Of the American Mathematical Society October 2018 Volume 65, Number 9
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Geometric Integration Theory Contents
Steven G. Krantz Harold R. Parks Geometric Integration Theory Contents Preface v 1 Basics 1 1.1 Smooth Functions . 1 1.2Measures.............................. 6 1.2.1 Lebesgue Measure . 11 1.3Integration............................. 14 1.3.1 Measurable Functions . 14 1.3.2 The Integral . 17 1.3.3 Lebesgue Spaces . 23 1.3.4 Product Measures and the Fubini–Tonelli Theorem . 25 1.4 The Exterior Algebra . 27 1.5 The Hausdorff Distance and Steiner Symmetrization . 30 1.6 Borel and Suslin Sets . 41 2 Carath´eodory’s Construction and Lower-Dimensional Mea- sures 53 2.1 The Basic Definition . 53 2.1.1 Hausdorff Measure and Spherical Measure . 55 2.1.2 A Measure Based on Parallelepipeds . 57 2.1.3 Projections and Convexity . 57 2.1.4 Other Geometric Measures . 59 2.1.5 Summary . 61 2.2 The Densities of a Measure . 64 2.3 A One-Dimensional Example . 66 2.4 Carath´eodory’s Construction and Mappings . 67 2.5 The Concept of Hausdorff Dimension . 70 2.6 Some Cantor Set Examples . 73 i ii CONTENTS 2.6.1 Basic Examples . 73 2.6.2 Some Generalized Cantor Sets . 76 2.6.3 Cantor Sets in Higher Dimensions . 78 3 Invariant Measures and the Construction of Haar Measure 81 3.1 The Fundamental Theorem . 82 3.2 Haar Measure for the Orthogonal Group and the Grassmanian 90 3.2.1 Remarks on the Manifold Structure of G(N,M).... 94 4 Covering Theorems and the Differentiation of Integrals 97 4.1 Wiener’s Covering Lemma and its Variants . -
Caucher Birkar — from Asylum Seeker to Fields Medal Winner at Cambridge
MATHS, 1 Caucher Birkar, 41, at VERSION Cambridge University, photographed by Jude Edginton REPR O OP HEARD THE ONE ABOUT THE ASYLUM SEEKER SUBS WHO WANDERED INTO A BRITISH UNIVERSITY... A RT AND CAME OUT A MATHS SUPERSTAR? PR ODUCTION CLIENT Caucher Birkar grew up in a Kurdish peasant family in a war zone and arrived in Nottingham as a refugee – now he has received the mathematics equivalent of the Nobel prize. By Tom Whipple BLACK YELLOW MAGENTA CYAN 91TTM1940232.pgs 01.04.2019 17:39 MATHS, 2 VERSION ineteen years ago, the mathematics Caucher Birkar in Isfahan, Receiving the Fields Medal If that makes sense, congratulations: you department at the University of Iran, in 1999 in Rio de Janeiro, 2018 now have a very hazy understanding of Nottingham received an email algebraic geometry. This is the field that from an asylum seeker who Birkar works in. wanted to talk to someone about The problem with explaining maths is REPR algebraic geometry. not, or at least not always, the stupidity of his They replied and invited him in. listeners. It is more fundamental than that: O OP N So it was that, shortly afterwards, it is language. Mathematics is not designed Caucher Birkar, the 21-year-old to be described in words. It is designed to be son of a Kurdish peasant family, described in mathematics. This is the great stood in front of Ivan Fesenko, a professor at triumph of the subject. It was why a Kurdish Nottingham, and began speaking in broken asylum seeker with bad English could convince SUBS English. -
The Singular Structure and Regularity of Stationary Varifolds 3
THE SINGULAR STRUCTURE AND REGULARITY OF STATIONARY VARIFOLDS AARON NABER AND DANIELE VALTORTA ABSTRACT. If one considers an integral varifold Im M with bounded mean curvature, and if S k(I) x ⊆ ≡ { ∈ M : no tangent cone at x is k + 1-symmetric is the standard stratification of the singular set, then it is well } known that dim S k k. In complete generality nothing else is known about the singular sets S k(I). In this ≤ paper we prove for a general integral varifold with bounded mean curvature, in particular a stationary varifold, that every stratum S k(I) is k-rectifiable. In fact, we prove for k-a.e. point x S k that there exists a unique ∈ k-plane Vk such