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| define conjecture in math: Uncle Petros and Goldbach's Conjecture Apostolos Doxiadis, 2012-11-15 Uncle Petros is a family joke. An ageing recluse, he lives alone in a suburb of Athens, playing chess and tending to his garden. If you didn't know better, you'd surely think he was one of life's failures. But his young nephew suspects otherwise. For Uncle Petros, he discovers, was once a celebrated mathematician, brilliant and foolhardy enough to stake everything on solving a problem that had defied all attempts at proof for nearly three centuries - Goldbach's Conjecture. His quest brings him into contact with some of the century's greatest mathematicians, including the Indian prodigy Ramanujan and the young Alan Turing. But his struggle is lonely and single-minded, and by the end it has apparently destroyed his life. Until that is a final encounter with his nephew opens up to Petros, once more, the deep mysterious beauty of mathematics. Uncle Petros and Goldbach's Conjecture is an inspiring novel of intellectual adventure, proud genius, the exhilaration of pure mathematics - and the rivalry and antagonism which torment those who pursue impossible goals. |
| define conjecture in math: The Poincare Conjecture Donal O'Shea, 2009-05-26 Henri Poincaré was one of the greatest mathematicians of the late nineteenth and early twentieth century. He revolutionized the field of topology, which studies properties of geometric configurations that are unchanged by stretching or twisting. The Poincaré conjecture lies at the heart of modern geometry and topology, and even pertains to the possible shape of the universe. The conjecture states that there is only one shape possible for a finite universe in which every loop can be contracted to a single point. Poincaré's conjecture is one of the seven millennium problems that bring a one-million-dollar award for a solution. Grigory Perelman, a Russian mathematician, has offered a proof that is likely to win the Fields Medal, the mathematical equivalent of a Nobel prize, in August 2006. He also will almost certainly share a Clay Institute millennium award. In telling the vibrant story of The Poincaré Conjecture, Donal O'Shea makes accessible to general readers for the first time the meaning of the conjecture, and brings alive the field of mathematics and the achievements of generations of mathematicians whose work have led to Perelman's proof of this famous conjecture. |
| define conjecture in math: How to Think Like a Mathematician Kevin Houston, 2009-02-12 Looking for a head start in your undergraduate degree in mathematics? Maybe you've already started your degree and feel bewildered by the subject you previously loved? Don't panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method, cases, induction, contradiction and contrapositive - are featured. Concrete examples are used throughout, and you'll get plenty of practice on topics common to many courses such as divisors, Euclidean algorithms, modular arithmetic, equivalence relations, and injectivity and surjectivity of functions. The material has been tested by real students over many years so all the essentials are covered. With over 300 exercises to help you test your progress, you'll soon learn how to think like a mathematician. |
| define conjecture in math: Mathematics and Computation Avi Wigderson, 2019-10-29 From the winner of the Turing Award and the Abel Prize, an introduction to computational complexity theory, its connections and interactions with mathematics, and its central role in the natural and social sciences, technology, and philosophy Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field, with strong links to most mathematical areas and to a growing number of scientific endeavors. Avi Wigderson takes a sweeping survey of complexity theory, emphasizing the field’s insights and challenges. He explains the ideas and motivations leading to key models, notions, and results. In particular, he looks at algorithms and complexity, computations and proofs, randomness and interaction, quantum and arithmetic computation, and cryptography and learning, all as parts of a cohesive whole with numerous cross-influences. Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. He ends with a comprehensive look at the theory of computation, its methodology and aspirations, and the unique and fundamental ways in which it has shaped and will further shape science, technology, and society. For further reading, an extensive bibliography is provided for all topics covered. Mathematics and Computation is useful for undergraduate and graduate students in mathematics, computer science, and related fields, as well as researchers and teachers in these fields. Many parts require little background, and serve as an invitation to newcomers seeking an introduction to the theory of computation. Comprehensive coverage of computational complexity theory, and beyond High-level, intuitive exposition, which brings conceptual clarity to this central and dynamic scientific discipline Historical accounts of the evolution and motivations of central concepts and models A broad view of the theory of computation's influence on science, technology, and society Extensive bibliography |
