Proofs from the BOOK
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Mathematical
2-12 JULY 2011 MATHEMATICAL HOST AND VENUE for Students Jacobs University Scientific Committee Étienne Ghys (École Normale The summer school is based on the park-like campus of Supérieure de Lyon, France), chair Jacobs University, with lecture halls, library, small group study rooms, cafeterias, and recreation facilities within Frances Kirwan (University of Oxford, UK) easy walking distance. Dierk Schleicher (Jacobs University, Germany) Alexei Sossinsky (Moscow University, Russia) Jacobs University is an international, highly selective, Sergei Tabachnikov (Penn State University, USA) residential campus university in the historic Hanseatic Anatoliy Vershik (St. Petersburg State University, Russia) city of Bremen. It features an attractive math program Wendelin Werner (Université Paris-Sud, France) with personal attention to students and their individual interests. Jean-Christophe Yoccoz (Collège de France) Don Zagier (Max Planck-Institute Bonn, Germany; › Home to approximately 1,200 students from over Collège de France) 100 different countries Günter M. Ziegler (Freie Universität Berlin, Germany) › English language university › Committed to excellence in higher education Organizing Committee › Has a special program with fellowships for the most Anke Allner (Universität Hamburg, Germany) talented students in mathematics from all countries Martin Andler (Université Versailles-Saint-Quentin, › Venue of the 50th International Mathematical Olympiad France) (IMO) 2009 Victor Kleptsyn (Université de Rennes, France) Marcel Oliver (Jacobs University, Germany) For more information about the mathematics program Stephanie Schiemann (Freie Universität Berlin, Germany) at Jacobs University, please visit: Dierk Schleicher (Jacobs University, Germany) math.jacobs-university.de Sergei Tabachnikov (Penn State University, USA) at Jacobs University, Bremen The School is an initiative in the framework of the European Campus of Excellence (ECE). -
Arithmetic Properties of the Herglotz Function
ARITHMETIC PROPERTIES OF THE HERGLOTZ FUNCTION DANYLO RADCHENKO AND DON ZAGIER Abstract. In this paper we study two functions F (x) and J(x), originally found by Herglotz in 1923 and later rediscovered and used by one of the authors in connection with the Kronecker limit formula for real quadratic fields. We discuss many interesting properties of these func- tions, including special values at rational or quadratic irrational arguments as rational linear combinations of dilogarithms and products of logarithms, functional equations coming from Hecke operators, and connections with Stark's conjecture. We also discuss connections with 1-cocycles for the modular group PSL(2; Z). Contents 1. Introduction 1 2. Elementary properties 2 3. Functional equations related to Hecke operators 4 4. Special values at positive rationals 8 5. Kronecker limit formula for real quadratic fields 10 6. Special values at quadratic units 11 7. Cohomological aspects 15 References 18 1. Introduction Consider the function Z 1 log(1 + tx) (1) J(x) = dt ; 0 1 + t defined for x > 0. Some years ago, Henri Cohen 1 showed one of the authors the identity p p π2 log2(2) log(2) log(1 + 2) J(1 + 2) = − + + : 24 2 2 In this note we will give many more similar identities, like p π2 log2(2) p J(4 + 17) = − + + log(2) log(4 + 17) 6 2 arXiv:2012.15805v1 [math.NT] 31 Dec 2020 and p 2 11π2 3 log2(2) 5 + 1 J = + − 2 log2 : 5 240 4 2 We will also investigate the connection to several other topics, such as the Kronecker limit formula for real quadratic fields, Hecke operators, Stark's conjecture, and cohomology of the modular group PSL2(Z). -
Proofs from the BOOK.Pdf
Martin Aigner Günter M. Ziegler Proofs from THE BOOK Fourth Edition Martin Aigner Günter M. Ziegler Proofs from THE BOOK Fourth Edition Including Illustrations by Karl H. Hofmann 123 Prof. Dr. Martin Aigner Prof.GünterM.Ziegler FB Mathematik und Informatik Institut für Mathematik, MA 6-2 Freie Universität Berlin Technische Universität Berlin Arnimallee 3 Straße des 17. Juni 136 14195 Berlin 10623 Berlin Deutschland Deutschland [email protected] [email protected] ISBN 978-3-642-00855-9 e-ISBN 978-3-642-00856-6 DOI 10.1007/978-3-642-00856-6 Springer Heidelberg Dordrecht London New York c Springer-Verlag Berlin Heidelberg 2010 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) Preface Paul Erdosliked˝ to talk aboutThe Book, in which God maintainsthe perfect proofsfor mathematical theorems, following the dictum of G. -