that every tangent cone at x is of the form V C for some cone C. n 1 n × In the case of minimizing hypersurfaces I − M we can go further. Indeed, we can show that the singular ⊆ set S (I), which is known to satisfy dim S (I) n 8, is in fact n 8 rectifiable with uniformly finite n 8 measure. ≤ − − − 7 An effective version of this allows us to prove that the second fundamental form A has apriori estimates in Lweak on I, an estimate which is sharp as A is not in L7 for the Simons cone. In fact, we prove the much stronger | | 7 estimate that the regularity scale rI has Lweak-estimates. The above results are in fact just applications of a new class of estimates we prove on the quantitative k k k k + stratifications S ǫ,r and S ǫ S ǫ,0. -
Birational Geometry of Algebraic Varieties
Proc. Int. Cong. of Math. – 2018 Rio de Janeiro, Vol. 1 (563–588) BIRATIONAL GEOMETRY OF ALGEBRAIC VARIETIES Caucher Birkar 1 Introduction This is a report on some of the main developments in birational geometry in recent years focusing on the minimal model program, Fano varieties, singularities and related topics, in characteristic zero. This is not a comprehensive survey of all advances in birational geometry, e.g. we will not touch upon the positive characteristic case which is a very active area of research. We will work over an algebraically closed field k of characteristic zero. Varieties are all quasi-projective. Birational geometry, with the so-called minimal model program at its core, aims to classify algebraic varieties up to birational isomorphism by identifying “nice” elements in each birational class and then classifying such elements, e.g study their moduli spaces. Two varieties are birational if they contain isomorphic open subsets. In dimension one, a nice element in a birational class is simply a smooth and projective element. In higher dimension though there are infinitely many such elements in each class, so picking a rep- resentative is a very challenging problem. Before going any further lets introduce the canonical divisor. 1.1 Canonical divisor. To understand a variety X one studies subvarieties and sheaves on it. Subvarieties of codimension one and their linear combinations, that is, divisors play a crucial role. Of particular importance is the canonical divisor KX . When X is smooth this is the divisor (class) whose associated sheaf OX (KX ) is the canonical sheaf !X := det ΩX where ΩX is the sheaf of regular differential forms. -
These Links No Longer Work. Springer Have Pulled the Free Plug
Sign up for a GitHub account Sign in Instantly share code, notes, and snippets. Create a gist now bishboria / springer-free-maths-books.md Last active 10 minutes ago Code Revisions 7 Stars 2104 Forks 417 Embed <script src="https://gist.githu b .coDmo/wbinslohabodr ZiIaP/8326b17bbd652f34566a.js"></script> Springer made a bunch of books available for free, these were the direct links springer-free-maths-books.md Raw These links no longer work. Springer have pulled the free plug. Graduate texts in mathematics duplicates = multiple editions A Classical Introduction to Modern Number Theory, Kenneth Ireland Michael Rosen A Classical Introduction to Modern Number Theory, Kenneth Ireland Michael Rosen A Course in Arithmetic, Jean-Pierre Serre A Course in Computational Algebraic Number Theory, Henri Cohen A Course in Differential Geometry, Wilhelm Klingenberg A Course in Functional Analysis, John B. Conway A Course in Homological Algebra, P. J. Hilton U. Stammbach A Course in Homological Algebra, Peter J. Hilton Urs Stammbach A Course in Mathematical Logic, Yu. I. Manin A Course in Number Theory and Cryptography, Neal Koblitz A Course in Number Theory and Cryptography, Neal Koblitz A Course in Simple-Homotopy Theory, Marshall M. Cohen A Course in p-adic Analysis, Alain M. Robert A Course in the Theory of Groups, Derek J. S. Robinson A Course in the Theory of Groups, Derek J. S. Robinson A Course on Borel Sets, S. M. Srivastava A Course on Borel Sets, S. M. Srivastava A First Course in Noncommutative Rings, T. Y. Lam A First Course in Noncommutative Rings, T. Y. Lam A Hilbert Space Problem Book, P. -