| define conjecture in math: Ricci Flow and the Poincare Conjecture John W. Morgan, Gang Tian, 2007 For over 100 years the Poincare Conjecture, which proposes a topological characterization of the 3-sphere, has been the central question in topology. Since its formulation, it has been repeatedly attacked, without success, using various topological methods. Its importance and difficulty were highlighted when it was chosen as one of the Clay Mathematics Institute's seven Millennium Prize Problems. in 2002 and 2003 Grigory Perelman posted three preprints showing how to use geometric arguments, in particular the Ricci flow as introduced and studied by Hamilton, to establish the Poincare Conjecture in the affirmative. This book provides full details of a complete proof of the Poincare Conjecture following Perelman's three preprints. After a lengthy introduction that outlines the entire argument, the book is divided into four parts. The first part reviews necessary results from Riemannian geometry and Ricci flow, including much of Hamilton's work. The second part starts with Perelman's length function, which is used to establish crucial non-collapsing theorems. Then it discusses the classification of non-collapsed, ancient solutions to the Ricci flow equation. The third part concerns the existence of Ricci flow with surgery for all positive time and an analysis of the topological and geometric changes introduced by surgery. The last part follows Perelman's third preprint to prove that when the initial Riemannian 3-manifold has finite fundamental group, Ricci flow with surgery becomes extinct after finite time. The proofs of the Poincare Conjecture and the closely related 3-dimensional spherical space-form conjectu The existence of Ricci flow with surgery has application to 3-manifolds far beyond the Poincare Conjecture. It forms the heart of the proof via Ricci flow of Thurston's Geometrization Conjecture. Thurston's Geometrization Conjecture, which classifies all compact 3-manifolds, will be the subject of a follow-up article. The organization of the material in this book differs from that given by Perelman. From the beginning the authors present all analytic and geometric arguments in the context of Ricci flow with surgery. in addition, the fourth part is a much-expanded version of Perelman's third preprint; it gives the first complete and detailed proof of the finite-time extinction theorem. With the large amount of background material that is presented and the detailed versions of the central arguments, this book is suitable for all mathematicians from advanced graduate students to specialists in geometry and topology. Clay Mathematics Institute Monograph Series The Clay Mathematics Institute Monograph Series publishes selected expositions of recent developments, both in emerging areas and in older subjects transformed by new insights or unifying ideas. Information for our distributors: Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA). |
| define conjecture in math: The Ultimate Challenge Jeffrey C. Lagarias, 2023-04-19 The $3x+1$ problem, or Collatz problem, concerns the following seemingly innocent arithmetic procedure applied to integers: If an integer $x$ is odd then “multiply by three and add one”, while if it is even then “divide by two”. The $3x+1$ problem asks whether, starting from any positive integer, repeating this procedure over and over will eventually reach the number 1. Despite its simple appearance, this problem is unsolved. Generalizations of the problem are known to be undecidable, and the problem itself is believed to be extraordinarily difficult. This book reports on what is known on this problem. It consists of a collection of papers, which can be read independently of each other. The book begins with two introductory papers, one giving an overview and current status, and the second giving history and basic results on the problem. These are followed by three survey papers on the problem, relating it to number theory and dynamical systems, to Markov chains and ergodic theory, and to logic and the theory of computation. The next paper presents results on probabilistic models for behavior of the iteration. This is followed by a paper giving the latest computational results on the problem, which verify its truth for $x < 5.4 cdot 10^{18}$. The book also reprints six early papers on the problem and related questions, by L. Collatz, J. H. Conway, H. S. M. Coxeter, C. J. Everett, and R. K. Guy, each with editorial commentary. The book concludes with an annotated bibliography of work on the problem up to the year 2000. |
| define conjecture in math: Number Theory and Its History Oystein Ore, 2012-07-06 Unusually clear, accessible introduction covers counting, properties of numbers, prime numbers, Aliquot parts, Diophantine problems, congruences, much more. Bibliography. |
| define conjecture in math: The Weil Conjectures Karen Olsson, 2019-07-16 A New York Times Editors' Pick and Paris Review Staff Pick A wonderful book. --Patti Smith I was riveted. Olsson is evocative on curiosity as an appetite of the mind, on the pleasure of glutting oneself on knowledge. --Parul Sehgal, The New York Times An eloquent blend of memoir and biography exploring the Weil siblings, math, and creative inspiration Karen Olsson’s stirring and unusual third book, The Weil Conjectures, tells the story of the brilliant Weil siblings—Simone, a philosopher, mystic, and social activist, and André, an influential mathematician—while also recalling the years Olsson spent studying math. As she delves into the lives of these two singular French thinkers, she grapples with their intellectual obsessions and rekindles one of her own. For Olsson, as a math major in college and a writer now, it’s the odd detours that lead to discovery, to moments of insight. Thus The Weil Conjectures—an elegant blend of biography and memoir and a meditation on the creative life. Personal, revealing, and approachable, The Weil Conjectures eloquently explores math as it relates to intellectual history, and shows how sometimes the most inexplicable pursuits turn out to be the most rewarding. |
