Notices of the American Mathematical Society

The problem of approximating a given quantity is one of the oldest challenges faced by mathematicians. Its increasing importance in contemporary mathematics has created an entirely new area known as Approximation Theory. The modern theory was initially developed along two divergent schools of thought: the Eastern or Russian group, employing almost exclusively algebraic methods, was headed by Chebyshev together with his coterie at the Saint Petersburg Mathematical School, while the Western mathematicians, adopting a more analytical approach, included Weierstrass, Hilbert, Klein, and others. This work traces the history of approximation theory from Leonhard Euler's cartographic investigations at the end of the 18th century to the early 20th century contributions of Sergei Bernstein in defining a new branch of function theory. One of the key strengths of this book is the narrative itself. The author combines a mathematical analysis of the subject with an engaging discussion of the differing philosophical underpinnings in approach as demonstrated by the various mathematicians. This exciting exposition integrates history, philosophy, and mathematics. While demonstrating excellent technical control of the underlying mathematics, the work is focused on essential results for the development of the theory. The exposition begins with a history of the forerunners of modern approximation theory, i.e., Euler, Laplace, and Fourier. The treatment then shifts to Chebyshev, his overall philosophy of mathematics, and the Saint Petersburg Mathematical School, stressing in particular the roles played by Zolotarev and the Markov brothers. A philosophical dialectic then unfolds, contrasting East vs. West, detailing the work of Weierstrass as well as that of the Goettingen school led by Hilbert and Klein. The final chapter emphasizes the important work of the Russian Jewish mathematician Sergei Bernstein, whose constructive proof of the Weierstrass theorem and extension of Chebyshev's work serve to unify East and West in their approaches to approximation theory. Appendices containing biographical data on numerous eminent mathematicians, explanations of Russian nomenclature and academic degrees, and an excellent index round out the presentation. A Tribute To The Vision And Legacy Of Israel Moiseevich Gelfand, The Invited Papers In This Volume Reflect The Unity Of Mathematics As A Whole, With Particular Emphasis On The Many Connections Among The Fields Of Geometry, Physics, And Representation Theory. Written By Leading Mathematicians, The Text Is Broadly Divided Into Two Sections: The First Is Devoted To Developments At The Intersection Of Geometry And Physics, And The Second To Representation Theory And Algebraic Geometry. Topics Include Conformal Field Theory, K-theory, Noncommutative Geometry, Gauge Theory, Representations Of Infinite-dimensional Lie Algebras, And Various Aspects Of The Langlands Program. Graduate Students And Researchers Will Benefit From And Find Inspiration In This Broad And Unique Work, Which Brings Together Fundamental Results In A Number Of Disciplines And Highlights The Rewards Of An Interdisciplinary Approach To Mathematics And Physics.--publisher's Website. Talk Given At The Dinner At Royal East Restaurant On September 3, 2003 / Israel M. Gelfand -- Mathematics As An Adequate Language / Israel M. Gelfand -- Interaction Between Geometry And Physics / Michael Atiyah -- Unlenbeck Spaces Via Affine Lie Algebras / Alexander Braverman, Michael Finkelberg And Dennis Gaitsgory -- New Questions Related To The Topological Degree / Haim Brezis -- Quantum Cobordisms And Formal Group Laws / Tom Coates And Alexander Givental -- On The Foundations Of Noncommutative Geometry / A. Connes -- Stable Distributions Supported On The Nilpotent Cone For The Group G[subscript 2] / Stephen Debacker And David Kazhdan -- Infinite-dimensional Vector Bundles In Algebraic Geometry : An Introduction / Vladimir Drinfeld -- Algebraic Lessons From The Theory Of Quantum Integrable Models / L. D. Faddeev -- Affine Structures And Non-archimedean Analytic Spaces / Maxim Kontsevich And Yan Soibelman -- Gelfand-zeitlin Theory