Nominations for President
ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society September 2013 Volume 60, Number 8 The Calculus Concept Inventory— Measurement of the Effect of Teaching Methodology in Mathematics page 1018 DML-CZ: The Experience of a Medium- Sized Digital Mathematics Library page 1028 Fingerprint Databases for Theorems page 1034 A History of the Arf-Kervaire Invariant Problem page 1040 About the cover: 63 years since ENIAC broke the ice (see page 1113) Solve the differential equation. Solve the differential equation. t ln t dr + r = 7tet dt t ln t dr + r = 7tet dt 7et + C r = 7et + C ln t ✓r = ln t ✓ WHO HAS THE #1 HOMEWORK SYSTEM FOR CALCULUS? THE ANSWER IS IN THE QUESTIONS. When it comes to online calculus, you need a solution that can grade the toughest open-ended questions. And for that there is one answer: WebAssign. WebAssign’s patent pending grading engine can recognize multiple correct answers to the same complex question. Competitive systems, on the other hand, are forced to use multiple choice answers because, well they have no choice. And speaking of choice, only WebAssign supports every major textbook from every major publisher. With new interactive tutorials and videos offered to every student, it’s not hard to see why WebAssign is the perfect answer to your online homework needs. It’s all part of the WebAssign commitment to excellence in education. Learn all about it now at webassign.net/math. 800.955.8275 webassign.net/math WA Calculus Question ad Notices.indd 1 11/29/12 1:06 PM Notices 1051 of the American Mathematical Society September 2013 Communications 1048 WHAT IS…the p-adic Mandelbrot Set? Joseph H. -
Bibliography
Bibliography [AK98] V. I. Arnold and B. A. Khesin, Topological methods in hydrodynamics, Springer- Verlag, New York, 1998. [AL65] Holt Ashley and Marten Landahl, Aerodynamics of wings and bodies, Addison- Wesley, Reading, MA, 1965, Section 2-7. [Alt55] M. Altman, A generalization of Newton's method, Bulletin de l'academie Polonaise des sciences III (1955), no. 4, 189{193, Cl.III. [Arm83] M.A. Armstrong, Basic topology, Springer-Verlag, New York, 1983. [Bat10] H. Bateman, The transformation of the electrodynamical equations, Proc. Lond. Math. Soc., II, vol. 8, 1910, pp. 223{264. [BB69] N. Balabanian and T.A. Bickart, Electrical network theory, John Wiley, New York, 1969. [BLG70] N. N. Balasubramanian, J. W. Lynn, and D. P. Sen Gupta, Differential forms on electromagnetic networks, Butterworths, London, 1970. [Bos81] A. Bossavit, On the numerical analysis of eddy-current problems, Computer Methods in Applied Mechanics and Engineering 27 (1981), 303{318. [Bos82] A. Bossavit, On finite elements for the electricity equation, The Mathematics of Fi- nite Elements and Applications IV (MAFELAP 81) (J.R. Whiteman, ed.), Academic Press, 1982, pp. 85{91. [Bos98] , Computational electromagnetism: Variational formulations, complementar- ity, edge elements, Academic Press, San Diego, 1998. [Bra66] F.H. Branin, The algebraic-topological basis for network analogies and the vector cal- culus, Proc. Symp. Generalised Networks, Microwave Research, Institute Symposium Series, vol. 16, Polytechnic Institute of Brooklyn, April 1966, pp. 453{491. [Bra77] , The network concept as a unifying principle in engineering and physical sciences, Problem Analysis in Science and Engineering (K. Husseyin F.H. Branin Jr., ed.), Academic Press, New York, 1977. -
A Century of Mathematics in America, Peter Duren Et Ai., (Eds.), Vol
