Spectrum – serie

This is an introduction to recent developments in algebraic combinatorics and an illustration of how research in mathematics actually progresses. The author recounts the story of the search for and discovery of a proof of a formula conjectured in the late 1970s: the number of n x n alternating sign matrices, objects that generalize permutation matrices. While apparent that the conjecture must be true, the proof was elusive. Researchers became drawn to this problem, making connections to aspects of invariant theory, to symmetric functions, to hypergeometric and basic hypergeometric series, and, finally, to the six-vertex model of statistical mechanics. All these threads are brought together in Zeilberger's 1996 proof of the original conjecture. The book is accessible to anyone with a knowledge of linear algebra. Students will learn what mathematicians actually do in an interesting and new area of mathematics, and even researchers in combinatorics will find something new here.

Flatland, Edwin Abbott's story of a two-dimensional universe, as told by one of its inhabitants who is introduced to the mysteries of three-dimensional space, has enjoyed an enduring popularity from the time of its publication in 1884. This fully annotated edition enables the modern-day reader to understand and appreciate the many 'dimensions' of this classic satire. Mathematical notes and illustrations enhance the usefulness of Flatland as an elementary introduction to higher-dimensional geometry. Historical notes show connections to late-Victorian England and to classical Greece. Citations from Abbott's other writings as well as the works of Plato and Aristotle serve to interpret the text. Commentary on language and literary style includes numerous definitions of obscure words. An appendix gives a comprehensive account of the life and work of Flatland's remarkable author.

Flatland, Edwin Abbott's story of a two-dimensional universe, as told by one of its inhabitants who is introduced to the mysteries of three-dimensional space, has enjoyed an enduring popularity from the time of its publication in 1884. This fully annotated edition enables the modern-day reader to understand and appreciate the many 'dimensions' of this classic satire. Mathematical notes and illustrations enhance the usefulness of Flatland as an elementary introduction to higher-dimensional geometry. Historical notes show connections to late-Victorian England and to classical Greece. Citations from Abbott's other writings as well as the works of Plato and Aristotle serve to interpret the text. Commentary on language and literary style includes numerous definitions of obscure words. An appendix gives a comprehensive account of the life and work of Flatland's remarkable author.

Changing the way students learn calculus at New Mexico State University. In the Spring of 1988, Marcus Cohen, Edward D. Gaughan, Arthur Knoebel, Douglas S. Kurtz, and David Penegelley began work on a student project approach to calculus. For the next two years, most of their waking hours (and some of their dreams) would be devoted to writing projects for their students and discovering how to make the use of projects in calculus classes not only successful, but practical as well. A grant from the National Science Foundation made it possible for this experiment to go forward on a large scale. The enthusiasm of the original group of five faculty was contagious, and soon other members of the department were also writing and using projects in their calculus classes. At the present time, about 80% of the calculus students at New Mexico State University are doing projects in their Calculus courses. Teachers can use their methods in teaching their own calculus courses.Student Research Projects in Calculus provides teachers with over 100 projects ready to assign to students in single and multivariable calculus. The authors have designed these projects with one goal in mind: to get students to think for themselves. Each project is a multistep, take-home problem, allowing students to work both individually and in groups. The projects resemble mini-research problems. Most of them require creative thought, and all of them engage the student's analytic and intuitive faculties. the projects often build from a specific example to the general case, and weave together ideas from many parts of the calculus. Project statements are clearly stated and contain a minimum of mathematical symbols. Students must draw their own diagrams, decide for themselves what the problem is about, and what toolsfrom the calculus they will use to solve it. This approach elicits from students an amazing level of sincere questioning, energetic research, dogged persistence, and conscientious communication. Each project has accompanying notes to the instructor, reporting students' experiences. The notes contain helpful information on prerequisites, list the main topics the project explores, and suggests helpful hints. The authors have also provided several introductory chapters to help instructors use projects successfully in their classes and begin to create their own.

Eric Temple Bell (1883-1960) was a distinguished mathematician and a best selling popularizer of mathematics. His Men of Mathematics, still in print after almost sixty years, inspired scores of young readers to become mathematicians. Under the name of John Taine, he also published science fiction novels (among them The Time Stream, Before the Dawn, and The Crystal Horde) that served to broaden the subject matter of that genre during its early years.In The Search for E. T. Bell, Constance Reid has given us a compelling account of this complicated, difficult man who never divulged to anyone, not even to his wife and son, the story of his early life and family background. Her book is thus more of a mystery than a traditional biography. It begins with the discovery of an unexpected inscription in an English churchyard and a series of cryptic notations in a boy's schoolbook. Then comes an inadvertent revelation, by Bell himself, in a respected mathematical journal. You will have to read the book to learn the rest.