From The Perspective Of Classical Mechanics. Ii / Bertram Kostant And Nolan Wallach -- Mirror Symmetry And Localizations / Chien-hao Liu, Kefeng Liu And Shing-tung Yau -- Character Sheaves And Generalizations / G. Lusztig -- Symplectomorphism Groups And Quantum Cohomology / Dusa Mcduff -- Algebraic Structure Of Yang-mills Theory / M. Movshev And A. Schwarz -- Seiberg-witten Theory And Random Partitions / Nikita A. Nekrasov And Andrei Okounkov -- Quantum Calabi-yau And Classical Crystals / Andrei Okounkov, Nikolai Reshetikhin And Cumrun Vafa -- Gelfand-tsetlin Algebras, Expectations, Inverse Limits, Fourier Analysis / A. M. Vershik. Pavel Etingof, Vladimir Retakh, I.m. Singer, Editors. Includes Bibliographical References. This book traces the history of approximation theory from Leonhard Euler's cartographic investigations at the end of the 18th century to the early 20th century work of Sergei Bernstein defining a new branch of function theory.Two different schools of thought are treated: the Russian group,employing almost exclusively algebraic methods, and Western mathematicians,who developed a more analytical approach. The author presents a mathematical analysis of the subject together with a discussion of the philosophical underpinnings demonstrated by the differences in approach. The expositionisexciting: history, philosophy, and mathematics are integrated. While demonstrating excellent technical control of the underlying mathematics, the work is focused on the essential resultsin the development of the theory. The exposition begins with a history of the forerunners of modern approximation theory, i.e. Euler, Laplace, and Fourier. It then plunges into the work of Chebyshev - his overall philosophy of mathematics - and the Saint Petersburg Mathematical School, with the key focus on the roles played by Zolotarev and the Markov brothers. The Russian approach is contrasted with that of Weierstrass' as well as the Gottingen school led by Hilbert and Klein. The final chapter stresses the important work ofSergei Bernstein who, in extending the work of Chebyshev and giving a constructive proof of the Weierstrass theorem, unifies East and West in their approaches to approximation theory. Biographical data about some sixty eminent mathematicians and an excellent index round out the work. Historians as well as students interested in the history of mathematics will find this book useful. TOC:Preface * Introduction* Forerunners * Pafnutii L'vovic Cebyshev * St. Petersburg Mathematical School * Developments Outside of Russia * Constructive Function Theory * Biographies of other Contributors * Explanations * Translations of Original Documents * Bibliography by Ivor Grattan-Guinness One of the distortions in most kinds of history is an imbalance between the study devoted to major figures and to lesser ones, concerning both achievements and influence: the Great Ones may be studied to death while the others are overly ignored and thereby remain underrated. In my own work in the history of mathematics I have noted at least a score of outstanding candidates for neglect, of whom Mario Pieri (1860–1913) is one. A most able contributor to geometry, arithmetic and mathematical analysis, and mat- matical logic during his rather short life, his work and its legacy are not well known. The main reason is that Pieri worked “in the shadow of giants,” to quote one of the authors 1 of this volume. Born into a scholarly family in Lucca, Pieri was educated briefly at the University of Bologna and principally at the prestigious Scuola Normale Superiore, in Pisa; under the influence of Luigi Bianchi (1856–1928) he wrote there his doctoral dissertations on al-braic and differential geometry. During his twenties came appointments in Turin, first at the military academy and then also at the university, where he fell under the sway of Corrado Segre (1863–1924) in algebraic geometry, and Giuseppe Peano (1858–1932) in the foundations of arithmetic, mathematical analysis, and mathematical logic. From 1900 to 1908 he held a chair at the University of Catania before moving to Parma, where he died from cancer.A list of errata can be found on the author Smith's personal webpage.