| define conjecture in math: What Is Mathematics, Really? Reuben Hersh, 1997-08-21 Most philosophers of mathematics treat it as isolated, timeless, ahistorical, inhuman. Reuben Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. Hersh pulls the screen back to reveal mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the humanist idea of the nature of mathematics more closely resembles how mathematicians actually work. At the heart of his book is a fascinating historical account of the mainstream of philosophy--ranging from Pythagoras, Descartes, and Spinoza, to Bertrand Russell, David Hilbert, and Rudolph Carnap--followed by the mavericks who saw mathematics as a human artifact, including Aristotle, Locke, Hume, Mill, and Lakatos. What is Mathematics, Really? reflects an insider's view of mathematical life, and will be hotly debated by anyone with an interest in mathematics or the philosophy of science. |
| define conjecture in math: Proofs from THE BOOK Martin Aigner, Günter M. Ziegler, 2013-06-29 According to the great mathematician Paul Erdös, God maintains perfect mathematical proofs in The Book. This book presents the authors candidates for such perfect proofs, those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems from number theory, geometry, analysis, combinatorics, and graph theory. As a result, this book will be fun reading for anyone with an interest in mathematics. |
| define conjecture in math: Weil's Conjecture for Function Fields Dennis Gaitsgory, Jacob Lurie, 2019-02-19 A central concern of number theory is the study of local-to-global principles, which describe the behavior of a global field K in terms of the behavior of various completions of K. This book looks at a specific example of a local-to-global principle: Weil’s conjecture on the Tamagawa number of a semisimple algebraic group G over K. In the case where K is the function field of an algebraic curve X, this conjecture counts the number of G-bundles on X (global information) in terms of the reduction of G at the points of X (local information). The goal of this book is to give a conceptual proof of Weil’s conjecture, based on the geometry of the moduli stack of G-bundles. Inspired by ideas from algebraic topology, it introduces a theory of factorization homology in the setting l-adic sheaves. Using this theory, Dennis Gaitsgory and Jacob Lurie articulate a different local-to-global principle: a product formula that expresses the cohomology of the moduli stack of G-bundles (a global object) as a tensor product of local factors. Using a version of the Grothendieck-Lefschetz trace formula, Gaitsgory and Lurie show that this product formula implies Weil’s conjecture. The proof of the product formula will appear in a sequel volume. |
| define conjecture in math: Street-Fighting Mathematics Sanjoy Mahajan, 2010-03-05 An antidote to mathematical rigor mortis, teaching how to guess answers without needing a proof or an exact calculation. In problem solving, as in street fighting, rules are for fools: do whatever works—don't just stand there! Yet we often fear an unjustified leap even though it may land us on a correct result. Traditional mathematics teaching is largely about solving exactly stated problems exactly, yet life often hands us partly defined problems needing only moderately accurate solutions. This engaging book is an antidote to the rigor mortis brought on by too much mathematical rigor, teaching us how to guess answers without needing a proof or an exact calculation. In Street-Fighting Mathematics, Sanjoy Mahajan builds, sharpens, and demonstrates tools for educated guessing and down-and-dirty, opportunistic problem solving across diverse fields of knowledge—from mathematics to management. Mahajan describes six tools: dimensional analysis, easy cases, lumping, picture proofs, successive approximation, and reasoning by analogy. Illustrating each tool with numerous examples, he carefully separates the tool—the general principle—from the particular application so that the reader can most easily grasp the tool itself to use on problems of particular interest. Street-Fighting Mathematics grew out of a short course taught by the author at MIT for students ranging from first-year undergraduates to graduate students ready for careers in physics, mathematics, management, electrical engineering, computer science, and biology. They benefited from an approach that avoided rigor and taught them how to use mathematics to solve real problems. Street-Fighting Mathematics will appear in print and online under a Creative Commons Noncommercial Share Alike license. |
| define conjecture in math: What is Mathematics? Richard Courant, Herbert Robbins, 1978 |
| define conjecture in math: Mathematics for Human Flourishing Francis Su, 2020-01-07 Winner of the Mathematics Association of America's 2021 Euler Book Prize, this is an inclusive vision of mathematics—its beauty, its humanity, and its power to build virtues that help us all flourish“This is perhaps the most important mathematics book of our time. Francis Su shows mathematics is an experience of the mind and, most important, of the heart.”—James Tanton, Global Math ProjectA good book is an entertaining read. A great book holds up a mirror that allows us to more clearly see ourselves and the world we live in. Francis Su’s Mathematics for Human Flourishing is both a good book and a great book.—MAA Reviews For mathematician Francis Su, a society without mathematical affection is like a city without concerts, parks, or museums. To miss out on mathematics is to live without experiencing some of humanity’s most beautiful ideas.In this profound book, written for a wide audience but especially for those disenchanted by their past experiences, an award‑winning mathematician and educator weaves parables, puzzles, and personal reflections to show how mathematics meets basic human desires—such as for play, beauty, freedom, justice, and love—and cultivates virtues essential for human flourishing. These desires and virtues, and the stories told here, reveal how mathematics is intimately tied to being human. Some lessons emerge from those who have struggled, including philosopher Simone Weil, whose own mathematical contributions were overshadowed by her brother’s, and Christopher Jackson, who discovered mathematics as an inmate in a federal prison. Christopher’s letters to the author appear throughout the book and show how this intellectual pursuit can—and must—be open to all. |