Do you expect to find articles about mathematics in your daily newspaper? If you are a reader of The Guardian you do, or at least you did during the second half of the 1980s. This volume collects many of the columns Keith Devlin wrote for The Guardian. Read them and assign them to your students to read. This is a book for delving in, and is accessible to anyone with an interest in things mathematical. Devlin takes mathematical discoveries and explains them to the interested lay reader. The topics range from computer discoveries dealing with large prime numbers to much deeper results, such as Fermat's Last Theorem. You will find articles on the traveling salesman problem, on cryptology, and on procedures for working out claims for traveling expenses. Although the individual pieces are short and easily read, many contain references to mathematical articles and can form the basis for student research papers.

Long known as a mathematical storyteller, Howard Eves writes his personal reminiscences, mostly mathematical, some not. The cast of characters includes Albert Einstein, Norbert Wiener, Julian Lowell Coolidge, Maurice Frechet, Nathan Altshiller-Court, G. H. Hardy, and many other interesting figures whom he encountered in a long and active life in mathematics. In Mathematical Reminiscences, we read of Eves' famous mathematical museum, with its lock of Einstein's hair, Hardy's scarf, Veblen's bright yellow lead pencil, Wiener's hat, and much, much, more.

The anthology is a collection of articles contiguous to the humanities written by renowned mathematicians of the twentieth century. The articles cover a variety of topics that, for want of a better name, shall be referred to as humanistic. An important criterion, thereby limiting the choice, is that the articles should be accessible to the literate reader who may or may not have technical knowledge of mathematics.The articles span roughly a century in time and a wide range in subject. They are by mathematicians acknowledged by their peers as outstanding creators whose work has added richly to the discipline. Each article is preceded by a brief biographical sketch of the author and a brief indication of the content. The material is accessible to a wide audience, lay as well as academic.

R. L. Moore: Mathematician and Teacher presents a full and frank biography of a mathematician recognized as one of the principal figures in the 20th Century progression of the American school of point set topology. He was equally well known as creator of the Moore Method (no textbooks, no lectures, no conferring) in which there is a current and growing revival of interest and modified application under inquiry-based learning projects in both the United States and the United Kingdom.Parker draws on oral history, with first-person recollections from many leading figures in the American mathematics community of the last half-century. The story embraces some of the most famous and influential mathematical names in America and Europe from the late 1900s in what is undoubtedly a lively account of this controversial figure, once described as Mr. Chips with Attitude. He was the first American to become a Visiting Lecturer for the American Mathematical Society, was a member of the National Academy of Sciences, published 68 papers and a book that is still referred to seventy years later and that has been the subject of literally hundreds of papers by other mathematicians around the globe. Three of Moore's students followed him as president of the American Mathematical Society, and three others became vice-presidents. Five served as president of the Mathematical Association of America, and three became members of the National Academy of Sciences.

Where can you hear an old man in a nursing home tell of how he disproved Goldbach's Conjecture as part of an undergraduate research project? Get inside the mind of James Clerk Maxwell as he discovers electro-magnetic waves? Witness the 19th century's greatest mathematician finally get the credit she was denied by sexism and murder? In Reality Conditions, a collection of short stories spanning a variety of genres, you can share in these and other fictional mathematical experiences.Each story is a mathematical journey designed to entertain, educate and tantalize. There is something here for everyone: humor, drama, the little details to the big picture, science fiction to true histories. Through these stories, those with little mathematical background will encounter some of the most interesting parts of the field of mathematics for the first time. Even professional mathematicians will be captivated by ideas that take us to the limits of knowledge, addressing the questions of how mathematics is related to the human mind and how it is related to reality. The book is perfect for leisure reading. Join one of the leading experts in the area of mathematical fiction as he takes us on a whirlwind tour of mathematics, both real and imaginary.