A Combinatorial Analysis of Topological Dissections
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ADVANCES IN MATHEMATICS 25, 267-285 (1977) A Combinatorial Analysis of Topological Dissections THOMAS ZASLAVSKY Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 From a topological space remove certain subspaces (cuts), leaving connected components (regions). We develop an enumerative theory for the regions in terms of the cuts, with the aid of a theorem on the Mijbius algebra of a subset of a distributive lattice. Armed with this theory we study dissections into cellular faces and dissections in the d-sphere. For example, we generalize known enumerations for arrangements of hyperplanes to convex sets and topological arrangements, enumerations for simple arrangements and the Dehn-Sommerville equations for simple polytopes to dissections with general intersection, and enumerations for arrangements of lines and curves and for plane convex sets to dissections by curves of the 2-sphere and planar domains. Contents. Introduction. 1. The fundamental relations for dissections and covers. 2. Algebraic combinatorics: Valuations and MGbius algebras. 3. Dissections into cells and properly cellular faces. 4. General position. 5. Counting low-dimensional faces. 6. Spheres and their subspaces. 7. Dissections by curves. INTRODUCTION A problem which arises from time to time in combinatorial geometry is to determine the number of pieces into which a certain geometric set is divided by given subsets. Think for instance of a plane cut by prescribed curves, as in a map of a land area crisscrossed by roads and hedgerows, or of a finite set of hyper- planes slicing up a convex domain in a Euclidean space. -
The Dehn-Sydler Theorem Explained
The Dehn-Sydler Theorem Explained Rich Schwartz February 16, 2010 1 Introduction The Dehn-Sydler theorem says that two polyhedra in R3 are scissors con- gruent iff they have the same volume and Dehn invariant. Dehn [D] proved in 1901 that equality of the Dehn invariant is necessary for scissors congru- ence. 64 years passed, and then Sydler [S] proved that equality of the Dehn invariant (and volume) is sufficient. In [J], Jessen gives a simplified proof of Sydler’s Theorem. The proof in [J] relies on two results from homolog- ical algebra that are proved in [JKT]. Also, to get the cleanest statement of the result, one needs to combine the result in [J] (about stable scissors congruence) with Zylev’s Theorem, which is proved in [Z]. The purpose of these notes is to explain the proof of Sydler’s theorem that is spread out in [J], [JKT], and [Z]. The paper [J] is beautiful and efficient, and I can hardly improve on it. I follow [J] very closely, except that I simplify wherever I can. The only place where I really depart from [J] is my sketch of the very difficult geometric Lemma 10.1, which is known as Sydler’s Fundamental Lemma. To supplement these notes, I made an interactive java applet that renders Sydler’s Fundamental Lemma transparent. The applet departs even further from Jessen’s treatment. [JKT] leaves a lot to be desired. The notation in [JKT] does not match the notation in [J] and the theorems proved in [JKT] are much more general than what is needed for [J]. -
The Bloch-Wigner-Ramakrishnan Polylogarithm Function
