An Invitation to Mathematics From Competitions to Research
This Invitation to Mathematics consists of 14 contributions, many from the world's leading mathematicians, that introduce the readers to exciting aspects of current mathematical research. The contributions are as varied as the personalities of active math
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Dierk Schleicher r Malte Lackmann Editors
An Invitation to Mathematics From Competitions to Research
Editors Dierk Schleicher Jacobs University Postfach 750 561 D-28725 Bremen Germany [email protected]
Malte Lackmann Immenkorv 13 24582 Bordesholm Germany [email protected]
ISBN 978-3-642-19532-7 e-ISBN 978-3-642-19533-4 DOI 10.1007/978-3-642-19533-4 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011928905 Mathematics Subject Classification (2010): 00-01, 00A09, 00A05 © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Contents
Preface: What is Mathematics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Welcome! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Structure and Randomness in the Prime Numbers . . . . . . . . . . . . Terence Tao
1
How to Solve a Diophantine Equation . . . . . . . . . . . . . . . . . . . . . . . . Michael Stoll
9
From Sex to Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Simon Norton Small Divisors: Number Theory in Dynamical Systems . . . . . . . 43 Jean-Christophe Yoccoz How do IMO Problems Compare with Research Problems? Ramsey Theory as a Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 W. Timothy Gowers How do Research Problems Compare with IMO Problems? A Walk Around Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Stanislav Smirnov Graph Theory Over 45 Years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 L´aszl´o Lov´ asz Communication Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Alexander A. Razborov Ten Digit Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Lloyd N. Trefethen
v
vi
Contents
The Ever-Elusive Blowup in the Mathematical Description of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Robert M. Kerr and M
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