Notices of the American Mathematical Society

This book is an outgrowth of lectures given on several occasions at Chalmers University of Technology and Goteborg University during the last ten years. As opposed to most introductory books on complex analysis, this one as­ sumes that the reader has previous knowledge of basic real analysis. This makes it possible to follow a rather quick route through the most fundamen­ tal material on the subject in order to move ahead to reach some classical highlights (such as Fatou theorems and some Nevanlinna theory), as well as some more recent topics (for example, the corona theorem and the HI_ BMO duality) within the time frame of a one-semester course. Sections 3 and 4 in Chapter 2, Sections 5 and 6 in Chapter 3, Section 3 in Chapter 5, and Section 4 in Chapter 7 were not contained in my original lecture notes and therefore might be considered special topics. In addition, they are completely independent and can be omitted with no loss of continuity. The order of the topics in the exposition coincides to a large degree with historical developments. The first five chapters essentially deal with theory developed in the nineteenth century, whereas the remaining chapters contain material from the early twentieth century up to the 1980s. Choosing methods of presentation and proofs is a delicate task. My aim has been to point out connections with real analysis and harmonic anal­ ysis, while at the same time treating classical complex function theory. On The Subject Of Differential Equations Many Elementary Books Have Been Written. This Book Bridges The Gap Between Elementary Courses And Research Literature. The Basic Concepts Necessary To Study Differential Equations - Critical Points And Equilibrium, Periodic Solutions, Invariant Sets And Invariant Manifolds - Are Discussed First. Stability Theory Is Then Developed Starting With Linearisation Methods Going Back To Lyapunov And Poincaré. In The Last Four Chapters More Advanced Topics Like Relaxation Oscillations, Bifurcation Theory, Chaos In Mappings And Differential Equations, Hamiltonian Systems Are Introduced, Leading Up To The Frontiers Of Current Research: Thus The Reader Can Start To Work On Open Research Problems, After Studying This Book. This New Edition Contains An Extensive Analysis Of Fractal Sets With Dynamical Aspects Like The Correlation- And Information Dimension. In Hamiltonian Systems, Topics Like Birkhoff Normal Forms And The Poincaré-birkhoff Theorem On Periodic Solutions Have Been Added. There Are Now 6 Appendices With New Material On Invariant Manifolds, Bifurcation Of Strongly Nonlinear Self-excited Systems And Normal Forms Of Hamiltonian Systems. The Subject Material Is Presented From Both The Qualitative And The Quantitative Point Of View, And Is Illustrated By Many Examples. Introduction -- Autonomous Equations -- Critical Points -- Periodic Solutions -- Introduction To The Theory Of Stability -- Linear Equations -- Stability By Linearisation -- Stability Analysis By The Direct Method -- Introduction To Perturbation Theory -- The Poincare-lindstedt Method -- The Method Of Averaging -- Relaxation Oscillations -- Bifurcation Theory -- Chaos -- Hamiltonian Systems. Ferdinand Verhulst. Includes Bibliographical References (p.295-300) And Index. Assuming Only Calculus And Linear Algebra, This Book Introduces The Reader In A Technically Complete Way To Measure Theory And Probability, Discrete Martingales, And Weak Convergence. It Is Self- Contained And Rigorous With A Tutorial Approach That Leads The Reader To Develop Basic Skills In Analysis And Probability. While The Original Goal Was To Bring Discrete Martingale Theory To A Wide Readership, It Has Been Extended So That The Book Also Covers The Basic Topics Of Measure Theory As Well As Giving An Introduction To The Central Limit Theory And Weak Convergence. Students Of Pure Mathematics And Statistics Can Expect To Acquire A Sound Introduction To Basic Measure Theory And Probability. A Reader With A Background In Finance, Business, Or Engineering Should Be Able To Acquire A Technical Understanding Of Discrete Martingales In The Equivalent Of One Semester. J. C. Taylor Is A Professor In The Department Of Mathematics And Statistics At Mcgill University In Montreal. He Is The Author Of Numerous Articles On Potential Theory, Both Probabilistic And Analytic, And Is Particularly Interested In The Potential Theory Of Symmetric Spaces. Ch. I. Probability Spaces -- Ch. Ii. Integration -- Ch. Iii. Independence And Product Measures -- Ch. Iv. Convergence Of Random Variables And Measurable Functions -- Ch. V. Conditional Expectation And An Introduction To Martingales -- Ch. Vi. An Introduction To Weak Convergence. J.c. Taylor. Includes Bibliographical References And Index.

