A NEW YORK TIMES BESTSELLER

The official book behind the Academy Award-winning film The Imitation Game, starring Benedict Cumberbatch and Keira Knightley

It is only a slight exaggeration to say that the British mathematician Alan Turing (1912-1954) saved the Allies from the Nazis, invented the computer and artificial intelligence, and anticipated gay liberation by decades--all before his suicide at age forty-one. This New York Times–bestselling biography of the founder of computer science, with a new preface by the author that addresses Turing's royal pardon in 2013, is the definitive account of an extraordinary mind and life.

Capturing both the inner and outer drama of Turing’s life, Andrew Hodges tells how Turing’s revolutionary idea of 1936--the concept of a universal machine--laid the foundation for the modern computer and how Turing brought the idea to practical realization in 1945 with his electronic design. The book also tells how this work was directly related to Turing’s leading role in breaking the German Enigma ciphers during World War II, a scientific triumph that was critical to Allied victory in the Atlantic. At the same time, this is the tragic account of a man who, despite his wartime service, was eventually arrested, stripped of his security clearance, and forced to undergo a humiliating treatment program--all for trying to live honestly in a society that defined homosexuality as a crime.

The inspiration for a major motion picture starring Benedict Cumberbatch and Keira Knightley, Alan Turing: The Enigma is a gripping story of mathematics, computers, cryptography, and homosexual persecution.

Today complex numbers have such widespread practical use--from electrical engineering to aeronautics--that few people would expect the story behind their derivation to be filled with adventure and enigma. In An Imaginary Tale, Paul Nahin tells the 2000-year-old history of one of mathematics' most elusive numbers, the square root of minus one, also known as i. He recreates the baffling mathematical problems that conjured it up, and the colorful characters who tried to solve them.

In 1878, when two brothers stole a mathematical papyrus from the ancient Egyptian burial site in the Valley of Kings, they led scholars to the earliest known occurrence of the square root of a negative number. The papyrus offered a specific numerical example of how to calculate the volume of a truncated square pyramid, which implied the need for i. In the first century, the mathematician-engineer Heron of Alexandria encountered I in a separate project, but fudged the arithmetic; medieval mathematicians stumbled upon the concept while grappling with the meaning of negative numbers, but dismissed their square roots as nonsense. By the time of Descartes, a theoretical use for these elusive square roots--now called "imaginary numbers"--was suspected, but efforts to solve them led to intense, bitter debates. The notorious i finally won acceptance and was put to use in complex analysis and theoretical physics in Napoleonic times.

Addressing readers with both a general and scholarly interest in mathematics, Nahin weaves into this narrative entertaining historical facts and mathematical discussions, including the application of complex numbers and functions to important problems, such as Kepler's laws of planetary motion and ac electrical circuits. This book can be read as an engaging history, almost a biography, of one of the most evasive and pervasive "numbers" in all of mathematics.

Some images inside the book are unavailable due to digital copyright restrictions.

Gamma-ray bursts are the brightest--and, until recently, among the least understood--cosmic events in the universe. Discovered by chance during the cold war, these evanescent high-energy explosions confounded astronomers for decades. But a rapid series of startling breakthroughs beginning in 1997 revealed that the majority of gamma-ray bursts are caused by the explosions of young and massive stars in the vast star-forming cauldrons of distant galaxies. New findings also point to very different origins for some events, serving to complicate but enrich our understanding of the exotic and violent universe. What Are Gamma-Ray Bursts? is a succinct introduction to this fast-growing subject, written by an astrophysicist who is at the forefront of today's research into these incredible cosmic phenomena.

Joshua Bloom gives readers a concise and accessible overview of gamma-ray bursts and the theoretical framework that physicists have developed to make sense of complex observations across the electromagnetic spectrum. He traces the history of remarkable discoveries that led to our current understanding of gamma-ray bursts, and reveals the decisive role these phenomena could play in the grand pursuits of twenty-first century astrophysics, from studying gravity waves and unveiling the growth of stars and galaxies after the big bang to surmising the ultimate fate of the universe itself.

What Are Gamma-Ray Bursts? is an essential primer to this exciting frontier of scientific inquiry, and a must-read for anyone seeking to keep pace with cutting-edge developments in physics today.