Translated from the original German by Peter Hilton and Jean Pedersen. The 99 points of intersection presented here were collected during a year-long search for surprising concurrence of lines. For each example we find compelling evidence for the sometimes startling fact that in a geometric figure three straight lines, or sometimes circles, pass through one and the same point. Of course, we are familiar with some examples of this from basic elementary geometry - the intersection of medians, altitudes, angle bisectors, and perpendicular bisectors of sides of a triangle. Here there are many more examples - some for figures other than triangles, some where even more than three straight lines pass through a common point. The main part of the book presents 99 points of intersection purely visually. They are developed in a sequence of figures, many without caption or verbal commentary. In addition the book contains general thoughts on and examples of the points of intersection, as well as some typical methods of proving their existence. Many of the examples shown in the book were inspired by questions and suggestions made by students and high-school teachers. Several of those examples have not only a geometrical, but also an intriguing aesthetic, aspect. The book addresses high-school students and students at the undergraduate level as well as their teachers, but will appeal to anyone interested in geometry.

Beautifully printed with 24 pages of full color. Ideal for Math Clubs. Math Horizons is a magazine that celebrates the people and ideas which are mathematics. Containing the editor.s selections from the first ten years of the magazine.s existence, this volume features exquisite expositions of undergraduate-level mathematics. Broad and appealing, the coverage also includes fiction with mathematical themes; literary, theatrical, and cinematic criticism; humor; history; and social history. Mathematics is shown as a human endeavor through biographies and interviews of mathematicians and users of mathematics including artists, writers, and scientists. The puzzles, games, and activities throughout make it a valuable resource for student math clubs. Though especially appealing to students of mathematics from high school to graduate school and their teachers, this collection is an eclectic and wide-ranging look at the culture of mathematics, and offers enjoyable reading for anyone with an interest in mathematics.

A fascinating collection of numerical curiosities and wonders. The Professor in Owen O Shea s book is the imaginary American Richard Stein. As Owen O Shea and the Professor travel through Ireland, O Shea notes the Professor s collection of amazing magic numbers in fascinating detail. His mathematical curiosities are wide ranging, concerning the 1915 sinking of the Lusitania to coincidences about Apollo 11 to the first moon walk to new numerical curiosities. The new curiosities, among many others, center on Presidents Lincoln and Kennedy, the USA and Ireland, the two World Wars, the King James Version of the Bible, and James Joyce.The Magic Numbers of the Professor reveals astonishing details about the year 1776; the year of American Independence. It contains discussions on prime numbers, gives some wonderful number patterns, and reveals many other eye-opening properties of numbers. It asks, for instance, if you know in how many different ways a US dollar can be changed. The Professor gives the answer to this and other currency questions. The number of the Beast 666, is discussed as well, as are many new equations involving that famous number all appearing here for the first time. And for those fascinated by games and gambling, a number of curious proposition bets involving dice, darts, and playing cards, and various mathematical puzzles are scattered throughout this singularly entertaining book.

Who would expect to find in the pages of Mathematics Magazine the first full treatment of one of the more important and oft-cited twentieth-century theorems in analysis, the Stone-Weierstrass Theorem in an article by Marshall Stone himself? Where else would one look for proofs of trigonometric identities using commutative ring theory? Or one of the earliest and best expository articles on the then new Jones knot polynomials, an article that won the prestigious Chauvenet Prize? Or an amusing article purporting to show that the value of pi has been time dependent over the years? These and much more are in this collection of the best from Mathematics Magazine. Readers are inundated with new material in the many mathematical journals. Gems from past issues of Mathematics Magazine or the Monthly or the College Mathematics Journal are read with pleasure when they appear, but get pushed into the background when the next issues arrive. So from time to time it is rewarding to go back and see just what marvelous material has been published over many years, articles now to some extent forgotten. There is history of mathematics (algebraic numbers, inequalities, probability and the Lebesgue integral, quaternions, Polya's enumeration theorem, and group theory) and stories of mathematicians (Hypatia, Gauss, E. T. Bell, Hamilton, and Euler). The list of authors is star-studded: E. T. Bell, Otto Neugebauer, D. H. Lehmer, Morris Kline, Einar Hille, Richard Bellman, Judith Grabiner, Paul Erdos, B. L. van der Waerden, Paul R. Halmos, Doris Schattschneider, J. J. Burckhardt, Branko Grunbaum, and many more.