This collection, based on several of Lang's "Files," deals with the area where science and academia meet the worlds of journalism and politics: social organization, government, and the roles that education and journalism play in shaping opinions leading to policy decisions. In discussing specific cases in which he became involved, Lang addresses general questions of standards: standards of journalism, standards of discourse, and standards of science. Recurring questions concern: - How people process information and how misinformation is spread and accepted - Inhibition of critical thinking and the role of education: teaching students to think clearly and independently — or conditioning them to accept dominant modes of perception uncritically - How to make corrections, and how attempts at corrections are sometimes obstructed - The extent to which we submit to the authority of those higher up, and whether one can keep the higher ups accountable, possibly in the face of evasions, stonewalling, and intimidation - The competence of so-called experts - Our responsibility for what we say or write - The use of editorial and academic power to suppress or marginalize ideas, evidence, or data that do not fit the tenets of certain establishments By dealing with case studies and providing extensive documentation, Lang challenges some individuals and establishments, at the same time that he challenges us to reconsider the ways they exercise their

In 1988 Shafarevich asked me to write a volume for the Encyclopaedia of Mathematical Sciences on Diophantine Geometry. I said yes, and here is the volume. By definition, diophantine problems concern the solutions of equations in integers, or rational numbers, or various generalizations, such as finitely generated rings over Z or finitely generated fields over Q. The word Geometry is tacked on to suggest geometric methods. This means that the present volume is not elementary. For a survey of some basic problems with a much more elementary approach, see [La 9Oc]. The field of diophantine geometry is now moving quite rapidly. Out standing conjectures ranging from decades back are being proved. I have tried to give the book some sort of coherence and permanence by em phasizing structural conjectures as much as results, so that one has a clear picture of the field. On the whole, I omit proofs, according to the boundary conditions of the encyclopedia. On some occasions I do give some ideas for the proofs when these are especially important. In any case, a lengthy bibliography refers to papers and books where proofs may be found. I have also followed Shafarevich's suggestion to give examples, and I have especially chosen these examples which show how some classical problems do or do not get solved by contemporary in sights. Fermat's last theorem occupies an intermediate position. Al though it is not proved, it is not an isolated problem any more. "On the subject of differential equations many elementary books have been written. This book bridges the gap between elementary courses and research literature. The basic concepts necessary to study differential equations - critical points and equilibrium, periodic solutions, invariant sets and invariant manifolds - are discussed first. Stability theory is then developed starting with linearisation methods going back to Lyapunov and Poincare." "In the last four chapters more advanced topics like relaxation oscillations, bifurcation theory, chaos in mappings and differential equations, Hamiltonian systems are introduced, leading up to the frontiers of current research: thus the reader can start to work on open research problems, after studying this book." "This new edition contains an extensive analysis of fractal sets with dynamical aspects like the correlation and information dimension. In Hamiltonian systems, topics like Birkhoff normal forms and the Poincare-Birkhoff theorem on periodic solutions have been added. There are now 6 appendices with new material on invariant manifolds, bifurcation of strongly nonlinear self-excited systems and normal forms of Hamiltonian systems." "The subject material is presented from both the qualitative and the quantitative point of view, and is illustrated by many examples."--Jacket This collection, based on several of Lang's "Files", deals with the area where the worlds of science and academia meet those of journalism and politics: social organisation, government, and the roles that education and journalism play in shaping opinions. In discussing specific cases in which he became involved, Lang addresses general questions of standards: standards of journalism, discourse, and of science. Recurring questions concern how people process information and misinformation; inhibition of critical thinking and the role of education; how to make corrections, and how attempts at corrections are sometimes obstructed; the extent to which we submit to authority, and whether we can hold the authorities accountable; the competence of so-called experts; and the use of editorial and academic power to suppress or marginalize ideas, evidence, or data that do not fit the tenets of certain establishments. By treating case studies and providing extensive documentation, Lang challenges some individuals and establishments to reconsider the ways they exercise their official or professional responsibilities. I am very thankful to Springer-Verlag for publishing a collection of some of my non-mathematical works-I call them political works. in the broad sense of the word political. Three of these have appeared in print: - My article on the Ladd-Lipset sUIvey. which appeared in the New York Review of Books. 18 May 1978; and also in The File (Springer-Verlag. 1981). - My article on the Baltimore case. which appeared in the JourƯ nal of Ethics and Behavior. February 1993. - My articles on HIV and AIDS. which appeared in the Yale SciƯ entific (Fall 1994 and Winter 1995). reprinted updated in the book AIDS: Virus-or drug induced? Kluwer Academic PubƯ lishers. 1996. pp. 271-307. The first item. "Academia. Journalism. and Politics." is itself a book based on my Huntington file. The "Background and MotivaƯ tion" section of this sub-book can be used as a foreword for all my "political" works. and also contains an explanation of how I use the word "political." In that section. readers will find a general discussion of the way I process information and some criteria I use in discourse