Circles Disturbed brings together important thinkers in mathematics, history, and philosophy to explore the relationship between mathematics and narrative. The book's title recalls the last words of the great Greek mathematician Archimedes before he was slain by a Roman soldier--"Don't disturb my circles"--words that seem to refer to two radically different concerns: that of the practical person living in the concrete world of reality, and that of the theoretician lost in a world of abstraction. Stories and theorems are, in a sense, the natural languages of these two worlds--stories representing the way we act and interact, and theorems giving us pure thought, distilled from the hustle and bustle of reality. Yet, though the voices of stories and theorems seem totally different, they share profound connections and similarities.

A book unlike any other, Circles Disturbed delves into topics such as the way in which historical and biographical narratives shape our understanding of mathematics and mathematicians, the development of "myths of origins" in mathematics, the structure and importance of mathematical dreams, the role of storytelling in the formation of mathematical intuitions, the ways mathematics helps us organize the way we think about narrative structure, and much more.

In addition to the editors, the contributors are Amir Alexander, David Corfield, Peter Galison, Timothy Gowers, Michael Harris, David Herman, Federica La Nave, G.E.R. Lloyd, Uri Margolin, Colin McLarty, Jan Christoph Meister, Arkady Plotnitsky, and Bernard Teissier.

The sign outside the conservative, white church in the small southern U.S. town announces that the church is part of the Episcopal Church--of Rwanda. In Anglican Communion in Crisis, Miranda Hassett tells the fascinating story of how a new alliance between conservative American Episcopalians and African Anglicans is transforming conflicts between American Episcopalians--especially over homosexuality--into global conflicts within the Anglican church.

In the mid-1990s, conservative American Episcopalians and Anglican leaders from Africa and other parts of the Southern Hemisphere began to forge ties in opposition to the American Episcopal Church's perceived liberalism and growing toleration of homosexuality. This resulted in dozens of American Episcopal churches submitting to the authority of African bishops.

Based on wide research, interviews with key participants and observers, and months Hassett spent in a southern U.S. parish of the Episcopal Church of Rwanda and in Anglican communities in Uganda, Anglican Communion in Crisis is the first anthropological examination of the coalition between American Episcopalians and African Anglicans. The book challenges common views--that the relationship between the Americans and Africans is merely one of convenience or even that the Americans bought the support of the Africans. Instead, Hassett argues that their partnership is a deliberate and committed movement that has tapped the power and language of globalization in an effort to move both the American Episcopal Church and the worldwide Anglican Communion to the right.

This collection of new and original papers on mathematical aspects of nonlinear dispersive equations includes both expository and technical papers that reflect a number of recent advances in the field. The expository papers describe the state of the art and research directions. The technical papers concentrate on a specific problem and the related analysis and are addressed to active researchers.

The book deals with many topics that have been the focus of intensive research and, in several cases, significant progress in recent years, including hyperbolic conservation laws, Schrödinger operators, nonlinear Schrödinger and wave equations, and the Euler and Navier-Stokes equations.

Hugh Everett III was an American physicist best known for his many-worlds interpretation of quantum mechanics, which formed the basis of his PhD thesis at Princeton University in 1957. Although counterintuitive, Everett's revolutionary formulation of quantum mechanics offers the most direct solution to the infamous quantum measurement problem--that is, how and why the singular world of our experience emerges from the multiplicities of alternatives available in the quantum world. The many-worlds interpretation postulates the existence of multiple universes. Whenever a measurement-like interaction occurs, the universe branches into relative states, one for each possible outcome of the measurement, and the world in which we find ourselves is but one of these many, but equally real, possibilities. Everett's challenge to the orthodox interpretation of quantum mechanics was met with scorn from Niels Bohr and other leading physicists, and Everett subsequently abandoned academia to conduct military operations research. Today, however, Everett's formulation of quantum mechanics is widely recognized as one of the most controversial but promising physical theories of the last century.

In this book, Jeffrey Barrett and Peter Byrne present the long and short versions of Everett's thesis along with a collection of his explanatory writings and correspondence. These primary source documents, many of them newly discovered and most unpublished until now, reveal how Everett's thinking evolved from his days as a graduate student to his untimely death in 1982. This definitive volume also features Barrett and Byrne's introductory essays, notes, and commentary that put Everett's extraordinary theory into historical and scientific perspective and discuss the puzzles that still remain.

