Mathematics : Selected Topics Beyond the Basic Courses PDF
Preview Mathematics : Selected Topics Beyond the Basic Courses
STUDENT MATHEMATICAL LIBRARY Volume 75 Mathematics++ Selected Topics Beyond the Basic Courses Ida Kantor Jiˇrí Matouˇsek Robert ˇSámal Mathematics++ Selected Topics Beyond the Basic Courses STUDENT MATHEMATICAL LIBRARY Volume 75 Mathematics++ Selected Topics Beyond the Basic Courses Ida Kantor Jirˇí Matousˇek Robert Sˇámal American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell(Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 14-01; 20Cxx, 28-01, 43-01, 52Axx, 54-01, 55-01. For additional information and updates on this book, visit www.ams.org/bookpages/stml-75 Library of Congress Cataloging-in-Publication Data Kantor,Ida,1981- Mathematics++: selectedtopicsbeyondthebasiccourses/IdaKantor,Jiˇr´ı Matouˇsek,RobertSˇ´amal. pagescm. –(Studentmathematicallibrary;volume75) Includesbibliographicalreferencesandindex. ISBN978-1-4704-2261-5(alk. paper) 1. Mathematics–Study and teaching (Graduate) 2. Computer science– Mathematics. I. Matouˇsek, Jiˇr´ı, 1963–2015 II. Sˇ´amal, Robert, 1977- III.Title. QA11.2.K36 2015 510–dc23 2015016136 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copyselectpagesforuseinteachingorresearch. Permissionisgrantedtoquotebrief passagesfromthispublicationinreviews,providedthecustomaryacknowledgmentof thesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthis publicationispermittedonlyunderlicensefromtheAmericanMathematicalSociety. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink(cid:2) service. For more information, please visit: http:// www.ams.org/rightslink. Send requests for translation rights and licensed reprints to reprint-permission @ams.org. Excluded from these provisions is material for which the author holds copyright. Insuchcases,requestsforpermissiontoreuseorreprintmaterialshouldbeaddressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article withinproceedings volumes. (cid:2)c2015AmericanMathematicalSociety. Allrightsreserved. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 201918171615 Contents Preface ix Chapter 1. Measure and Integral 1 §1. Measure 6 §2. The Lebesgue Integral 22 §3. Foundations of Probability Theory 31 §4. Literature 36 Bibliography 37 Chapter 2. High-Dimensional Geometry and Measure Concentration 39 §1. Peculiarities of Large Dimensions 41 §2. The Brunn–Minkowski Inequality and Euclidean Isoperimetry 44 §3. The Standard Normal Distribution and the Gaussian Measure 53 §4. Measure Concentration 61 §5. Literature 81 Bibliography 82 Chapter 3. Fourier Analysis 85 §1. Characters 87 §2. The Fourier Transform 94 v vi Contents §3. Two Unexpected Applications 99 §4. Convolution 106 §5. Poisson Summation Formula 109 §6. Influence of Variables 113 §7. Infinite Groups 124 §8. Literature 137 Bibliography 138 Chapter 4. Representations of Finite Groups 141 §1. Basic Definitions and Examples 142 §2. Decompositions into Irreducible Representations 145 §3. Irreducible Decompositions, Characters, Orthogonality 150 §4. Irreducible Representations of the Symmetric Group 160 §5. An Application in Communication Complexity 163 §6. More Applications and Literature 168 Bibliography 169 Chapter 5. Polynomials 173 §1. Rings, Fields, and Polynomials 173 §2. The Schwartz–Zippel Theorem 175 §3. Polynomial Identity Testing 176 §4. Interpolation, Joints, and Contagious Vanishing 180 §5. Varieties, Ideals, and the Hilbert Basis Theorem 185 §6. The Nullstellensatz 188 §7. B´ezout’s Inequality in the Plane 195 §8. More Properties of Varieties 200 §9. B´ezout’s Inequality in Higher Dimensions 219 §10. Bounding the Number of Connected Components 226 §11. Literature 232 Bibliography 232 Contents vii Chapter 6. Topology 235 §1. Topological Spaces and Continuous Maps 236 §2. Bits of General Topology 240 §3. Compactness 247 §4. Homotopy and Homotopy Equivalence 253 §5. The Borsuk–Ulam Theorem 257 §6. Operations on Topological Spaces 262 §7. Simplicial Complexes and Relatives 271 §8. Non-embeddability 283 §9. Homotopy Groups 289 §10. Homology of Simplicial Complexes 301 §11. Simplicial Approximation 309 §12. Homology Does Not Depend on Triangulation 314 §13. A Quick Harvest and Two More Theorems 318 §14. Manifolds 321 §15. Literature 330 Bibliography 330 Index 333 Preface This bookintroduces six selectedareasof mostly 20thcentury math- ematics. We assume that the reader has gone through the usual un- dergraduate courses and is used to rigorous presentation withproofs. Mathematicsisbeautiful,andusefulallover,butextensive. Even in computer science, one of the most mathematical fields besides mathematicsitself, university curriculamostly teachonlymathemat- ics developed prior to the 20th century (with the exception of areas more directly related to computing, such as logic or discrete mathe- matics). This is not because of lack of modernity, but because build- ing proper foundations takes a lot of time and there is hardly room for anything else, even when the mathematical courses occupy the maximum politically acceptable part of the curriculum. This observation was the starting point of a project resulting in this book. Contemporary research in computer science (but in otherfieldsaswell)usesnumerous mathematicaltoolsnotcoveredin the basic courses. We had the experience of struggling with papers containing mathematical terminology unknown to us, and we saw a number of other people having similar problems. We decided to teach a course, mainly for Ph.D. students of the- oretical computer science, introducing various mathematical areas in a concise and accessible way. With expected periodicity of three semesters, the course covered one to three areas per semester. ix
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