from The Reviews: Between Number Theory And Geometry There Have Been Several Stimulating Influences, And This Book Records Of These Enterprises. This Author, Who Has Been At The Centre Of Such Research For Many Years, Is One Of The Best Guides A Reader Can Hope For. The Book Is Full Of Beautiful Results, Open Questions, Stimulating Conjectures And Suggestions Where To Look For Future Developments. This Volume Bears Witness Of The Braod Scope Of Knowledge Of The Author, And The Influence Of Several People Who Have Commented On The Manuscript Before Publication... Although In The Series Of Number Theory, This Volume Is On Diophantine Geometry, The Reader Will Notice That Algebraic Geometry Is Present In Every Chapter. ...the Style Of The Book Is Clear. Ideas Are Well Explained, And The Author Helps The Reader To Pass By Several Technicalities.
mededelingen Van Het Wiskundig Genootschap

This text presents a systematic and well-motivated development of differential geometry leading to the global version of Cartan connections. The material is presented at a level accessible to a first-year graduate student. The first four chapters provide a complete development of the fundamentals of differential topology, foliations, Lie groups, and homogeneous spaces. Chapter 5 studies Cartan geometries which generalize homogeneous spaces in the same way that Riemannian geometry generalizes Euclidean geometry. One of the beautiful facets of Cartan geometries is that curvature appears as an exact local measurement of "broken symmetry." The last three chapters study Riemannian geometry, conformal geometry, and projective geometry. Topics included in the five appendices are a comparison of Cartan and Ehresmann connections, and the derivation of the divergence and curl operators from symmetry considerations.

Cartan geometries were the first examples of connections on a principal bundle. They seem to be almost unknown these days, in spite of the great beauty and conceptual power they confer on geometry. The aim of the present book is to fill the gap in the literature on differential geometry by the missing notion of Cartan connections. Although the author had in mind a book accessible to graduate students, potential readers would also include working differential geometers who would like to know more about what Cartan did, which was to give a notion of "espaces généralisés" (= Cartan geometries) generalizing homogeneous spaces (= Klein geometries) in the same way that Riemannian geometry generalizes Euclidean geometry. In addition, physicists will be interested to see the fully satisfying way in which their gauge theory can be truly regarded as geometry.