Math. Ann. 286, 613424 (1990) Springer-Verlag 1990 The Bloch-Wigner-Ramakrishnan polylogarithm function Don Zagier Max-Planck-Insfitut fiir Mathematik, Gottfried-Claren-Strasse 26, D-5300 Bonn 3, Federal Republic of Germany To Hans Grauert The polylogarithm function co ~n appears in many parts of mathematics and has an extensive literature [2]. It can be analytically extended to the cut plane ~\[1, ~) by defining Lira(x) inductively as x [ Li m_ l(z)z-tdz but then has a discontinuity as x crosses the cut. However, for 0 m = 2 the modified function O(x) = ~(Liz(x)) + arg(1 -- x) loglxl extends (real-) analytically to the entire complex plane except for the points x=0 and x= 1 where it is continuous but not analytic. This modified dilogarithm function, introduced by Wigner and Bloch [1], has many beautiful properties. In particular, its values at algebraic argument suffice to express in closed form the volumes of arbitrary hyperbolic 3-manifolds and the values at s= 2 of the Dedekind zeta functions of arbitrary number fields (cf. [6] and the expository article [7]). It is therefore natural to ask for similar real-analytic and single-valued modification of the higher polylogarithm functions Li,. Such a function Dm was constructed, and shown to satisfy a functional equation relating D=(x-t) and D~(x), by Ramakrishnan E3]. His construction, which involved monodromy arguments for certain nilpotent subgroups of GLm(C), is completely explicit, but he does not actually give a formula for Dm in terms of the polylogarithm. In this note we write down such a formula and give a direct proof of the one-valuedness and functional equation. -
Exploring Topics of the Art Gallery Problem
The College of Wooster Open Works Senior Independent Study Theses 2019 Exploring Topics of the Art Gallery Problem Megan Vuich The College of Wooster, [email protected] Follow this and additional works at: https://openworks.wooster.edu/independentstudy Recommended Citation Vuich, Megan, "Exploring Topics of the Art Gallery Problem" (2019). Senior Independent Study Theses. Paper 8534. This Senior Independent Study Thesis Exemplar is brought to you by Open Works, a service of The College of Wooster Libraries. It has been accepted for inclusion in Senior Independent Study Theses by an authorized administrator of Open Works. For more information, please contact [email protected]. © Copyright 2019 Megan Vuich Exploring Topics of the Art Gallery Problem Independent Study Thesis Presented in Partial Fulfillment of the Requirements for the Degree Bachelor of Arts in the Department of Mathematics and Computer Science at The College of Wooster by Megan Vuich The College of Wooster 2019 Advised by: Dr. Robert Kelvey Abstract Created in the 1970’s, the Art Gallery Problem seeks to answer the question of how many security guards are necessary to fully survey the floor plan of any building. These floor plans are modeled by polygons, with guards represented by points inside these shapes. Shortly after the creation of the problem, it was theorized that for guards whose positions were limited to the polygon’s j n k vertices, 3 guards are sufficient to watch any type of polygon, where n is the number of the polygon’s vertices. Two proofs accompanied this theorem, drawing from concepts of computational geometry and graph theory. -
Paul Erdős Mathematical Genius, Human (In That Order)
Paul Erdős Mathematical Genius, Human (In That Order) By Joshua Hill MATH 341-01 4 June 2004 "A Mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent that theirs, it is because the are made with ideas... The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colours of the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics." --G.H. Hardy "Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is." -- Paul Erdős "One of the first people I met in Princeton was Paul Erdős. He was 26 years old at the time, and had his Ph.D. for several years, and had been bouncing from one postdoctoral fellowship to another... Though I was slightly younger, I considered myself wiser in the ways of the world, and I lectured Erdős "This fellowship business is all well and good, but it can't go on for much longer -- jobs are hard to get -- you had better get on the ball and start looking for a real honest job." ... Forty years after my sermon, Erdős hasn't found it necessary to look for an "honest" job yet." -- Paul Halmos Introduction Paul Erdős (said "Air-daish") was a brilliant and prolific mathematician, who was central to the advancement of several major branches of mathematics. -