Heavenly Mathematics traces the rich history of spherical trigonometry, revealing how the cultures of classical Greece, medieval Islam, and the modern West used this forgotten art to chart the heavens and the Earth. Once at the heart of astronomy and ocean-going navigation for two millennia, the discipline was also a mainstay of mathematics education for centuries and taught widely until the 1950s. Glen Van Brummelen explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography; Islamic religious rituals; celestial navigation; polyhedra; stereographic projection; and more. He conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation of its elegant proofs and often surprising conclusions. Heavenly Mathematics is illustrated throughout with stunning historical images and informative drawings and diagrams. This unique compendium also features easy-to-use appendixes as well as exercises that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

Totally nonnegative matrices arise in a remarkable variety of mathematical applications. This book is a comprehensive and self-contained study of the essential theory of totally nonnegative matrices, defined by the nonnegativity of all subdeterminants. It explores methodological background, historical highlights of key ideas, and specialized topics.

The book uses classical and ad hoc tools, but a unifying theme is the elementary bidiagonal factorization, which has emerged as the single most important tool for this particular class of matrices. Recent work has shown that bidiagonal factorizations may be viewed in a succinct combinatorial way, leading to many deep insights. Despite slow development, bidiagonal factorizations, along with determinants, now provide the dominant methodology for understanding total nonnegativity. The remainder of the book treats important topics, such as recognition of totally nonnegative or totally positive matrices, variation diminution, spectral properties, determinantal inequalities, Hadamard products, and completion problems associated with totally nonnegative or totally positive matrices. The book also contains sample applications, an up-to-date bibliography, a glossary of all symbols used, an index, and related references.

The emigration of mathematicians from Europe during the Nazi era signaled an irrevocable and important historical shift for the international mathematics world. Mathematicians Fleeing from Nazi Germany is the first thoroughly documented account of this exodus. In this greatly expanded translation of the 1998 German edition, Reinhard Siegmund-Schultze describes the flight of more than 140 mathematicians, their reasons for leaving, the political and economic issues involved, the reception of these emigrants by various countries, and the emigrants' continuing contributions to mathematics. The influx of these brilliant thinkers to other nations profoundly reconfigured the mathematics world and vaulted the United States into a new leadership role in mathematics research.

Based on archival sources that have never been examined before, the book discusses the preeminent emigrant mathematicians of the period, including Emmy Noether, John von Neumann, Hermann Weyl, and many others. The author explores the mechanisms of the expulsion of mathematicians from Germany, the emigrants' acculturation to their new host countries, and the fates of those mathematicians forced to stay behind. The book reveals the alienation and solidarity of the emigrants, and investigates the global development of mathematics as a consequence of their radical migration.

An in-depth yet accessible look at mathematics both as a scientific enterprise and human endeavor, Mathematicians Fleeing from Nazi Germany provides a vivid picture of a critical chapter in the history of international science.

Based on extensive research in Sanskrit sources, Mathematics in India chronicles the development of mathematical techniques and texts in South Asia from antiquity to the early modern period. Kim Plofker reexamines the few facts about Indian mathematics that have become common knowledge--such as the Indian origin of Arabic numerals--and she sets them in a larger textual and cultural framework. The book details aspects of the subject that have been largely passed over in the past, including the relationships between Indian mathematics and astronomy, and their cross-fertilizations with Islamic scientific traditions. Plofker shows that Indian mathematics appears not as a disconnected set of discoveries, but as a lively, diverse, yet strongly unified discipline, intimately linked to other Indian forms of learning.

Far more than in other areas of the history of mathematics, the literature on Indian mathematics reveals huge discrepancies between what researchers generally agree on and what general readers pick up from popular ideas. This book explains with candor the chief controversies causing these discrepancies--both the flaws in many popular claims, and the uncertainties underlying many scholarly conclusions. Supplementing the main narrative are biographical resources for dozens of Indian mathematicians; a guide to key features of Sanskrit for the non-Indologist; and illustrations of manuscripts, inscriptions, and artifacts. Mathematics in India provides a rich and complex understanding of the Indian mathematical tradition.

**Author's note: The concept of "computational positivism" in Indian mathematical science, mentioned on p. 120, is due to Prof. Roddam Narasimha and is explored in more detail in some of his works, including "The Indian half of Needham's question: some thoughts on axioms, models, algorithms, and computational positivism" (Interdisciplinary Science Reviews 28, 2003, 1-13).

Although political and legal institutions are essential to any nation's economic development, the forces that have shaped these institutions are poorly understood. Drawing on rich evidence about the development of the American states from the mid-nineteenth to the late twentieth century, this book documents the mechanisms through which geographical and historical conditions--such as climate, access to water transportation, and early legal systems--impacted political and judicial institutions and economic growth.