Garrett Birkhoff has had a lifelong connection with Harvard mathematics. He was an infant when his father, the famous mathematician G. D. Birkhoff, joined the Harvard faculty. He has had a long academic career at Harvard: A.B. in 1932, Society of Fellows in 1933-1936, and a faculty appointmentfrom 1936 until his retirement in 1981. His research has ranged widely through alge bra, lattice theory, hydrodynamics, differential equations, scientific computing, and history of mathematics. Among his many publications are books on lattice theory and hydrodynamics, and the pioneering textbook A Survey of Modern Algebra, written jointly with S. Mac Lane. He has served as president ofSIAM and is a member of the National Academy of Sciences. Mathematics at Harvard, 1836-1944 GARRETT BIRKHOFF O. OUTLINE As my contribution to the history of mathematics in America, I decided to write a connected account of mathematical activity at Harvard from 1836 (Harvard's bicentennial) to the present day. During that time, many mathe maticians at Harvard have tried to respond constructively to the challenges and opportunities confronting them in a rapidly changing world. This essay reviews what might be called the indigenous period, lasting through World War II, during which most members of the Harvard mathe matical faculty had also studied there. Indeed, as will be explained in §§ 1-3 below, mathematical activity at Harvard was dominated by Benjamin Peirce and his students in the first half of this period. Then, from 1890 until around 1920, while our country was becoming a great power economically, basic mathematical research of high quality, mostly in traditional areas of analysis and theoretical celestial mechanics, was carried on by several faculty members. -
The Vile, Dopy, Evil and Odious Game Players
The vile, dopy, evil and odious game players Aviezri S. Fraenkel¤ Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot 76100, Israel February 7, 2010 Wil: We have worked and published jointly, for example on graph focality. Now that one of us is already an octogenarian and the other is only a decade away, let's have some fun; let's play a game. Gert: I'm game. Wil: In the Fall 2009 issue of the MSRI gazette Emissary, Elwyn Berlekamp and Joe Buhler proposed the following puzzle: \Nathan and Peter are playing a game. Nathan always goes ¯rst. The players take turns changing a positive integer to a smaller one and then passing the smaller number back to their op- ponent. On each move, a player may either subtract one from the integer or halve it, rounding down if necessary. Thus, from 28 the legal moves are to 27 or to 14; from 27, the legal moves are to 26 or to 13. The game ends when the integer reaches 0. The player who makes the last move wins. For example, if the starting integer is 15, a legal sequence of moves might be to 7, then 6, then 3, then 2, then 1, and then to 0. (In this sample game one of the players could have played better!) Assuming both Nathan and Peter play according to the best possible strategy, who will win if the starting integer is 1000? 2000?" Let's dub it the Mark game, since it's due to Mark Krusemeyer according to Berlekamp and Buhler. -
Introduction to Representation Theory
STUDENT MATHEMATICAL LIBRARY Volume 59 Introduction to Representation Theory Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina with historical interludes by Slava Gerovitch http://dx.doi.org/10.1090/stml/059 STUDENT MATHEMATICAL LIBRARY Volume 59 Introduction to Representation Theory Pavel Etingof Oleg Golberg Sebastian Hensel Tiankai Liu Alex Schwendner Dmitry Vaintrob Elena Yudovina with historical interludes by Slava Gerovitch American Mathematical Society Providence, Rhode Island Editorial Board Gerald B. Folland Brad G. Osgood (Chair) Robin Forman John Stillwell 2010 Mathematics Subject Classification. Primary 16Gxx, 20Gxx. For additional information and updates on this book, visit www.ams.org/bookpages/stml-59 Library of Congress Cataloging-in-Publication Data Introduction to representation theory / Pavel Etingof ...[et al.] ; with historical interludes by Slava Gerovitch. p. cm. — (Student mathematical library ; v. 59) Includes bibliographical references and index. ISBN 978-0-8218-5351-1 (alk. paper) 1. Representations of algebras. 2. Representations of groups. I. Etingof, P. I. (Pavel I.), 1969– QA155.I586 2011 512.46—dc22 2011004787 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 copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904- 2294 USA. -
