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Commentary on the Kervaire–Milnor Correspondence 1958–1961
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 52, Number 4, October 2015, Pages 603–609 http://dx.doi.org/10.1090/bull/1508 Article electronically published on July 1, 2015 COMMENTARY ON THE KERVAIRE–MILNOR CORRESPONDENCE 1958–1961 ANDREW RANICKI AND CLAUDE WEBER Abstract. The extant letters exchanged between Kervaire and Milnor during their collaboration from 1958–1961 concerned their work on the classification of exotic spheres, culminating in their 1963 Annals of Mathematics paper. Michel Kervaire died in 2007; for an account of his life, see the obituary by Shalom Eliahou, Pierre de la Harpe, Jean-Claude Hausmann, and Claude We- ber in the September 2008 issue of the Notices of the American Mathematical Society. The letters were made public at the 2009 Kervaire Memorial Confer- ence in Geneva. Their publication in this issue of the Bulletin of the American Mathematical Society is preceded by our commentary on these letters, provid- ing some historical background. Letter 1. From Milnor, 22 August 1958 Kervaire and Milnor both attended the International Congress of Mathemati- cians held in Edinburgh, 14–21 August 1958. Milnor gave an invited half-hour talk on Bernoulli numbers, homotopy groups, and a theorem of Rohlin,andKer- vaire gave a talk in the short communications section on Non-parallelizability of the n-sphere for n>7 (see [2]). In this letter written immediately after the Congress, Milnor invites Kervaire to join him in writing up the lecture he gave at the Con- gress. The joint paper appeared in the Proceedings of the ICM as [10]. Milnor’s name is listed first (contrary to the tradition in mathematics) since it was he who was invited to deliver a talk. -
Arxiv:1303.6028V2 [Math.DG] 29 Dec 2014 B Ahrglrlvlhprufc Scle an Called Is Hypersurface Level Regular Each E Scle the Called Is Set E.[T3 )
ISOPARAMETRIC FUNCTIONS ON EXOTIC SPHERES CHAO QIAN AND ZIZHOU TANG Abstract. This paper extends widely the work in [GT13]. Existence and non-existence results of isoparametric functions on exotic spheres and Eells-Kuiper projective planes are established. In particular, every homotopy n-sphere (n > 4) carries an isoparametric function (with certain metric) with 2 points as the focal set, in strong contrast to the classification of cohomogeneity one actions on homotopy spheres [St96] ( only exotic Kervaire spheres admit cohomogeneity one actions besides the standard spheres ). As an application, we improve a beautiful result of B´erard-Bergery [BB77] ( see also pp.234-235 of [Be78] ). 1. Introduction Let N be a connected complete Riemannian manifold. A non-constant smooth function f on N is called transnormal, if there exists a smooth function b : R R such that f 2 = → |∇ | b( f ), where f is the gradient of f . If in addition, there exists a continuous function a : ∇ R R so that f = a( f ), where f is the Laplacian of f , then f is called isoparametric. → △ △ Each regular level hypersurface is called an isoparametric hypersurface and the singular level set is called the focal set. The two equations of the function f mean that the regular level hypersurfaces of f are parallel and have constant mean curvatures, which may be regarded as a geometric generalization of cohomogeneity one actions in the theory of transformation groups ( ref. [GT13] ). Owing to E. Cartan and H. F. M¨unzner [M¨u80], the classification of isoparametric hy- persurfaces in a unit sphere has been one of the most challenging problems in submanifold geometry. -
EXOTIC SPHERES and CURVATURE 1. Introduction Exotic
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 45, Number 4, October 2008, Pages 595–616 S 0273-0979(08)01213-5 Article electronically published on July 1, 2008 EXOTIC SPHERES AND CURVATURE M. JOACHIM AND D. J. WRAITH Abstract. Since their discovery by Milnor in 1956, exotic spheres have pro- vided a fascinating object of study for geometers. In this article we survey what is known about the curvature of exotic spheres. 