| define conjecture in math: An Introduction to Abstract Mathematics Robert J. Bond, William J. Keane, 2007-08-24 Bond and Keane explicate the elements of logical, mathematical argument to elucidate the meaning and importance of mathematical rigor. With definitions of concepts at their disposal, students learn the rules of logical inference, read and understand proofs of theorems, and write their own proofs all while becoming familiar with the grammar of mathematics and its style. In addition, they will develop an appreciation of the different methods of proof (contradiction, induction), the value of a proof, and the beauty of an elegant argument. The authors emphasize that mathematics is an ongoing, vibrant disciplineits long, fascinating history continually intersects with territory still uncharted and questions still in need of answers. The authors extensive background in teaching mathematics shines through in this balanced, explicit, and engaging text, designed as a primer for higher- level mathematics courses. They elegantly demonstrate process and application and recognize the byproducts of both the achievements and the missteps of past thinkers. Chapters 1-5 introduce the fundamentals of abstract mathematics and chapters 6-8 apply the ideas and techniques, placing the earlier material in a real context. Readers interest is continually piqued by the use of clear explanations, practical examples, discussion and discovery exercises, and historical comments. |
| define conjecture in math: The Mordell Conjecture Hideaki Ikoma, Shu Kawaguchi, Atsushi Moriwaki, 2022-02-03 This book provides a self-contained proof of the Mordell conjecture (Faltings's theorem) and a concise introduction to Diophantine geometry. |
| define conjecture in math: The Geometrization Conjecture John Morgan, Gang Tian, 2014-05-21 This book gives a complete proof of the geometrization conjecture, which describes all compact 3-manifolds in terms of geometric pieces, i.e., 3-manifolds with locally homogeneous metrics of finite volume. The method is to understand the limits as time goes to infinity of Ricci flow with surgery. The first half of the book is devoted to showing that these limits divide naturally along incompressible tori into pieces on which the metric is converging smoothly to hyperbolic metrics and pieces that are locally more and more volume collapsed. The second half of the book is devoted to showing that the latter pieces are themselves geometric. This is established by showing that the Gromov-Hausdorff limits of sequences of more and more locally volume collapsed 3-manifolds are Alexandrov spaces of dimension at most 2 and then classifying these Alexandrov spaces. In the course of proving the geometrization conjecture, the authors provide an overview of the main results about Ricci flows with surgery on 3-dimensional manifolds, introducing the reader to this difficult material. The book also includes an elementary introduction to Gromov-Hausdorff limits and to the basics of the theory of Alexandrov spaces. In addition, a complete picture of the local structure of Alexandrov surfaces is developed. All of these important topics are of independent interest. Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA). |
| define conjecture in math: Rational Points on Varieties Bjorn Poonen, 2017-12-13 This book is motivated by the problem of determining the set of rational points on a variety, but its true goal is to equip readers with a broad range of tools essential for current research in algebraic geometry and number theory. The book is unconventional in that it provides concise accounts of many topics instead of a comprehensive account of just one—this is intentionally designed to bring readers up to speed rapidly. Among the topics included are Brauer groups, faithfully flat descent, algebraic groups, torsors, étale and fppf cohomology, the Weil conjectures, and the Brauer-Manin and descent obstructions. A final chapter applies all these to study the arithmetic of surfaces. The down-to-earth explanations and the over 100 exercises make the book suitable for use as a graduate-level textbook, but even experts will appreciate having a single source covering many aspects of geometry over an unrestricted ground field and containing some material that cannot be found elsewhere. |
| define conjecture in math: The Riemann Hypothesis Peter B. Borwein, 2008 The Riemann Hypothesis has become the Holy Grail of mathematics in the century and a half since 1859 when Bernhard Riemann, one of the extraordinary mathematical talents of the 19th century, originally posed the problem. While the problem is notoriously difficult, and complicated even to state carefully, it can be loosely formulated as the number of integers with an even number of prime factors is the same as the number of integers with an odd number of prime factors. The Hypothesis makes a very precise connection between two seemingly unrelated mathematical objects, namely prime numbers and the zeros of analytic functions. If solved, it would give us profound insight into number theory and, in particular, the nature of prime numbers. This book is an introduction to the theory surrounding the Riemann Hypothesis. Part I serves as a compendium of known results and as a primer for the material presented in the 20 original papers contained in Part II. The original papers place the material into historical context and illustrate the motivations for research on and around the Riemann Hypothesis. Several of these papers focus on computation of the zeta function, while others give proofs of the Prime Number Theorem, since the Prime Number Theorem is so closely connected to the Riemann Hypothesis. The text is suitable for a graduate course or seminar or simply as a reference for anyone interested in this extraordinary conjecture. |