Program Cincinnati Solving the Biggest Challenges in the Digital Universe
July 31-Aug 3, 2019 PROGRAM CINCINNATI SOLVING THE BIGGEST CHALLENGES IN THE DIGITAL UNIVERSE. At Akamai, we thrive on solving complex challenges for businesses, helping them digitally transform, outpace competitors, and achieve their goals. Cloud delivery and security. Video streaming. Secure application access. Our solutions make it easier for many of the world’s top brands to deliver the best, most secure digital experiences — in industries like entertainment, sports, gaming, nance, retail, software, and others. We helped broadcasters deliver high-quality live streaming during the 2018 Pyeongchang Games. We mitigated a record-breaking, memcached-fueled 1.3 Tbps DDoS attack. We’ve managed Black Friday web trafc for the biggest retailers on the planet. SECURE AND GROW YOUR BUSINESS. AKAMAI.COM WELCOME TO MAA MATHFEST! The MAA is pleased that you have joined us in Cincinnati for the math event of the summer. What are my favorite things to do at MAA MathFest? Attend the Invited Addresses! When I think back on prior MAA MathFest meetings, the Invited Addresses are the talks that I still remember and that have renewed my excitement for mathematics. This year will continue that tradition. We have excellent speakers presenting on a variety of exciting topics. If you see an Invited Address title that looks interesting, go to that talk. It will be worth it. Remember to attend the three 20-minute talks given by the MAA Adler Teaching Award winners on Friday afternoon. Jumpstart your passion for teaching and come hear these great educators share their TABLE OF CONTENTS insights on teaching, connecting with students, and the answer to “life, the universe and everything” (okay, maybe they won’t talk about 3 EARLE RAYMOND HEDRICK LECTURE SERIES the last item, but I am sure they will give inspiring and motivating presentations). -
A "Rule of Five" Framework for Models and Modeling to Unify Mathematicians and Biologists and Improve Student Learning
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/305119086 A "Rule of Five" Framework for Models and Modeling to Unify Mathematicians and Biologists and Improve Student Learning Article · June 2016 CITATIONS READS 12 533 6 authors, including: Carrie Diaz Eaton Hannah Callender Highlander Bates College University of Portland 38 PUBLICATIONS 87 CITATIONS 35 PUBLICATIONS 195 CITATIONS SEE PROFILE SEE PROFILE Michael Drew LaMar Richard Schugart College of William and Mary Western Kentucky University 36 PUBLICATIONS 237 CITATIONS 14 PUBLICATIONS 378 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: Inclusion and Diversity in STEM View project Interdisciplinary Mathematics Education View project All content following this page was uploaded by Carrie Diaz Eaton on 10 May 2017. The user has requested enhancement of the downloaded file. Framework for Teaching Models and Modeling 1 A “Rule-of-Five” Framework for Models and Modeling to Unify Mathematicians and Biologists and Improve Student Learning C. D. Eaton Environmental Literacy Program Unity College Unity, ME 04444, USA [email protected] H. C. Highlander Department of Mathematics University of Portland Portland, OR 97203, USA [email protected] K. D. Dahlquist Department of Biology Loyola Marymount University Los Angeles, CA 90045, USA [email protected] M. D. LaMar Department of Biology College of William and Mary Williamsburg, VA 23185, USA [email protected] 2 CD. Eaton