103 Trigonometry Problems contains highly-selected problems and solutions used in the training and testing of the USA International Mathematical Olympiad (IMO) team. Though many problems may initially appear impenetrable to the novice, most can be solved using only elementary high school mathematics techniques.

Key features:

* Gradual progression in problem difficulty builds and strengthens mathematical skills and techniques

* Basic topics include trigonometric formulas and identities, their applications in the geometry of the triangle, trigonometric equations and inequalities, and substitutions involving trigonometric functions

* Problem-solving tactics and strategies, along with practical test-taking techniques, provide in-depth enrichment and preparation for possible participation in various mathematical competitions

* Comprehensive introduction (first chapter) to trigonometric functions, their relations and functional properties, and their applications in the Euclidean plane and solid geometry expose advanced students to college level material

103 Trigonometry Problems is a cogent problem-solving resource for advanced high school students, undergraduates, and mathematics teachers engaged in competition training.

Other books by the authors include 102 Combinatorial Problems: From the Training of the USA IMO Team (0-8176-4317-6, 2003) and A Path to Combinatorics for Undergraduates: Counting Strategies (0-8176-4288-9, 2004).

A tribute to the vision and legacy of Israel Moiseevich Gelfand, the invited papers in this volume reflect the unity of mathematics as a whole, with particular emphasis on the many connections among the fields of geometry, physics, and representation theory. Written by leading mathematicians, the text is broadly divided into two sections: the first is devoted to developments at the intersection of geometry and physics, and the second to representation theory and algebraic geometry. Topics include conformal field theory, K-theory, noncommutative geometry, gauge theory, representations of infinite-dimensional Lie algebras, and various aspects of the Langlands program. Graduate students and researchers will benefit from and find inspiration in this broad and unique work, which brings together fundamental results in a number of disciplines and highlights the rewards of an interdisciplinary approach to mathematics and physics. Contributors: M. Atiyah; A. Braverman; H. Brezis; T. Coates; A. Connes; S. Debacker; V. Drinfeld; L.D. Faddeev; M. Finkelberg; D. Gaitsgory; I.M. Gelfand; A. Givental; D. Kazhdan; M. Kontsevich; B. Kostant; C-H. Liu; K. Liu; G. Lusztig; D. McDuff; M. Movshev; N.A. Nekrasov; A. Okounkov; N. Reshetikhin; A. Schwarz; Y. Soibelman; C. Vafa; A.M. Vershik; N. Wallach; and S-T. Yau.

The congruences of a lattice form the congruence lattice. In the past half-century, the study of congruence lattices has become a large and important field with a great number of interesting and deep results and many open problems. This self-contained exposition by one of the leading experts in lattice theory, George Grätzer, presents the major results on congruence lattices of finite lattices featuring the author's signature Proof-by-Picture method and its conversion to transparencies.

Key features:

* Includes the latest findings from a pioneering researcher in the field

* Insightful discussion of techniques to construct nice finite lattices with given congruence lattices and nice congruence-preserving extensions

* Contains complete proofs, an extensive bibliography and index, and nearly 80 open problems

* Additional information provided by the author online at:

http://www.maths.umanitoba.ca/homepages/gratzer.html/

The book is appropriate for a one-semester graduate course in lattice theory, yet is also designed as a practical reference for researchers studying lattices.