| define conjecture in math: Geometric Topology in Dimensions 2 and 3 E.E. Moise, 2013-06-29 Geometric topology may roughly be described as the branch of the topology of manifolds which deals with questions of the existence of homeomorphisms. Only in fairly recent years has this sort of topology achieved a sufficiently high development to be given a name, but its beginnings are easy to identify. The first classic result was the SchOnflies theorem (1910), which asserts that every 1-sphere in the plane is the boundary of a 2-cell. In the next few decades, the most notable affirmative results were the Schonflies theorem for polyhedral 2-spheres in space, proved by J. W. Alexander [Ad, and the triangulation theorem for 2-manifolds, proved by T. Rad6 [Rd. But the most striking results of the 1920s were negative. In 1921 Louis Antoine [A ] published an extraordinary paper in which he 4 showed that a variety of plausible conjectures in the topology of 3-space were false. Thus, a (topological) Cantor set in 3-space need not have a simply connected complement; therefore a Cantor set can be imbedded in 3-space in at least two essentially different ways; a topological 2-sphere in 3-space need not be the boundary of a 3-cell; given two disjoint 2-spheres in 3-space, there is not necessarily any third 2-sphere which separates them from one another in 3-space; and so on and on. The well-known horned sphere of Alexander [A ] appeared soon thereafter. |
| define conjecture in math: A Concise Course in Algebraic Topology J. P. May, 1999-09 Algebraic topology is a basic part of modern mathematics, and some knowledge of this area is indispensable for any advanced work relating to geometry, including topology itself, differential geometry, algebraic geometry, and Lie groups. This book provides a detailed treatment of algebraic topology both for teachers of the subject and for advanced graduate students in mathematics either specializing in this area or continuing on to other fields. J. Peter May's approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields. But he also retains the classical presentations of various topics where appropriate. Most chapters end with problems that further explore and refine the concepts presented. The final four chapters provide sketches of substantial areas of algebraic topology that are normally omitted from introductory texts, and the book concludes with a list of suggested readings for those interested in delving further into the field. |
| define conjecture in math: The Geometry of Schemes David Eisenbud, Joe Harris, 2006-04-06 Grothendieck’s beautiful theory of schemes permeates modern algebraic geometry and underlies its applications to number theory, physics, and applied mathematics. This simple account of that theory emphasizes and explains the universal geometric concepts behind the definitions. In the book, concepts are illustrated with fundamental examples, and explicit calculations show how the constructions of scheme theory are carried out in practice. |
| define conjecture in math: Conjectures and Refutations Karl Popper, 2014-05-01 Conjectures and Refutations is one of Karl Popper's most wide-ranging and popular works, notable not only for its acute insight into the way scientific knowledge grows, but also for applying those insights to politics and to history. It provides one of the clearest and most accessible statements of the fundamental idea that guided his work: not only our knowledge, but our aims and our standards, grow through an unending process of trial and error. |
| define conjecture in math: The Novikov Conjecture Matthias Kreck, Wolfgang Lück, 2005-12-05 These lecture notes contain a guided tour to the Novikov Conjecture and related conjectures due to Baum-Connes, Borel and Farrell-Jones. They begin with basics about higher signatures, Whitehead torsion and the s-Cobordism Theorem. Then an introduction to surgery theory and a version of the assembly map is presented. Using the solution of the Novikov conjecture for special groups some applications to the classification of low dimensional manifolds are given. |
| define conjecture in math: Connect Geometry P E McGraw-Hill, 1999-01-14 Contains a collection of resources for teachers on geometry. |
| define conjecture in math: The Four-Color Problem , 2011-08-29 The Four-Color Problem |
| define conjecture in math: Character Theory of Finite Groups I. Martin Isaacs, 2006-11-21 Character theory is a powerful tool for understanding finite groups. In particular, the theory has been a key ingredient in the classification of finite simple groups. Characters are also of interest in their own right, and their properties are closely related to properties of the structure of the underlying group. The book begins by developing the module theory of complex group algebras. After the module-theoretic foundations are laid in the first chapter, the focus is primarily on characters. This enhances the accessibility of the material for students, which was a major consideration in the writing. Also with students in mind, a large number of problems are included, many of them quite challenging. In addition to the development of the basic theory (using a cleaner notation than previously), a number of more specialized topics are covered with accessible presentations. These include projective representations, the basics of the Schur index, irreducible character degrees and group structure, complex linear groups, exceptional characters, and a fairly extensive introduction to blocks and Brauer characters. This is a corrected reprint of the original 1976 version, later reprinted by Dover. Since 1976 it has become the standard reference for character theory, appearing in the bibliography of almost every research paper in the subject. It is largely self-contained, requiring of the reader only the most basic facts of linear algebra, group theory, Galois theory and ring and module theory. |
