Mathematical Analysis During The 20th Century

For several centuries, analysis has been one of the most prestigious and important subjects in mathematics. The present book sets off by tracing the evolution of mathematical analysis, and then endeavours to understand the developments of main trends, problems, and conjectures. It features chapters on general topology, 'classical' integration and measure theory, functional analysis, harmonic analysis and Lie groups, theory of functions and analytic geometry, differential and partial differential equations, topological and differential geometry. The ubiquitous presence of analysis also requires the consideration of related topics such as probability theory or algebraic geometry. Each chapter features a comprehensive first part on developments during the period 1900-1950, and then provides outlooks on representative achievements during the later part of the century. The book provides many original quotations from outstanding mathematicians as well as an extensive bibliography of the seminal publications. It will be an interesting and useful reference work for graduate students, lecturers, and all professional mathematicians and other scientists with an interest in the history of mathematics.

'Will be a valuable source book for analysts interested in the history of the main ideas of analysis, as well as for others wanting to know about developments in other fields.' -EMS'This is a superb history of 20th century mathematical analysis.' -Zentralblatt MathematikThis book studies the 20th century evolution of essential ideas in mathematical analysis, a field that since the times of Newton and Leibnitz has been one of the most important and prestigious in mathematics. Each chapter features a comprehensive first part on developments during the period 1900-1950, and then provides outlooks on representative achievements during the later part of the century. The book will be an interesting and useful reference for graduate students and lecturers in mathematics, professional mathematicians and historians of science, as well as the interested layperson.

A glorious period of Hungarian mathematics started in 1900 when Lipót Fejér discovered the summability of Fourier series.This was followed by the discoveries of his disciples in Fourier analysis and in the theory of analytic functions. At the same time Frederic (Frigyes) Riesz created functional analysis and Alfred Haar gave the first example of wavelets. Later the topics investigated by Hungarian mathematicians broadened considerably, and included topology, operator theory, differential equations, probability, etc. The present volume, the first of two, presents some of the most remarkable results achieved in the twentieth century by Hungarians in analysis, geometry and stochastics. The book is accessible to anyone with a minimum knowledge of mathematics. It is supplemented with an essay on the history of Hungary in the twentieth century and biographies of those mathematicians who are no longer active. A list of all persons referred to in the chapters concludes the volume.

Numerical analysis has witnessed many significant developments in the 20th century. This book brings together 16 papers dealing with historical developments, survey papers and papers on recent trends in selected areas of numerical analysis, such as: approximation and interpolation, solution of linear systems and eigenvalue problems, iterative methods, quadrature rules, solution of ordinary-, partial- and integral equations. The papers are reprinted from the 7-volume project of the Journal of Computational and Applied Mathematics on '/homepage/sac/cam/na2000/index.htmlNumerical Analysis 2000'. An introductory survey paper deals with the history of the first courses on numerical analysis in several countries and with the landmarks in the development of important algorithms and concepts in the field.

Analysis as an independent subject was created as part of the scientific revolution in the seventeenth century. Kepler, Galileo, Descartes, Fermat, Huygens, Newton, and Leibniz, to name but a few, contributed to its genesis. Since the end of the seventeenth century, the historical progress of mathematical analysis has displayed unique vitality and momentum. No other mathematical field has so profoundly influenced the development of modern scientific thinking. Describing this multidimensional historical development requires an in-depth discussion which includes a reconstruction of general trends and an examination of the specific problems. This volume is designed as a collective work of authors who are proven experts in the history of mathematics. It clarifies the conceptual change that analysis underwent during its development while elucidating the influence of specific applications and describing the relevance of biographical and philosophical backgrounds. The first ten chapters of the book outline chronological development and the last three chapters survey the history of differential equations, the calculus of variations, and functional analysis. Special features are a separate chapter on the development of the theory of complex functions in the nineteenth century and two chapters on the influence of physics on analysis. One is about the origins of analytical mechanics, and one treats the development of boundary-value problems of mathematical physics (especially potential theory) in the nineteenth century. The book presents an accurate and very readable account of the history of analysis. Each chapter provides a comprehensive bibliography. Mathematical examples have been carefully chosen so that readers with a modest background in mathematics can follow them. It is suitable for mathematical historians and a general mathematical audience.

From the Calculus to Set Theory traces the development of the calculus from the early seventeenth century through its expansion into mathematical analysis to the developments in set theory and the foundations of mathematics in the early twentieth century. It chronicles the work of mathematicians from Descartes and Newton to Russell and Hilbert and many, many others while emphasizing foundational questions and underlining the continuity of developments in higher mathematics. The other contributors to this volume are H. J. M. Bos, R. Bunn, J. W. Dauben, T. W. Hawkins, and K. Møller-Pedersen.