The book shows how a state's geography and climate influenced whether elites based their wealth in agriculture or trade. States with more occupationally diverse elites in 1860 had greater levels of political competition in their legislature from 1866 to 2000. The book also examines the effects of early legal systems. Because of their colonial history, thirteen states had an operational civil-law legal system prior to statehood. All of these states except Louisiana would later adopt common law. By the late eighteenth century, the two legal systems differed in their balances of power. In civil-law systems, judiciaries were subordinate to legislatures, whereas in common-law systems, the two were more equal. Former civil-law states and common-law states exhibit persistent differences in the structure of their courts, the retention of judges, and judicial budgets. Moreover, changes in court structures, retention procedures, and budgets occur under very different conditions in civil-law and common-law states.

The Evolution of a Nation illustrates how initial geographical and historical conditions can determine the evolution of political and legal institutions and long-run growth.

How do you draw a straight line? How do you determine if a circle is really round? These may sound like simple or even trivial mathematical problems, but to an engineer the answers can mean the difference between success and failure. How Round Is Your Circle? invites readers to explore many of the same fundamental questions that working engineers deal with every day--it's challenging, hands-on, and fun.

John Bryant and Chris Sangwin illustrate how physical models are created from abstract mathematical ones. Using elementary geometry and trigonometry, they guide readers through paper-and-pencil reconstructions of mathematical problems and show them how to construct actual physical models themselves--directions included. It's an effective and entertaining way to explain how applied mathematics and engineering work together to solve problems, everything from keeping a piston aligned in its cylinder to ensuring that automotive driveshafts rotate smoothly. Intriguingly, checking the roundness of a manufactured object is trickier than one might think. When does the width of a saw blade affect an engineer's calculations--or, for that matter, the width of a physical line? When does a measurement need to be exact and when will an approximation suffice? Bryant and Sangwin tackle questions like these and enliven their discussions with many fascinating highlights from engineering history. Generously illustrated, How Round Is Your Circle? reveals some of the hidden complexities in everyday things.

The interest earned on a bank account, the arrangement of seeds in a sunflower, and the shape of the Gateway Arch in St. Louis are all intimately connected with the mysterious number e. In this informal and engaging history, Eli Maor portrays the curious characters and the elegant mathematics that lie behind the number. Designed for a reader with only a modest mathematical background, this biography brings out the central importance of e to mathematics and illuminates a golden era in the age of science.

What are your chances of dying on your next flight, being called for jury duty, or winning the lottery? We all encounter probability problems in our everyday lives. In this collection of twenty-one puzzles, Paul Nahin challenges us to think creatively about the laws of probability as they apply in playful, sometimes deceptive, ways to a fascinating array of speculative situations. Games of Russian roulette, problems involving the accumulation of insects on flypaper, and strategies for determining the odds of the underdog winning the World Series all reveal intriguing dimensions to the workings of probability. Over the years, Nahin, a veteran writer and teacher of the subject, has collected these and other favorite puzzles designed to instruct and entertain math enthusiasts of all backgrounds.

If idiots A and B alternately take aim at each other with a six-shot revolver containing one bullet, what is the probability idiot A will win? What are the chances it will snow on your birthday in any given year? How can researchers use coin flipping and the laws of probability to obtain honest answers to embarrassing survey questions? The solutions are presented here in detail, and many contain a profound element of surprise. And some puzzles are beautiful illustrations of basic mathematical concepts: "The Blind Spider and the Fly," for example, is a clever variation of a "random walk" problem, and "Duelling Idiots" and "The Underdog and the World Series" are straightforward introductions to binomial distributions.

Written in an informal way and containing a plethora of interesting historical material, Duelling Idiots is ideal for those who are fascinated by mathematics and the role it plays in everyday life and in our imaginations.

This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gábor Szego's classic 1915 theorem and its 1920 extension. Barry Simon emphasizes necessary and sufficient conditions, and provides mathematical background that until now has been available only in journals. Topics include background from the theory of meromorphic functions on hyperelliptic surfaces and the study of covering maps of the Riemann sphere with a finite number of slits removed. This allows for the first book-length treatment of orthogonal polynomials for measures supported on a finite number of intervals on the real line.