This book is an English translation of a collection of papers in Russian from a conference held in Moscow and St. Petersburg in 1983, 200 years after Euler's death. Two of the Russian papers in the collection are themselves translations from the German. The present English translation appears in 2007, 300 years after his birth. We speak of the ""Age of Euler"". A justification of this term is provided by a list of scientific terms connected with Euler s name and his many contributions to pure mathematics, well-known in the mathematical community and, in part, covered in this volume. What makes this collection remarkable, though, is the extensive treatment of Euler's contributions outside pure mathematics-to astronomy, celestial mechanics, ballistics, history of science, instruments and technology, physics, mechanics, hydromechanics, mechanics of elastic systems, variational principles of mechanics, and a mathematical theory of music. In addition to this survey of his contributions to science, we find here material otherwise very difficult to find detailed accounts of Euler's family life and the careers pursued by his children and grandchildren. Readers otherwise well-informed about Euler and his work will find here much to enhance their appreciation of this extraordinary scientist and human being. For significant and factual knowledge of Euler's contributions to science past and present, this book is easily the best available choice.

This is a collection of gems from the literature of mathematics that shine as brightly today as when they first appeared in print. They deserve to be seen and admired. The selections include two opposing views on the purpose of mathematics, The Strong Law of Small Numbers, the treatment of calculus in the 1771 Encyclopaedia Britannica, several proofs that the number of legs on a horse is infinite, a deserved refutation of the ridiculous Euler-Diderot anecdote, the real story of pi and the Indiana Legislature, the reason why Theodorus stopped proving that square roots were irrational when he got to the square root of 17, an excerpt from Mathematics Made Difficult; a glimpse into the mind of a calculating prodigy.There will be something of interest here for almost anyone interested in mathematics. Underwood Dudley is the bestselling author of several MAA books: Mathematical Cranks, Numerology, and the Trisectors. He has an Erdos number of 1.

This book picks up the history of mathematics from where Sherlock Holmes in Babylon left it. The forty articles of Who Gave You the Epsilon? continue the story of the development of mathematics into the nineteenth and twentieth centuries. The articles have all been published in the Mathematical Association of America journals and are in many cases written by distinguished mathematicians such as G. H. Hardy and B. van der Waerden. The articles are arranged thematically to show the development of analysis, geometry, algebra and number theory through this period of time. Each chapter is preceded by a foreword, giving the historical background and setting and the scene, and is followed by an afterword, reporting on advances in our historical knowledge and understanding since the articles first appeared. This book is ideal for anyone wanting to explore the history of mathematics.

This book is the second volume based on lectures for pre-college students given by prominent mathematicians in the Bay Area Mathematical Adventures (BAMA). This book reflects the flavor of the BAMA lectures and the excitement they have generated among the high school and middle school students in the Silicon Valley. The topics cover a wide range of mathematical subjects each treated by a leading proponent of the subject at levels designed to challenge and attract students whose mathematical interests are just beginning. In addition, the treatments given here will intrigue and enchant a more mature mathematician. It is hoped that the publication of these lectures will expose students outside of the San Francisco Bay Area to interesting mathematical topics and treatments outside of their normal experience in the classroom. Mathematical educators are encouraged to offer the students in their own localities similar opportunities to come into contact with exciting adventures in mathematics.

Judith Grabiner, the author of A Historian Looks Back, has long been interested in investigating what mathematicians actually do, and how mathematics actually has developed. She addresses the results of her investigations not principally to other historians, but to mathematicians and teachers of mathematics. This book brings together much of what she has had to say to this audience. The centerpiece of the book is The Calculus as Algebra: J.-L. Lagrange, 1736-1813. The book describes the achievements, setbacks, and influence of Lagrange's pioneering attempt to reduce the calculus to algebra. Nine additional articles round out the book describing the history of the derivative; the origin of delta-epsilon proofs; Descartes and problem solving; the contrast between the calculus of Newton and Maclaurin, and that of Lagrange; Maclaurin s way of doing mathematics and science and his surprisingly important influence; some widely held myths about the history of mathematics; Lagrange s attempt to prove Euclid s parallel postulate; and the central role that mathematics has played throughout the history of western civilization. The development of mathematics cannot be programmed or predicted. Still, seeing how ideas have been formed over time and what the difficulties were can help teachers find new ways to explain mathematics. Appreciating its cultural background can humanize mathematics for students. And famous mathematicians' struggles and successes should interest - and perhaps inspire - researchers. Readers will see not only what the mathematical past was like, but also how important parts of the mathematical present came to be.