The congruences of a lattice form the congruence lattice. In the past half-century, the study of congruence lattices has become a large and important field with a great number of interesting and deep results and many open problems. This self-contained exposition by one of the leading experts in lattice theory, George Grätzer, presents the major results on congruence lattices of finite lattices featuring the author's signature "Proof-by-Picture" method and its conversion to transparencies. Key features: * Includes the latest findings from a pioneering researcher in the field * Insightful discussion of techniques to construct "nice" finite lattices with given congruence lattices and "nice" congruence-preserving extensions * Contains complete proofs, an extensive bibliography and index, and nearly 80 open problems * Additional information provided by the author online at: http://www.maths.umanitoba.ca/homepages/gratzer.html/ The book is appropriate for a one-semester graduate course in lattice theory, yet is also designed as a practical reference for researchers studying lattices 103 Trigonometry Problems contains highly-selected problems and solutions used in the training and testing of the USA International Mathematical Olympiad (IMO) team. Though many problems may initially appear impenetrable to the novice, most can be solved using only elementary high school mathematics techniques. Key features: Gradual progression in problem difficulty builds and strengthens mathematical skill and techniques ; Basic topics include trigonometric formulas and identities, their applications in the geometry of the triangle, trigonometric equations and inequalities, and substitutions involving trigonometric functions ; Problem-solving tactics and strategies, along with practical test-taking techniques, provide in-depth enrichment and preparation for possible participation in various mathematical competitions ; Comprehensive introduction (first chapter) to trigonometric functions, their relations and functional properties, and their applications in the Euclidean plane and solid geometry expose advanced students to college level material. -- from back cover A tribute to the vision and legacy of Israel Moiseevich Gelfand, the invited papers in this volume reflect the unity of mathematics as a whole, with particular emphasis on the many connections between the fields of geometry, physics, and representation theory. Written by distinguished mathematicians, the text is divided as follows: several articles are devoted to developments at the intersection of geometry and physics, while others treat the interaction of representation theory, Poisson- and algebraic geometry. Topics include conformal field theory, K-theory, noncommutative geometry, gauge theory, representations of infinite-dimensional Lie algebras, Gelfand-Kirillov theory, and various aspects of the Langlands program.Graduate students and researchers will benefit from and find inspiration in this broad and unique work, which brings together fundamental results in a number of disciplines and highlights the rewards of an interdisciplinary approach to mathematics and physics. Cover Table of Contents Feature Articles Topological Fluid Dynamics Foolproof: A Sampling of Mathematical Folk Humor Patterns of Research in Mathematics Communications WHAT IS...an Open Book? Mathematicians Are from Mars, Math Educators Are from Venus: The Story of a Successful Collaboration Ten Years of the "New" Notices Brought to You by...AMS Staff MathSciNet Matters Commentary Opinion Letters to the Editor Kepler's Conjecture and Hales's Proof—A Book Review Departments Crossword Puzzle Mathematics People Mathematics Opportunities For Your Information Inside the AMS Reference and Book List Mathematics Calendar New Publications Offered by the AMS Classified Advertisements General Information Regarding Meetings & Conferences of the AMS Meetings & Conferences of the AMS Presenters of Papers, Atlanta Meeting Program of the Sessions, Atlanta Meeting Meetings and Conferences Table of Contents "This book introduces readers to Mario Pieri's career and his studies in foundations, both from historical and modern viewpoints, placing his life and research in context and tracing his influence on his contemporaries as well as more recent mathematicians. The text also provides a glimpse of the Italian academic world of Pieri's time, and its relationship with the developing international mathematics community. Included in this volume are the first English translations, along with analyses, of two of his most important axiomatizations - his postulates for arithmetic, which Peano judged superior to his own; and his foundation of elementary geometry on the basis of point and sphere, which Alfred Tarski used as a basis for his own system."--BOOK JACKET The congruences of a lattice form the congruence lattice. In the past half-century, the study of congruence lattices has become a large and important field with a great number of interesting and deep results and many open problems. This self-contained exposition by one of the leading experts in lattice theory, George Gratzer, presents the major results on congruence lattices of finite lattices featuring the author's signature Proof-by-Picture method and its conversion to transparencies. The book is appropriate for a one-semester graduate course in lattice theory, yet is also designed as a practical reference for researchers studying lattices. This book is the first in a series of three volumes that comprehensively examine Mario Pieris life, mathematical work and influence. The book introduces readers to Pieris career and his studies in foundations, from both historical and modern viewpoints. Included in this volume are the first English translations, along with analyses, of two of his most important axiomatizations one in arithmetic and one in geometry. The book combines an engaging exposition, little-known historical notes, exhaustive references and an excellent index. And yet the book requires no specialized experience in mathematical logic or the foundations of geometry. Tribute to the vision and legacy of Israel Moiseevich Gel'fand Written by leading mathematicians, these invited papers reflect the unity of mathematics as a whole, with particular emphasis on the many connections among the fields of geometry, physics, and representation theory Topics include conformal field theory, K-theory, noncommutative geometry, gauge theory, representations of infinite-dimensional Lie algebras, and various aspects of the Langlands program * Exciting exposition integrates history, philosophy, and mathematics * Combines a mathematical analysis of approximation theory with an engaging discussion of the differing philosophical underpinnings behind its development * Appendices containing biographical data on numerous eminent mathematicians, explanations of Russian nomenclature and academic degrees, and an excellent index round out the presentation "103 Trigonometry Problems is a cogent problem solving resource for advanced high school students, undergraduate and mathematics teachers engaged in competition training."--Jacket Karl-georg Steffens. Includes Bibliographical References (p. [201]-216) And Index.