| define conjecture in math: Analysis of Boolean Functions Ryan O'Donnell, 2014-06-05 This graduate-level text gives a thorough overview of the analysis of Boolean functions, beginning with the most basic definitions and proceeding to advanced topics. |
| define conjecture in math: The Science of Conjecture James Franklin, 2015-08-01 How did we make reliable predictions before Pascal and Fermat's discovery of the mathematics of probability in 1654? What methods in law, science, commerce, philosophy, and logic helped us to get at the truth in cases where certainty was not attainable? In The Science of Conjecture, James Franklin examines how judges, witch inquisitors, and juries evaluated evidence; how scientists weighed reasons for and against scientific theories; and how merchants counted shipwrecks to determine insurance rates. The Science of Conjecture provides a history of rational methods of dealing with uncertainty and explores the coming to consciousness of the human understanding of risk. |
| define conjecture in math: Algebraic Geometry and Arithmetic Curves Qing Liu, Reinie Erne, 2006-06-29 This book is a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck's duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo's criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the funadmental theorem of stable reduction of Deligne-Mumford. The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly 600 exercises. |
| define conjecture in math: Dense Sphere Packings Thomas Callister Hales, 2012-09-06 The definitive account of the recent computer solution of the oldest problem in discrete geometry. |
| define conjecture in math: Automated Deduction - CADE 28 André Platzer, 2021 This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions. |
| define conjecture in math: Four Colors Suffice Robin J. Wilson, 2002 On October 23, 1852, Professor Augustus De Morgan wrote a letter to a colleague, unaware that he was launching one of the most famous mathematical conundrums in history--one that would confound thousands of puzzlers for more than a century. This is the amazing story of how the map problem was solved. The problem posed in the letter came from a former student: What is the least possible number of colors needed to fill in any map (real or invented) so that neighboring counties are always colored differently? This deceptively simple question was of minimal interest to cartographers, who saw little need to limit how many colors they used. But the problem set off a frenzy among professional mathematicians and amateur problem solvers, among them Lewis Carroll, an astronomer, a botanist, an obsessive golfer, the Bishop of London, a man who set his watch only once a year, a California traffic cop, and a bridegroom who spent his honeymoon coloring maps. In their pursuit of the solution, mathematicians painted maps on doughnuts and horseshoes and played with patterned soccer balls and the great rhombicuboctahedron. It would be more than one hundred years (and countless colored maps) later before the result was finally established. Even then, difficult questions remained, and the intricate solution--which involved no fewer than 1,200 hours of computer time--was greeted with as much dismay as enthusiasm. Providing a clear and elegant explanation of the problem and the proof, Robin Wilson tells how a seemingly innocuous question baffled great minds and stimulated exciting mathematics with far-flung applications. This is the entertaining story of those who failed to prove, and those who ultimately did prove, that four colors do indeed suffice to color any map. |
| define conjecture in math: Book of Proof Richard H. Hammack, 2016-01-01 This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity. |
| define conjecture in math: Discrete Mathematics Oscar Levin, 2016-08-16 This gentle introduction to discrete mathematics is written for first and second year math majors, especially those who intend to teach. The text began as a set of lecture notes for the discrete mathematics course at the University of Northern Colorado. This course serves both as an introduction to topics in discrete math and as the introduction to proof course for math majors. The course is usually taught with a large amount of student inquiry, and this text is written to help facilitate this. Four main topics are covered: counting, sequences, logic, and graph theory. Along the way proofs are introduced, including proofs by contradiction, proofs by induction, and combinatorial proofs. The book contains over 360 exercises, including 230 with solutions and 130 more involved problems suitable for homework. There are also Investigate! activities throughout the text to support active, inquiry based learning. While there are many fine discrete math textbooks available, this text has the following advantages: It is written to be used in an inquiry rich course. It is written to be used in a course for future math teachers. It is open source, with low cost print editions and free electronic editions. |