This book provides a concise treatment of topics in complex analysis, suitable for a one-semester course. It is an outgrowth of lectures given by the author over the last ten years at the University of Goteborg and Chalmers University of Technology. While treating classical complex function theory, the author emphasizes connections to real and harmonic analysis, and presents general tools that might be useful in other areas of analysis. The book introduces all of the basic ideas in beginning complex analysis and still has time to reach many topics near the frontier of the subject. The reader is expected to have an understanding of basic integration theory and functional analysis. Many exercises illustrate and sharpen the theory, and extended exercises give the reader an active part in complementing the material presented in the text. Cover Table of Contents Articles Groups and Physics—Dogmatic Opinions of a Senior Citizen Communications Mathematicians as Educators Ten Lessons I Wish I Had Been Taught Review of BBC's Horizon Program, “Fermat's Last Theorem” Mathematical Journals: Past, Present and Future—A Personal View Recompetition of the NSF-Funded Mathematics Institutes Departments Mathematics People Mathematics Opportunities For Your lnformation Reference Mathematics Calendar New Publications Offered by the AMS Publications of Continuing Interest Classifieds Membership Forms Meetings and Conferences of the AMS Meetings and Conferences Table of Contents Commentary From the Editor Letters to the Editor From the AMS 1997 AMS Election Reclassification of Mathematics Departments (Corrected List) AMS Standard Cover Sheet Assuming only calculus and linear algebra, Professor Taylor introduces readers to measure theory and probability, discrete martingales, and weak convergence. This is a technically complete, self-contained and rigorous approach that helps the reader to develop basic skills in analysis and probability. Students of pure mathematics and statistics can thus expect to acquire a sound introduction to basic measure theory and probability, while readers with a background in finance, business, or engineering will gain a technical understanding of discrete martingales in the equivalent of one semester. J. C. Taylor is the author of numerous articles on potential theory, both probabilistic and analytic, and is particularly interested in the potential theory of symmetric spaces.

This book is an outgrowth of lectures given at several occasions at the University of Göteborg and Chalmers University of Technology during the last ten years. Contrary to most introductory texts on complex analysis, it preassumes knowledge of basic analysis. This makes it possible to move rather quickly through the most fundamental material and to reach within a one-semester course some classical highlights such as Fatou theorems and some Nevanlinna theory, as well as more recent topics, for example the corona theorem and the H1-BMO duality.

This book is an outgrowth of lectures given at several occasions at the University of Göteborg and Chalmers University of Technology during the last ten years. Contrary to most introductory texts on complex analysis, it preassumes knowledge of basic analysis. This makes it possible to move rather quickly through the most fundamental material and to reach within a one-semester course some classical highlights such as Fatou theorems and some Nevanlinna theory, as well as more recent topics, for example the corona theorem and the H1-BMO duality Ch. 1. Vector Spaces -- Ch. 2. Finite-dimensional Vector Spaces -- Ch. 3. Linear Maps -- Ch. 4. Polynomials -- Ch. 5. Eigenvalues And Eigenvectors -- Ch. 6. Inner-product Spaces -- Ch. 7. Operators On Inner-product Spaces -- Ch. 8. Operators On Complex Vector Spaces -- Ch. 9. Operators On Real Vector Spaces -- Ch. 10. Trace And Determinant. Sheldon Axler. Includes Indexes. For lecture courses that cover the classical theory of nonlinear differential equations associated with Poincare and Lyapunov and introduce the student to the ideas of bifurcation theory and chaos, this text is ideal. Its excellent pedagogical style typically consists of an insightful overview followed by theorems, illustrative examples, and exercises. Although several mathematicians, especially C.F. Gauss, studied the notion of a smooth manifold in special cases, the idea of an abstract manifold of arbitrary finite dimension seems to be due to Riemann. Intended for a second course in pursuing mathematics, this volume discusses topics such as the existence of eigenvalues on complex vector spaces, upper-triangular matrices and orthogonal projections. An introduction to analysis usually begins with a study of properties of R, the set of real numbers.