1. Introduction Exotic spheres are manifolds which are homeomorphic but not diffeomorphic to a standard sphere. In this introduction our aims are twofold: First, to give a brief account of the discovery of exotic spheres and to make some general remarks about the structure of these objects as smooth manifolds. Second, to outline the basics of curvature for Riemannian manifolds which we will need later on. In subsequent sections, we will explore the interaction between topology and geometry for exotic spheres. We will use the term differentiable to mean differentiable of class C∞,and all diffeomorphisms will be assumed to be smooth. As every graduate student knows, a smooth manifold is a topological manifold that is equipped with a smooth (differentiable) structure, that is, a smooth maximal atlas. Recall that an atlas is a collection of charts (homeomorphisms from open neighbourhoods in the manifold onto open subsets of some Euclidean space), the domains of which cover the manifold. Where the chart domains overlap, we impose a smooth compatibility condition for the charts [doC, chapter 0] if we wish our manifold to be smooth. Such an atlas can then be extended to a maximal smooth atlas by including all possible charts which satisfy the compatibility condition with the original maps. -
Souhrnný Rejstřík Za Léta 1986–2000
Pokroky matematiky, fyziky a astronomie Souhrnný rejstřík PMFA za léta 1986–2000 Pokroky matematiky, fyziky a astronomie, Vol. 45 (2000), No. 4, 313--352 Persistent URL: http://dml.cz/dmlcz/141055 Terms of use: © Jednota českých matematiků a fyziků, 2000 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz Souhrnný rejstřík PMFA za léta 1986–2000 Obsah Matematika a informatika ..................................................... 313 Fyzika ........................................................................ 318 Astronomie ................................................................... 322 Diskuse ....................................................................... 322 Vyučování .................................................................... 323 Jubilea českých a slovenských matematiků a fyziků ............................ 325 Nekrology .................................................................... 329 Ze života JČMF, JSMF a JČSMF ............................................. 330 Zprávy ....................................................................... 335 Různé ........................................................................ 340 Rejstřík českých a slovenských -
La Mathématicienne Française Du Jour
LaLa mathématiciennemathématicienne françaisefrançaise dudu jourjour Nalini Anantharaman Née à Paris en 1976 Biographie Elle étudie les mathématiques à l'École normale supérieure , puis enseigne à l’École normale supérieure (ENS) de Lyon et à l’École polytechnique. Elle est actuellement professeur à l'université Paris-Sud (Orsay) Vice-présidente de la société Mathématique de France Distinctions Prix Henri-Poincaré 2012 : l'une des quatre lauréats. Médaille d'argent du CNRS en 2013. Prix Jacques Herbrand en 2011. Thème de recherche Ses travaux «se situent à l’interface entre la théorie des systèmes dynamiques classiques et l’analyse des équations aux dérivées partielles», selon le CNRS. La mathématicienne Nalini Anantharaman, enseignante- mouvements pouvaient en fait être désordonnés sur le très chercheuse fait dans la simplicité lorsqu’elle évoque son long terme, et non réguliers comme l’observation conduisait métier. Le prix Henri Poincaré, du nom du célèbre à le supposer. Il a ainsi élaboré une théorie abstraite sur la mathématicien français du début du XXème siècle notion d’évolution ordonnée et chaotique. Tout au long du récompense traditionnellement des chercheurs en “physique XXème siècle, de nombreux chercheurs ont développé des mathématique”. Nalini Anantharaman fait partie des quatre concepts autour de cette idée pour les appliquer à d’autres chercheurs à avoir été salués pour leurs travaux. Le prix a domaines. Comme par exemple les évolutions biologiques. été remis au début du mois d’août 2012, mais Nalini Mais cette théorie du régulier et du chaos ne s’applique pas Anantharaman n’y était pas : la jeune chercheuse donnait au monde quantique. -
Newsletter · Thursday 18Th
KEEP UPDATED NEWSLETTER · THURSDAY 18TH ICIAM'S JOURNEY Like its predecessors, a congress such as ICIAM 2019 would have never been countries like South Africa. ICIAM possible without the organization and supervision of the International Council has the ability to promote for Industrial and Applied Mathematics (ICIAM), a formal entity created in 1990 mathematics all over the world". to supervise these quadrennial meetings. From Paris 1987 to Valencia 2019 many things have changed, and not just in the field of Industrial and Applied The current and future presidents Mathematics. of the Council agree in that ICIAM 2019 in Valencia has been an The original societies that founded ICIAM (GAMM, IMA, SIAM, and SMAI) are absolute success. "In my opinion now surrounded by many other members from around the world and the everything has worked very well Council has expanded its activities significantly since 2004. despite the huge number of participants", says