| define conjecture in math: The Millennium Prize Problems James Carlson, Arthur Jaffe, Andrew Wiles, 2023-09-14 On August 8, 1900, at the second International Congress of Mathematicians in Paris, David Hilbert delivered his famous lecture in which he described twenty-three problems that were to play an influential role in mathematical research. A century later, on May 24, 2000, at a meeting at the Collège de France, the Clay Mathematics Institute (CMI) announced the creation of a US$7 million prize fund for the solution of seven important classic problems which have resisted solution. The prize fund is divided equally among the seven problems. There is no time limit for their solution. The Millennium Prize Problems were selected by the founding Scientific Advisory Board of CMI—Alain Connes, Arthur Jaffe, Andrew Wiles, and Edward Witten—after consulting with other leading mathematicians. Their aim was somewhat different than that of Hilbert: not to define new challenges, but to record some of the most difficult issues with which mathematicians were struggling at the turn of the second millennium; to recognize achievement in mathematics of historical dimension; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; and to emphasize the importance of working towards a solution of the deepest, most difficult problems. The present volume sets forth the official description of each of the seven problems and the rules governing the prizes. It also contains an essay by Jeremy Gray on the history of prize problems in mathematics. |
| define conjecture in math: How Numbers Work New Scientist, 2018-03-21 Think of a number between one and ten. No, hang on, let's make this interesting. Between zero and infinity. Even if you stick to the whole numbers, there are a lot to choose from - an infinite number in fact. Throw in decimal fractions and infinity suddenly gets an awful lot bigger (is that even possible?) And then there are the negative numbers, the imaginary numbers, the irrational numbers like pi which never end. It literally never ends. The world of numbers is indeed strange and beautiful. Among its inhabitants are some really notable characters - pi, e, the imaginary number i and the famous golden ratio to name just a few. Prime numbers occupy a special status. Zero is very odd indeed: is it a number, or isn't it? How Numbers Work takes a tour of this mind-blowing but beautiful realm of numbers and the mathematical rules that connect them. Not only that, but take a crash course on the biggest unsolved problems that keep mathematicians up at night, find out about the strange and unexpected ways mathematics influences our everyday lives, and discover the incredible connection between numbers and reality itself. ABOUT THE SERIES New Scientist Instant Expert books are definitive and accessible entry points to the most important subjects in science; subjects that challenge, attract debate, invite controversy and engage the most enquiring minds. Designed for curious readers who want to know how things work and why, the Instant Expert series explores the topics that really matter and their impact on individuals, society, and the planet, translating the scientific complexities around us into language that's open to everyone, and putting new ideas and discoveries into perspective and context. |
| define conjecture in math: Forcing For Mathematicians Nik Weaver, 2014-01-24 Ever since Paul Cohen's spectacular use of the forcing concept to prove the independence of the continuum hypothesis from the standard axioms of set theory, forcing has been seen by the general mathematical community as a subject of great intrinsic interest but one that is technically so forbidding that it is only accessible to specialists. In the past decade, a series of remarkable solutions to long-standing problems in C*-algebra using set-theoretic methods, many achieved by the author and his collaborators, have generated new interest in this subject. This is the first book aimed at explaining forcing to general mathematicians. It simultaneously makes the subject broadly accessible by explaining it in a clear, simple manner, and surveys advanced applications of set theory to mainstream topics. |
| define conjecture in math: A Spiral Workbook for Discrete Mathematics Harris Kwong, 2015-11-06 A Spiral Workbook for Discrete Mathematics covers the standard topics in a sophomore-level course in discrete mathematics: logic, sets, proof techniques, basic number theory, functions,relations, and elementary combinatorics, with an emphasis on motivation. The text explains and claries the unwritten conventions in mathematics, and guides the students through a detailed discussion on how a proof is revised from its draft to a nal polished form. Hands-on exercises help students understand a concept soon after learning it. The text adopts a spiral approach: many topics are revisited multiple times, sometimes from a dierent perspective or at a higher level of complexity, in order to slowly develop the student's problem-solving and writing skills. |
| define conjecture in math: Euler's Gem David S. Richeson, 2019-07-23 How a simple equation reshaped mathematics Leonhard Euler’s polyhedron formula describes the structure of many objects—from soccer balls and gemstones to Buckminster Fuller’s buildings and giant all-carbon molecules. Yet Euler’s theorem is so simple it can be explained to a child. From ancient Greek geometry to today’s cutting-edge research, Euler’s Gem celebrates the discovery of Euler’s beloved polyhedron formula and its far-reaching impact on topology, the study of shapes. Using wonderful examples and numerous illustrations, David Richeson presents this mathematical idea’s many elegant and unexpected applications, such as showing why there is always some windless spot on earth, how to measure the acreage of a tree farm by counting trees, and how many crayons are needed to color any map. Filled with a who’s who of brilliant mathematicians who questioned, refined, and contributed to a remarkable theorem’s development, Euler’s Gem will fascinate every mathematics enthusiast. This paperback edition contains a new preface by the author. |
DEFINE Definition & Meaning - Merriam-Webster
The meaning of DEFINE is to determine or identify the essential qualities or meaning of. How to use define in a sentence.