Calculus and Its Origins is an overview of calculus as an intellectual pursuit having a 2,000-year history. Author David Perkins examines the extent to which mathematicians and scholars from Egypt, Persia, and India absorbed and nourished Greek geometry, and details how the scholars wove their inquiries into a unified theory. Chapters cover the story of Archimedes discovery of the area of a parabolic segment; ibn Al-Haytham s calculation of the volume of a revolved area; Jyesthadeva s explanation of the infinite series for sine and cosine; Wallis s deduction of the link between hyperbolas and logarithms; Newton s generalization of the binomial theorem; Leibniz s discovery of integration by parts-and much more. Each chapter also contains exercises by such mathematical luminaries as Pascal, Maclaurin, Barrow, Cauchy, and Euler.Requiring only a basic knowledge of geometry and algebra-similar triangles, polynomials, factoring-and a willingness to treat the infinite as metaphor-Calculus and Its Origins is a treasure of the human intellect, pearls strung together by mathematicians across cultures and centuries.

This book is about beautiful mathematical concepts and creations. Mathematical ideas have an aesthetic appeal that can be appreciated by those who have the time and dedication to investigate. Mathematical topics are presented in the categories of words, images, formulas, theorems, proofs, solutions, and unsolved problems. Readers will investigate exciting mathematical topics ranging from complex numbers to arithmetic progressions, from Alcuin's sequence to the zeta function, and from hypercubes to infinity squared. Do you know that a lemniscate curve is the circular inversion of a hyperbola? That Sierpinski's triangle has fractal dimension 1.585....? That a regular septagon can be constructed with straightedge, compass, and an angle trisector? Do you know how to prove Lagrange s theorem that every positive integer is the sum of four squares? Can you find the first three digits of the millionth Fibonacci number? Discover the keys to these and many other mathematical problems. In each case, the mathematics is compelling, elegant, simple, and beautiful. Who should read this book? There is something new for any mathematically-minded person. High school and college students will find motivation for their mathematical studies. Professional mathematicians will find fresh examples of mathematical beauty to pass along to others. Within each chapter, the topics require progressively more prerequisite knowledge. An appendix gives background definitions and theorems, while another gives challenging exercises (with solutions).

This book is about the concept of mathematical maturity. Mathematical maturity is central to a mathematics education. The goal of a mathematics education is to transform the student from someone who treats mathematical ideas empirically and intuitively to someone who treats mathematical ideas analytically and can control and manipulate them effectively.Put more directly, a mathematically mature person is one who can read, analyze, and evaluate proofs. And, most significantly, he/she is one who can create proofs. For this is what modern mathematics is all about: coming up with new ideas and validating them with proofs.The book provides background, data, and analysis for understanding the concept of mathematical maturity. It turns the idea of mathematical maturity from a topic for coffee-room conversation to a topic for analysis and serious consideration.

Beginning with one of the most remarkable ecological collapses of recent times, that of the passenger pigeon, this book surveys collapse processes across the entire spectrum of the natural and man-made world. The reader is taken through extreme weather events, crashing financial markets and companies, the chaotic nature of Earth's orbit, revolutionary political change, the spread of disease and many other fascinating cases. The author's key observation is the consistent occurrence of one or more of six fundamental dynamics ('six sources of collapse'), all of which can be best described and investigated using fundamental mathematical concepts which include low probability events, group dynamics, evolutionary games, instability, nonlinearity and network effects. The exposition assumes a minimal mathematical background, but even professional mathematicians are likely to get rich insights from the range of examples. This book will also be of interest to readers in business and finance, engineering, ecology and political science.

Misfortune struck one June day in 1944, when a five-year-old boy was blinded following an accident with a paring knife. That boy, Larry W. Baggett, grew up to become an internationally renowned research mathematician and a successful university professor. At every stage in his life, Baggett broke new ground: he was the first blind student enrolled in the Orlando public school system, the first blind student admitted to Davidson College, and the first blind doctoral student in mathematics at the University of Washington. This memoir describes his successes and failures as a blind person living and learning in the sighted world. In addition, he reflects on his two great passions in life - mathematics and music - with short musings on both topics, such as discussing how to figure out how many dominoes are in a set, the intricacies of jazz chord progressions, and the mysterious Comma of Pythagoras.