In the 20th century, many mathematicians in Russia made great contributions to the field of mathematics. This invaluable book, which presents the main achievements of Russian mathematicians in that century, is the first most comprehensive book on Russian mathematicians. It has been produced as a gesture of respect and appreciation for those mathematicians and it will serve as a good reference and an inspiration for future mathematicians. It presents differences in mathematical styles and focuses on Soviet mathematicians who often discussed “what to do” rather than “how to do it”. Thus, the book will be valued beyond historical documentation. The editor, Professor Yakov Sinai, a distinguished Russian mathematician, has taken pains to select leading Russian mathematicians — such as Lyapunov, Luzin, Egorov, Kolmogorov, Pontryagin, Vinogradov, Sobolev, Petrovski and Krein — and their most important works. One can, for example, find works of Lyapunov, which parallel those of Poincaré; and works of Luzin, whose analysis plays a very important role in the history of Russian mathematics; Kolmogorov has established the foundations of probability based on analysis. The editor has tried to provide some parity and, at the same time, included papers that are of interest even today. The original works of the great mathematicians will prove to be enjoyable to readers and useful to the many researchers who are preserving the interest in how mathematics was done in the former Soviet Union. Contents:Lyapunov (A New Case of Integrability of Differential Equations of Motion of a Solid Body in Liquid)Luzin (Sur l'absolue convergence des series trigonometriques)SteklovEgorov (Mathematics and Religion in Moscow, by C E Ford)Smirnov (Sur les polynomes orthogonaux a une veriable complexe)Bernstein (Sur la meilleure approximation sur tout l'axe reel des fonctions continues par des fonctions entieres de degre fini)UrysohnChebotaryovVinogradov (Representation of an Odd Number as the Sum of Three Primes)Aleksandrov (Sur la notion de dimension des ensembles fermes)MenshovGelfond (Sur le septierie probleme de Hilbert)Khinchin (Three Pearls of Number Theory)Kolmogorov (Local Structure of Turbulence in an Incompressible Viscous Fluid at Very Large Reynolds Numbers)Pontryagin (Homotopic Classification of an (n+2)-Dimensional Spheres into an n-Dimensional Spheres)Gelfand (On Identities for Eigenvalues of a Second Order Differential Operators)Sobolev (On a Theorem of Functional Analysis)Petrovsky (On Problem of some PDE's)Krein (On Extreme Points of Regularly Convex Sets)Liusternik (Topology and Variational Problem)Rokhlin (Proof of Gudkov's Hypothesis)Novikov (Periodic Groups)Bogoliubov (Mathematical Problems of Quantum Field Theory)Aleksandrov (Neue ungleichungen fur die mischvolumen konvexer korper)Kantorovich (A New Method of Solving of Some Classes of Extremal Problems)Malcev (Free Topological Algebras)Linnik (An Application of the Theory of Matrices and of Lobatschevskian Geometry to the Theory of Dirichlet's Real Characters)Markov (The Theory of Algorithms)Lavrentev (On the Theory of Quasi-Conformal Mapping of Three-Dimensional Domains)Tikhonov (Ueber die Erweiteung von Raumen)Delone (Sur le nombre de representations d'un nombre par une forme eubique a discriminent negatif)Keldysh (On the Completeness of the Eigenfunctions of Some Classes of Non-Self Adjoint Linear Operators)Faddeevand other articles Readership: General mathematicians. Keywords:Geometry & Topology;Analysis & Differential Equations;Algebra & Number TheoryReviews:“For anyone who wants an overview of mathematics in Russia during the 20th century there is now the volume Russian Mathematicians in the 20th century … It shall remain on my book shelf as a monument over a heroic generation.”Professor Lennart Carleson Institute of Mathematics, The Royal Institute of Technology, Stockholm, Sweden “The list selected is very representative both topically and geographically. It covers research in all areas of mathematics … The 33 persons in the list worked not only in Moscow and Leningrad (now Saint Petersburg), but also in Kiev, Odessa, Kazan, and Novosibirsk. Most of the work presented in this volume was done during the Soviet era when the Russian mathematical community was artificially isolated from the international one for political reasons. Thus to develop their subjects, Soviet mathematicians needed to be self-sufficient. And this volume shows that they indeed succeeded in it. The originality of the Russian mathematical school is clearly seen when one reads the papers included in the book. Altogether this volume gives a very strong impression of the versatility, originality and strength of the Russian mathematical school.”L D Faddeev Petersburg Department of the Steklov Institute of Mathematics , Russian Academy of Sciences “This book is fascinating … It shows the greatness of Russian or Soviet mathematicians and the foundations on which younger mathematicians could build up, leading to world leadership until the end of the Soviet Union when the exodus started.”F Hirzebruch Emeritus Professor of Mathematics University of Bonn

This book contains several contributions on the most outstanding events in the development of twentieth century mathematics, representing a wide variety of specialities in which Russian and Soviet mathematicians played a considerable role. The articles are written in an informal style, from mathematical philosophy to the description of the development of ideas, personal memories and give a unique account of personal meetings with famous representatives of twentieth century mathematics who exerted great influence in its development. This book will be of great interest to mathematicians, who will enjoy seeing their own specialities described with some historical perspective. Historians will read it with the same motive, and perhaps also to select topics for future investigation.