In addition to the Szego and Killip-Simon theorems for orthogonal polynomials on the unit circle (OPUC) and orthogonal polynomials on the real line (OPRL), Simon covers Toda lattices, the moment problem, and Jacobi operators on the Bethe lattice. Recent work on applications of universality of the CD kernel to obtain detailed asymptotics on the fine structure of the zeros is also included. The book places special emphasis on OPRL, which makes it the essential companion volume to the author's earlier books on OPUC.

In today's unpredictable and chaotic world, we look to science to provide certainty and answers--and often blame it when things go wrong. The Blind Spot reveals why our faith in scientific certainty is a dangerous illusion, and how only by embracing science's inherent ambiguities and paradoxes can we truly appreciate its beauty and harness its potential.

Crackling with insights into our most perplexing contemporary dilemmas, from climate change to the global financial meltdown, this book challenges our most sacredly held beliefs about science, technology, and progress. At the same time, it shows how the secret to better science can be found where we least expect it--in the uncertain, the ambiguous, and the inevitably unpredictable. William Byers explains why the subjective element in scientific inquiry is in fact what makes it so dynamic, and deftly balances the need for certainty and rigor in science with the equally important need for creativity, freedom, and downright wonder. Drawing on an array of fascinating examples--from Wall Street's overreliance on algorithms to provide certainty in uncertain markets, to undecidable problems in mathematics and computer science, to Georg Cantor's paradoxical but true assertion about infinity--Byers demonstrates how we can and must learn from the existence of blind spots in our scientific and mathematical understanding.

The Blind Spot offers an entirely new way of thinking about science, one that highlights its strengths and limitations, its unrealized promise, and, above all, its unavoidable ambiguity. It also points to a more sophisticated approach to the most intractable problems of our time.

Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive exploration of Mumford-Tate groups and domains. Containing basic theory and a wealth of new views and results, it will become an essential resource for graduate students and researchers.

Although Mumford-Tate groups can be defined for general structures, their theory and use to date has mainly been in the classical case of abelian varieties. While the book does examine this area, it focuses on the nonclassical case. The general theory turns out to be very rich, such as in the unexpected connections of finite dimensional and infinite dimensional representation theory of real, semisimple Lie groups. The authors give the complete classification of Hodge representations, a topic that should become a standard in the finite-dimensional representation theory of noncompact, real, semisimple Lie groups. They also indicate that in the future, a connection seems ready to be made between Lie groups that admit discrete series representations and the study of automorphic cohomology on quotients of Mumford-Tate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject.

The physics of neutrinos--uncharged elementary particles that are key to helping us better understand the nature of our universe--is one of the most exciting frontiers of modern science. This book provides a comprehensive overview of neutrino physics today and explores promising new avenues of inquiry that could lead to future breakthroughs.

The Physics of Neutrinos begins with a concise history of the field and a tutorial on the fundamental properties of neutrinos, and goes on to discuss how the three neutrino types interchange identities as they propagate from their sources to detectors. The book shows how studies of neutrinos produced by such phenomena as cosmic rays in the atmosphere and nuclear reactions in the solar interior provide striking evidence that neutrinos have mass, and it traces our astounding progress in deciphering the baffling experimental findings involving neutrinos. The discovery of neutrino mass offers the first indication of a new kind of physics that goes beyond the Standard Model of elementary particles, and this book considers the unanticipated patterns in the masses and mixings of neutrinos in the framework of proposed new theoretical models.

The Physics of Neutrinos maps out the ambitious future facilities and experiments that will advance our knowledge of neutrinos, and explains why the way forward in solving the outstanding questions in neutrino science will require the collective efforts of particle physics, nuclear physics, astrophysics, and cosmology.

This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the question turns out to be closely related to porous sets in Banach spaces, it provides a bridge between descriptive set theory and the classical topic of existence of derivatives of vector-valued Lipschitz functions. The topic is relevant to classical analysis and descriptive set theory on Banach spaces. The book opens several new research directions in this area of geometric nonlinear functional analysis.

The new methods developed here include a game approach to perturbational variational principles that is of independent interest. Detailed explanation of the underlying ideas and motivation behind the proofs of the new results on Fréchet differentiability of vector-valued functions should make these arguments accessible to a wider audience. The most important special case of the differentiability results, that Lipschitz mappings from a Hilbert space into the plane have points of Fréchet differentiability, is given its own chapter with a proof that is independent of much of the work done to prove more general results. The book raises several open questions concerning its two main topics.