Proofs from the BOOK Third Edition Springer-Verlag Berlin Heidelberg Gmbh Martin Aigner Gunter M
Martin Aigner Gunter M. Ziegler Proofs from THE BOOK Third Edition Springer-Verlag Berlin Heidelberg GmbH Martin Aigner Gunter M. Ziegler Proofs from THE BOOK Third Edition With 250 Figures Including Illustrations by Karl H. Hofmann Springer Martin Aigner Gunter M. Ziegler Freie Universitat Berlin Technische Universitat Berlin Institut flir Mathematik II (WE2) Institut flir Mathematik, MA 6-2 Arnimallee 3 StraBe des 17. Juni 136 14195 Berlin, Germany 10623 Berlin, Germany email: [email protected] email: [email protected] Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de. Mathematics Subject Classification (2000): 00-01 (General) ISBN 978-3-662-05414-7 ISBN 978-3-662-05412-3 (eBook) DOl 10 .1007/978-3 -662-05412-3 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-Verlag. Viola tions are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1998,2001,2004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
Oberwolfach Jahresbericht Annual Report 2008 Herausgeber / Published By
titelbild_2008:Layout 1 26.01.2009 20:19 Seite 1 Oberwolfach Jahresbericht Annual Report 2008 Herausgeber / Published by Mathematisches Forschungsinstitut Oberwolfach Direktor Gert-Martin Greuel Gesellschafter Gesellschaft für Mathematische Forschung e.V. Adresse Mathematisches Forschungsinstitut Oberwolfach gGmbH Schwarzwaldstr. 9-11 D-77709 Oberwolfach-Walke Germany Kontakt http://www.mfo.de [email protected] Tel: +49 (0)7834 979 0 Fax: +49 (0)7834 979 38 Das Mathematische Forschungsinstitut Oberwolfach ist Mitglied der Leibniz-Gemeinschaft. © Mathematisches Forschungsinstitut Oberwolfach gGmbH (2009) JAHRESBERICHT 2008 / ANNUAL REPORT 2008 INHALTSVERZEICHNIS / TABLE OF CONTENTS Vorwort des Direktors / Director’s Foreword ......................................................................... 6 1. Besondere Beiträge / Special contributions 1.1 Das Jahr der Mathematik 2008 / The year of mathematics 2008 ................................... 10 1.1.1 IMAGINARY - Mit den Augen der Mathematik / Through the Eyes of Mathematics .......... 10 1.1.2 Besuch / Visit: Bundesministerin Dr. Annette Schavan ............................................... 17 1.1.3 Besuche / Visits: Dr. Klaus Kinkel und Dr. Dietrich Birk .............................................. 18 1.2 Oberwolfach Preis / Oberwolfach Prize ....................................................................... 19 1.3 Oberwolfach Vorlesung 2008 .................................................................................... 27 1.4 Nachrufe .............................................................................................................. -
The Art Gallery Theorem
Outline The players The theorem The proof from THE BOOK Variations The Art Gallery Theorem Vic Reiner, Univ. of Minnesota Monthly Math Hour at UW May 18, 2014 Vic Reiner, Univ. of Minnesota The Art Gallery Theorem Outline The players The theorem The proof from THE BOOK Variations 1 The players 2 The theorem 3 The proof from THE BOOK 4 Variations Vic Reiner, Univ. of Minnesota The Art Gallery Theorem Outline The players The theorem The proof from THE BOOK Variations Victor Klee, formerly of UW Vic Reiner, Univ. of Minnesota The Art Gallery Theorem Outline The players The theorem The proof from THE BOOK Variations Klee’s question posed to V. Chvátal Given the floor plan of a weirdly shaped art gallery having N straight sides, how many guards will we need to post, in the worst case, so that every bit of wall is visible to a guard? Can one do it with N=3 guards? Vic Reiner, Univ. of Minnesota The Art Gallery Theorem Outline The players The theorem The proof from THE BOOK Variations Klee’s question posed to V. Chvátal Given the floor plan of a weirdly shaped art gallery having N straight sides, how many guards will we need to post, in the worst case, so that every bit of wall is visible to a guard? Can one do it with N=3 guards? Vic Reiner, Univ. of Minnesota The Art Gallery Theorem Outline The players The theorem The proof from THE BOOK Variations Vasek Chvátal: Yes, I can prove that! Vic Reiner, Univ.