to readers working on classical and quantum gravity, paying attention both to difficult calculations and to existence theorems, and discussing in detail the current literature. The first aim of the book is to describe recent work on the problem of boundary conditions in one-loop quantum cosmology. The motivation of this research was to under stand whether supersymmetric theories are one-loop finite in the presence of boundaries, with application to the boundary-value problemsoccurring in quantum cosmology. Indeed, higher-loop calculations in the absence of boundaries are already available in the litera ture, showing that supergravity is not finite. I believe, however, that one-loop calculations in the presence of boundaries are more fundamental, in that they provide a more direct check of the inconsistency of supersymmetric quantum cosmology from the perturbative point of view. It therefore appears that higher-order calculations are not strictly needed, if the one-loop test already yields negative results. Even though the question is not yet settled, this research has led to many interesting, new applications of areas of theoretical and mathematical physics such as twistor theory in flat space, self-adjointness theory, the generalized Riemann zeta-function, and the theory of boundary counterterms in super gravity. I have also compared in detail my work with results by other authors, explaining, whenever possible, the origin of different results, the limits of my work and the unsolved problems."

Dedicated to the memory of Chih-Han Sah, this volume continues a long tradition of one of the most influential mathematical seminars of this century. A number of topics are covered, including combinatorial geometry, connections between logic and geometry, Lie groups, algebras and their representations. An additional area of importance is noncommutative algebra and geometry, and its relations to modern physics. Distinguished mathematicians contributing to this work: T.V. Alekseevskaya V. Kac A.V. Borovik A. Kazarnovsky-Krol C.-H. Sah* M. Kontsevich G. Cherlin A. Radul J.L. Dupont A.L. Rosenberg I.M. Gelfand N. White The Gelfand Mathematical Seminar volumes stimulate the birth of significant ideas in contemporary mathematics and remain invaluable reference material. * indicates deceased contributor (Production: please ensure that appropriate symbol be incorporated onto the final back cover design)