DEFINE Definition & Meaning | Dictionary.com
Define definition: to state or set forth the meaning of (a word, phrase, etc.).. See examples of DEFINE used in a sentence.
DEFINE | English meaning - Cambridge Dictionary
DEFINE definition: 1. to say what the meaning of something, especially a word, is: 2. to explain and describe the…. Learn more.
DEFINE definition and meaning | Collins English Dictionary
If you define something, you show, describe, or state clearly what it is and what its limits are, or what it is like. We were unable to define what exactly was wrong with him. [ VERB wh ]
Define - definition of define by The Free Dictionary
define - show the form or outline of; "The tree was clearly defined by the light"; "The camera could define the smallest object"
DEFINE - Definition & Meaning - Reverso English Dictionary
Define definition: state the meaning of a word or phrase. Check meanings, examples, usage tips, pronunciation, domains, related words.
define - Wiktionary, the free dictionary
May 13, 2025 · define (third-person singular simple present defines, present participle defining, simple past and past participle defined) To determine with precision; to mark out with …
Define: Definition, Meaning, and Examples - usdictionary.com
Dec 24, 2024 · The word "define" means to explain or clarify the meaning of something or to establish boundaries and parameters. It is a versatile word used in many contexts, from everyday …
Define Definition & Meaning - YourDictionary
Define Sentence Examples The child's eagerness and interest carry her over many obstacles that would be our undoing if we stopped to define and explain everything. It will not be welfare (or, at …
DEFINITION Definition & Meaning - Merriam-Webster
The meaning of DEFINITION is a statement of the meaning of a word or word group or a sign or symbol. How to use definition in a sentence.
DEFINE Definition & Meaning - Merriam-Webster
The meaning of DEFINE is to determine or identify the essential qualities or meaning of. How to use define in a sentence.
DEFINE Definition & Meaning | Dictionary.com
Define definition: to state or set forth the meaning of (a word, phrase, etc.).. See examples of DEFINE used in a sentence.
DEFINE | English meaning - Cambridge Dictionary
DEFINE definition: 1. to say what the meaning of something, especially a word, is: 2. to explain and describe the…. Learn more.
DEFINE definition and meaning | Collins English Dictionary
If you define something, you show, describe, or state clearly what it is and what its limits are, or what it is like. We were unable to define what exactly was wrong with him. [ VERB wh ]
Define - definition of define by The Free Dictionary
define - show the form or outline of; "The tree was clearly defined by the light"; "The camera could define the smallest object"
DEFINE - Definition & Meaning - Reverso English Dictionary
Define definition: state the meaning of a word or phrase. Check meanings, examples, usage tips, pronunciation, domains, related words.
define - Wiktionary, the free dictionary
May 13, 2025 · define (third-person singular simple present defines, present participle defining, simple past and past participle defined) To determine with precision; to mark out with …
Define: Definition, Meaning, and Examples - usdictionary.com
Dec 24, 2024 · The word "define" means to explain or clarify the meaning of something or to establish boundaries and parameters. It is a versatile word used in many contexts, from …
Define Definition & Meaning - YourDictionary
Define Sentence Examples The child's eagerness and interest carry her over many obstacles that would be our undoing if we stopped to define and explain everything. It will not be welfare (or, …
DEFINITION Definition & Meaning - Merriam-Webster
The meaning of DEFINITION is a statement of the meaning of a word or word group or a sign or symbol. How to use definition in a sentence.