Esteban. "The The Council is responsible for choosing the venue and the 27 invited speakers, conferences are of high quality and which are chosen with a diverse criteria, not only mathematically and in general, especially in the main geographicall, but also with respect to academic vs. industrial work, gender or conferences —invited and prizes— type of academic institution. Furthermore, it is also responsible for overseeing the lecturers have made a real the selection of the five ICIAM prizes —Collatz, Lagrange, Maxwell, Su Buchin, Ya-Xiang Yuan recalls attending ICIAM in effort to present their results in a and Pioneer— awarded every congress. its 1995 edition in Hamburg. "We were comprehensible and pleasant way only 20 Chinese and now we're like 300, for a great variety of participants as well as create a database with all so the same thing can happen to working in very diverse areas ". -
Number Theory, Analysis and Geometry
Number Theory, Analysis and Geometry Dorian Goldfeld • Jay Jorgenson • Peter Jones Dinakar Ramakrishnan • Kenneth A. Ribet John Tate Editors Number Theory, Analysis and Geometry In Memory of Serge Lang 123 Editors Dorian Goldfeld Jay Jorgenson Department of Mathematics Department of Mathematics Columbia University City University of New York New York, NY 10027 New York, NY 10031 USA USA [email protected] [email protected] Peter Jones Dinakar Ramakrishnan Department of Mathematics Department of Mathematics Yale University California Institute of Technology New Haven, CT 06520 Pasadena, CA 91125 USA USA [email protected] [email protected] Kenneth A. Ribet John Tate Department of Mathematics Department of Mathematics University of California at Berkeley Harvard University Berkeley, CA 94720 Cambridge, MA 02138 USA USA [email protected] [email protected] ISBN 978-1-4614-1259-5 e-ISBN 978-1-4614-1260-1 DOI 10.1007/978-1-4614-1260-1 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011941121 © Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. -
From the Associate Executive Director's Desk
FROM THE ASSOCIATE Executive Director’s Desk Joseph Khoury University of Ottawa, Ottawa IN THIS ISSUE DANS CE NUMÉRO for the second year in a December 2008, the Board of row, and my decision not to Directors approved the move Editorial ....................................2 assume the Executive Director to a new location owned by position did not make things the Pharmacists Association of Book Review: Lewis Carroll in easier. The determination and Canada. A one year renewable Numberland and Kelvin: Life, hard work of the Executive lease was signed in December Labours and Legacy ..............4 Office staff, the efforts made and the move took place over Book Review: Mathematics by the Dean of Science and the Holiday period. The new the Chair of the Department address is: and Technology .....................5 This was supposed to be my first message as the new Executive of Mathematics and Statistics Brief Book Reviews ..................6 Director of the Canadian at the University of Ottawa Canadian Mathematical Society and their staff got us through 1785 Alta Vista Drive, Suite 105 Education Notes .......................7 Mathematical Society. As many of you know by now, I informed this hard time. On behalf Ottawa, ON K1G 3Y6 Canadian Mathematics the Executive Committee of the CMS, I would like to CANADA Education Forum 2009 ...........10 and the Board of Directors thank the dedicated and hard working Executive Office staff, The main telephone number CMS/CSHPM Summer Meeting in October 2008 about my decision not to assume the André Lalonde (Dean), Victor is (613) 733-2662. Staff e- 2009 / Réunion d’été 2009 Leblanc (Chair) and their staff mail addresses remain the SMC/SCHPM ............................11 position effective January 1, 2009 for reasons beyond my for their help on the many same. -
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. -
Downbeat.Com September 2010 U.K. £3.50
downbeat.com downbeat.com september 2010 2010 september £3.50 U.K. DownBeat esperanza spalDing // Danilo pérez // al Di Meola // Billy ChilDs // artie shaw septeMBer 2010 SEPTEMBER 2010 � Volume 77 – Number 9 President Kevin Maher Publisher Frank Alkyer Editor Ed Enright Associate Editor Aaron Cohen Art Director Ara Tirado Production Associate Andy Williams Bookkeeper Margaret Stevens Circulation Manager Kelly Grosser AdVertisiNg sAles Record Companies & Schools Jennifer Ruban-Gentile 630-941-2030 [email protected] Musical Instruments & East Coast Schools Ritche Deraney 201-445-6260 [email protected] Classified Advertising Sales Sue Mahal 630-941-2030 [email protected] offices 102 N. Haven Road Elmhurst, IL 60126–2970 