Theorems of the 21st Century Volume I
This book consists of short descriptions of 106 mathematical theorems, which belong to the great achievements of 21st century mathematics but require relatively little mathematical background to understand their formulation and appreciate their impor
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Theorems of the 21st Century Volume I
Theorems of the 21st Century
Bogdan Grechuk
Theorems of the 21st Century Volume I
123
Bogdan Grechuk Department of Mathematics University of Leicester Leicester, UK
ISBN 978-3-030-19095-8 ISBN 978-3-030-19096-5 https://doi.org/10.1007/978-3-030-19096-5
(eBook)
Mathematics Subject Classification (2010): 00-02, 00A05, 00A06, 00A09, 01-02, 01A61 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface
Most of the theorems published recently in leading mathematical journals are so difficult that even their exact formulation is hard to understand for non-mathematicians, and, in some cases, even for mathematicians working in unrelated areas. For example, in 2009 Ngô Båo Châu proved the result called the “Fundamental lemma of the Langlands program”, which was listed by Time magazine as one of the Top 10 scientific discoveries of 2009, and earned him a Fields Medal, one of the most prestigious awards in mathematics. However, even the exact formulation of the lemma (with all the notations defined) takes several pages to write down, and requires a high-level mathematical education to understand. We can compare this result with an abstract painting or a piece of modern art, which can be fully appreciated by a relatively small group of people. However, music, poetry, paintings, films, etc., tend to be described as “best” if they can be appreciated by millions. Do there exist mathematical theorems like the songs of The Beatles, who had stadiums of fans at their concerts? That is, theorems which are sufficiently difficult and important to be accepted for publication into a leading mathematical journal, but at the same time with a
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