The Princeton Companion to Mathematics

The Princeton Companion to Mathematics

Language: English

Pages: 1034

ISBN: 0691118809

Format: PDF / Kindle (mobi) / ePub


This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world's leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and music--and much, much more.

Unparalleled in its depth of coverage, The Princeton Companion to Mathematics surveys the most active and exciting branches of pure mathematics, providing the context and broad perspective that are vital at a time of increasing specialization in the field. Packed with information and presented in an accessible style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties.

  • Features nearly 200 entries, organized thematically and written by an international team of distinguished contributors
  • Presents major ideas and branches of pure mathematics in a clear, accessible style
  • Defines and explains important mathematical concepts, methods, theorems, and open problems
  • Introduces the language of mathematics and the goals of mathematical research
  • Covers number theory, algebra, analysis, geometry, logic, probability, and more
  • Traces the history and development of modern mathematics
  • Profiles more than ninety-five mathematicians who influenced those working today
  • Explores the influence of mathematics on other disciplines
  • Includes bibliographies, cross-references, and a comprehensive index

Contributors incude:

Graham Allan, Noga Alon, George Andrews, Tom Archibald, Sir Michael Atiyah, David Aubin, Joan Bagaria, Keith Ball, June Barrow-Green, Alan Beardon, David D. Ben-Zvi, Vitaly Bergelson, Nicholas Bingham, Béla Bollobás, Henk Bos, Bodil Branner, Martin R. Bridson, John P. Burgess, Kevin Buzzard, Peter J. Cameron, Jean-Luc Chabert, Eugenia Cheng, Clifford C. Cocks, Alain Connes, Leo Corry, Wolfgang Coy, Tony Crilly, Serafina Cuomo, Mihalis Dafermos, Partha Dasgupta, Ingrid Daubechies, Joseph W. Dauben, John W. Dawson Jr., Francois de Gandt, Persi Diaconis, Jordan S. Ellenberg, Lawrence C. Evans, Florence Fasanelli, Anita Burdman Feferman, Solomon Feferman, Charles Fefferman, Della Fenster, José Ferreirós, David Fisher, Terry Gannon, A. Gardiner, Charles C. Gillispie, Oded Goldreich, Catherine Goldstein, Fernando Q. Gouvêa, Timothy Gowers, Andrew Granville, Ivor Grattan-Guinness, Jeremy Gray, Ben Green, Ian Grojnowski, Niccolò Guicciardini, Michael Harris, Ulf Hashagen, Nigel Higson, Andrew Hodges, F. E. A. Johnson, Mark Joshi, Kiran S. Kedlaya, Frank Kelly, Sergiu Klainerman, Jon Kleinberg, Israel Kleiner, Jacek Klinowski, Eberhard Knobloch, János Kollár, T. W. Körner, Michael Krivelevich, Peter D. Lax, Imre Leader, Jean-François Le Gall, W. B. R. Lickorish, Martin W. Liebeck, Jesper Lützen, Des MacHale, Alan L. Mackay, Shahn Majid, Lech Maligranda, David Marker, Jean Mawhin, Barry Mazur, Dusa McDuff, Colin McLarty, Bojan Mohar, Peter M. Neumann, Catherine Nolan, James Norris, Brian Osserman, Richard S. Palais, Marco Panza, Karen Hunger Parshall, Gabriel P. Paternain, Jeanne Peiffer, Carl Pomerance, Helmut Pulte, Bruce Reed, Michael C. Reed, Adrian Rice, Eleanor Robson, Igor Rodnianski, John Roe, Mark Ronan, Edward Sandifer, Tilman Sauer, Norbert Schappacher, Andrzej Schinzel, Erhard Scholz, Reinhard Siegmund-Schultze, Gordon Slade, David J. Spiegelhalter, Jacqueline Stedall, Arild Stubhaug, Madhu Sudan, Terence Tao, Jamie Tappenden, C. H. Taubes, Rüdiger Thiele, Burt Totaro, Lloyd N. Trefethen, Dirk van Dalen, Richard Weber, Dominic Welsh, Avi Wigderson, Herbert Wilf, David Wilkins, B. Yandell, Eric Zaslow, Doron Zeilberger

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

geometric-like) foundation. Important developments in nineteenthcentury mathematics, mainly in connection with the rise of non-euclidean geometries [II.2 §§6–10] and with problems in the foundations of analysis [II.5], eventually led to a fundamental change of orientation, where arithmetic (and eventually set theory [IV.1]) became the bastion of certainty and clarity from which other mathematical disciplines, geometry included, drew their legitimacy and their clarity. (See the crisis in the

understanding of the role of the logical quantifiers [I.2 §3.2] (universal, ∀, and existential, ∃) in the process of formulating a modern mathematical proof. This understanding emerged in an informal, but increasingly clear, fashion as part of the process of the rigorization of analysis and the distancing from visual intuition, especially at the hands of Cauchy, bolzano [VI.28], and Weierstrass. It was formally defined and systematically codified for the first time by Frege in his 1879 Begriffsschrift

and foreign. Brouwer was not worried, for, as he would say in 1933, “the spheres of truth are less transparent than those of illusion.” But Weyl, although convinced that Brouwer had delineated the domain of mathematical intuition in a completely satisfactory way, remarked in 1925: “the mathematician watches with pain the largest part of his towering theories dissolve into mist before his eyes.” Weyl seems to have abandoned intuitionism shortly thereafter. Fortunately, there was an alternative

operator on the real tangent space that squares to −1. This operator has eigenvalues ±i, which can be thought of as “holomorphic” and “anti-holomorphic” directions. The Hermitian property states that these directions are orthogonal, and we say that the manifold is Kähler if they remain so after 1. The notation gab¯ indicates the conjugate-linear property of a Hermitian inner product. PUP: I can confirm that bold 1 is OK. 166 III. Mathematical Concepts transport around loops. This means that

assertions follows from the fact that there is a different element of the Cantor set for every subset A of the natural numbers (just take the ternary number 0.a1 a2 a3 . . . , where ai = 2 whenever i ∈ A and ai = 0 otherwise), and there are uncountably many subsets of the natural numbers. To justify the second, note that the total length of the intervals making up Xn is ( 23 )n (since one removes a third of Xn−1 to produce Xn ). Since the Cantor set is contained in every Xn , its measure must be

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