C. Edward Sandifer has been studying Euler for decades and is one of the world's leading experts on his work. This book is the second collection of Sandifer's 'How Euler Did It' columns. Each is a jewel of historical and mathematical exposition. The sum total of years of work and study of the most prolific mathematician in history, this volume will leave the reader marveling at Euler's clever inventiveness, which Sandifer explicates and puts in context.

What do mathematicians think of themselves, and what do others think of them? These musings, the theme of a special session at the San Jose MathFest of 2007, resonate with a large and diverse group of mathematicians and students, for these are the questions that govern the way that mathematicians live. A large and diverse group of mathematicians and mathematical people were assembled to offer their views on these matters. The contributions represent a vast array of perspectives on the psychology of the mathematician. It is hoped that readers will find the thoughts assembled here stimulating and cause for further rumination.

Founded in 1961 and still thriving today, the recreational magazine Pythagoras has given generations of high school students in the Netherlands a perspective on many branches of mathematics that are not taught in schools. This book is a selection of the best and most inspiring articles from Pythagoras magazine. It contains a mix of problems - some easy, some challenging, always original - as well as playful introductions to subjects in algebra, geometry, topology, number theory, and more. Concepts like sudoku and the magic square are given a whole new dimension. Articles on Escher's 'impossible objects', a recurring subject over the years, are part of a special section on mathematics and art. Smart high school students and anyone else with a sharp and inquisitive mind will find in this book a treasure trove rich enough to keep one's mind engaged for many weeks and months.

The MAA was founded in 1915 to serve as a home for The American Mathematical Monthly. The mission of the Association-to advance mathematics, especially at the collegiate level-has, however, always been larger than merely publishing world-class mathematical exposition. MAA members have explored more than just mathematics; we have, as this volume tries to make evident, investigated mathematical connections to pedagogy, history, the arts, technology, literature, every field of intellectual endeavor. Essays, all commissioned for this volume, include exposition by Bob Devaney, Robin Wilson, and Frank Morgan; history from Karen Parshall, Della Dumbaugh, and Bill Dunham; pedagogical discussion from Paul Zorn, Joe Gallian, and Michael Starbird, and cultural commentary from Bonnie Gold, Jon Borwein, and Steve Abbott.This volume contains 35 essays by all-star writers and expositors writing to celebrate an extraordinary century for mathematics-more mathematics has been created and published since 1915 than in all of previous recorded history. We've solved age-old mysteries, created entire new fields of study, and changed our conception of what mathematics is. Many of those stories are told in this volume as the contributors paint a portrait of the broad cultural sweep of mathematics during the MAA's first century. Mathematics is the most thrilling, the most human, area of intellectual inquiry; you will find in this volume compelling proof of that claim.

At the turn of the twentieth century, mathematical scholarship in the United States underwent a stunning transformation. In 1890, no American professor was producing mathematical research worthy of international attention. Graduate students were then advised to pursue their studies abroad. By the start of World War I, the standing of American mathematics had radically changed. George David Birkhoff, Leonard Dickson, and others were turning out cutting edge investigations that attracted notice in the intellectual centers of Europe. Harvard, Chicago, and Princeton maintained graduate programs comparable to those overseas. This book explores the people, timing, and factors behind this rapid advance.Through the mid-nineteenth century, most American colleges followed a classical curriculum that, in mathematics, rarely reached beyond calculus. With no doctoral programs of any sort in the United States until 1860, mathematical scholarship lagged far behind that in Europe. After the Civil War, visionary presidents at Harvard and Johns Hopkins broadened and deepened the opportunities for study. The breakthrough for mathematics began in 1890 with the hiring, in consecutive years, of William F. Osgood and Maxime Bôcher at Harvard and E. H. Moore at Chicago. Each of these young men had studied in Germany where they acquired vital mathematical knowledge and taste. Over the next few years, Osgood, Bôcher, and Moore established their own research programs and introduced new graduate courses. Working with other like-minded individuals through the nascent American Mathematical Society, the infrastructure of meetings and journals were created. In the early twentieth century, Princeton dramatically upgraded its faculty to give the United States the stability of a third mathematics center. The publication by Birkhoff, in 1913, of the solution to a famous conjecture served notice that American mathematics had earned consideration with the European powers of Germany, France, Italy, England, and Russia.