630-941-2030 Fax: 630-941-3210 http://downbeat.com [email protected] customer serVice 877-904-5299 [email protected] coNtributors Senior Contributors: Michael Bourne, John McDonough, Howard Mandel Atlanta: Jon Ross; Austin: Michael Point; Boston: Fred Bouchard, Frank-John Hadley; Chicago: John Corbett, Alain Drouot, Michael Jackson, Peter Margasak, Bill Meyer, Mitch Myers, Paul Natkin, How- ard Reich; Denver: Norman Provizer; Indiana: Mark Sheldon; Iowa: Will Smith; Los Angeles: Earl Gibson, Todd Jenkins, Kirk Silsbee, Chris Walker, Joe Woodard; Michigan: John Ephland; Minneapolis: Robin James; Nashville: Robert Doerschuk; New Orleans: Erika Goldring, David Kunian; New York: Alan Bergman, Herb Boyd, Bill Douthart, Ira Gitler, Eugene Gologursky, Norm Harris, D.D. Jackson, Jimmy Katz, Jim Macnie, Ken Micallef, Jennifer -
Studies in Basidiodendron Eyrei and Similar-Looking Taxa (Auriculariales, Basidiomycota)
Botany Studies in Basidiodendron eyrei and similar-looking taxa (Auriculariales, Basidiomycota) Journal: Botany Manuscript ID cjb-2020-0045.R2 Manuscript Type: Article Date Submitted by the 14-Jul-2020 Author: Complete List of Authors: Spirin , Viacheslav ; Botany Unit (Mycology), Finnish Museum of Natural History, University of Helsinki., Malysheva, Vera; Komarov Botanical Institute RAS, Laboratory of Systematic and Geography of Fungi Mendes Alvarenga,Draft Renato Lúcio ; Universidade Federal de Pernambuco Centro de Biociencias, Micologia Kotiranta, Heikki; Finnish Environment Institute Larsson, Karl-Henrik; Natural History Museum , University of Oslo, Keyword: heterobasidiomycetes, phylogeny, taxonomy Is the invited manuscript for consideration in a Special Not applicable (regular submission) Issue? : https://mc06.manuscriptcentral.com/botany-pubs Page 1 of 40 Botany Studies in Basidiodendron eyrei and similar-looking taxa (Auriculariales, Basidiomycota) Viacheslav Spirin1, Vera Malysheva2, Renato Lucio Mendes-Alvarenga3, Heikki Kotiranta4, Karl-Henrik Larsson5,6 1Botany Unit (Mycology), Finnish Museum of Natural History, University of Helsinki. P.O. Box 7, FI-00014 Helsinki, Finland. e-mail: [email protected] 2Komarov Botanical Institute, Russian Academy of Sciences, Prof. Popova Str., 197376 St. Petersburg, Russia. e-mail: [email protected] 3Departamento de Micologia, Centro de Biociências, Universidade Federal de Pernambuco, Av. da Engenharia s/n, Recife, Pernambuco,Draft 50740-570, Brazil. e-mail: [email protected] 4Finnish Environment Institute, Latokartanonkaari 11, 00790 Helsinki, Finland. e-mail: [email protected] 5Natural History Museum, University of Oslo, P.O. Box 1172, Blindern, 0318 Oslo, Norway. e- mail: [email protected] 6Gothenburg Global Biodiversity Centre, Post Box 461, 40530 Gothenburg, Sweden Corresponding author: Viacheslav Spirin. -
Probability and Random Processes ECS 315
Probability and Random Processes ECS 315 Asst. Prof. Dr. Prapun Suksompong [email protected] 4 Combinatorics Office Hours: BKD, 6th floor of Sirindhralai building Tuesday 9:00-10:00 Wednesday 14:20-15:20 1 Thursday 9:00-10:00 Supplementary References Mathematics of Choice How to count without counting By Ivan Niven permutations, combinations, binomial coefficients, the inclusion-exclusion principle, combinatorial probability, partitions of numbers, generating polynomials, the pigeonhole principle, and much more A Course in Combinatorics By J. H. van Lint and R. M. Wilson 2 Cartesian product The Cartesian product A × B is the set of all ordered pairs where and . Named after René Descartes His best known philosophical statement is “Cogito ergo sum” (French: Je pense, donc je suis; I think, therefore I am) 3 [http://mathemagicalworld.weebly.com/rene-descartes.html] Heads, Bodies and Legs flip-book 4 Heads, Bodies and Legs flip-book (2) 5 Interactive flipbook: Miximal 6 [http://www.fastcodesign.com/3028287/miximal-is-the-new-worlds-cutest-ipad-app] One Hundred Thousand Billion Poems 7 [https://www.youtube.com/watch?v=2NhFoSFNQMQ] One Hundred Thousand Billion Poems Cent mille milliards de poèmes 8 Pokemon Go: Designing how your avatar looks 9 How many chess games are possible? 10 [https://www.youtube.com/watch?v=Km024eldY1A] An Estimate: The Shannon Number 11 [https://www.youtube.com/watch?v=Km024eldY1A] An Estimate: The Shannon Number 12 [https://www.youtube.com/watch?v=Km024eldY1A] An Estimate: The Shannon Number 13 [https://www.youtube.com/watch?v=Km024eldY1A] The Shannon Number: Just an Estimate 14 [https://www.youtube.com/watch?v=Km024eldY1A] Example: Sock It Two Me 15 [Greenes, 1977] Example: Sock It Two Me Jack is so busy that he's always throwing his socks into his top drawer without pairing them.