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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. -
Identity Evropa
AGAINST IDENTITY EVROPA AGAINST THE ALT RIGHT Big Nazi On Campus May 15, 2016 ON FRIDAY, May 6th, white nationalist Richard Spencer, President and director of National Policy Institute (NPI), (a think tank aimed at mil- lennials and educated adults that puts on conferences), and head of its publishing arm Washington Summit Publishers, arrived just before 3pm at UC Berkeley. Encircled by three other white nationalists, Spen- cer walked from the street through several corridors and hallways until finally making his way to Sproul Plaza where a group of other supporters had already gathered and started to live-stream and hold signs. In doing so, Spencer was stepping out of the world of paid con- ferences and weekly podcasts and into the terrain of street activism. Having announced the event on his twitter 48 hours before hand and working with Red Ice Radio, a live-streaming and in home studio run by a white nationalist married couple, the National Policy Institute along with Identity Europa, the youth wing of the American Freedom Party, (a key organizer for ANP is David Duke’s former right-hand man, Jamie Kelso), a Neo-Nazi formation, was working to create a “virtual rally.” The event itself was billed as a “Safe Space” to talk about race in America, using language common among left-wing, ac- tivist, and anarchist spaces. Before the rally even began, Spencer’s fellow white nationalists at Red Ice were already playing up what they imagined was going to happen that day. “Here is is, the birth of the free speech movement, and all of these liberals aren’t going to be able to stand white people talking about race,” they stated, (as if somehow Further resources Berkeley was devoid of white people doing just that). -
Fcaglp, Unlp, 2018
Scientific Philosophy Gustavo E. Romero IAR-CONICET/UNLP, Argentina FCAGLP, UNLP, 2018 Epistemology Episteme, as distinguished from techne, is etymologically derived from the Ancient Greek word ἐπιστήμη for knowledge or science, which comes from the verb ἐπίσταμαι, "to know". In Plato's terminology episteme means knowledge, as in "justified true belief", in contrast to doxa, common belief or opinion. The word epistemology, meaning the study of knowledge, is derived from episteme. Plato Epistemology is the general study of cognitive processes and their outcome: knowledge. Knowledge is the product of cognitive operations made by an inquiring subject. It is not a thing or a substance, but a series of brain changes in the knower. Knowledge is not independent of the knowing subject, although we often feign it is for practical reasons. Knowledge is different from belief: I can know a story, for instance, but do not believe it. Belief implies a psychological adherence to some propositions. It is possible to believe something without understanding it, so belief is not necessary associated with neither truth nor justification. Knowledge acquisition requires a modification of the brain of the knower. This can be done in different ways, hence there are different kinds of knowledge. (i) Sensory-motor knowledge: the result of learning from actions. (ii) Perceptual knowledge: the result of perceiving events, either internal or external to the subject. (iii) Conceptual or propositional knowledge: the result of ideation, conjecturing, testing, correcting. Notice that not all knowledge is beneficial: we can learn trivialities, falsehoods, or highly harmful habits The three kind of knowledge are interrelated: conceptual knowledge can improve motor skills and perception; perception is used to evaluate conjectures; motor skills can help to improve perception and build instruments such as books, that enhance the ability to learn. -
George W. Whitehead Jr
George W. Whitehead Jr. 1918–2004 A Biographical Memoir by Haynes R. Miller ©2015 National Academy of Sciences. Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. GEORGE WILLIAM WHITEHEAD JR. August 2, 1918–April 12 , 2004 Elected to the NAS, 1972 Life George William Whitehead, Jr., was born in Bloomington, Ill., on August 2, 1918. Little is known about his family or early life. Whitehead received a BA from the University of Chicago in 1937, and continued at Chicago as a graduate student. The Chicago Mathematics Department was somewhat ingrown at that time, dominated by L. E. Dickson and Gilbert Bliss and exhibiting “a certain narrowness of focus: the calculus of variations, projective differential geometry, algebra and number theory were the main topics of interest.”1 It is possible that Whitehead’s interest in topology was stimulated by Saunders Mac Lane, who By Haynes R. Miller spent the 1937–38 academic year at the University of Chicago and was then in the early stages of his shift of interest from logic and algebra to topology. Of greater importance for Whitehead was the appearance of Norman Steenrod in Chicago. Steenrod had been attracted to topology by Raymond Wilder at the University of Michigan, received a PhD under Solomon Lefschetz in 1936, and remained at Princeton as an Instructor for another three years. He then served as an Assistant Professor at the University of Chicago between 1939 and 1942 (at which point he moved to the University of Michigan). -
Multiple Mixing from Weak Hyperbolicity by the Hopf Argument Yves Coudène, Boris Hasselblatt, Serge Troubetzkoy
,pdfcreator=HAL,pdfproducer=PDFLaTeX,pdfsubject=Mathematics [math]/Dynamical Systems [math.DS] Multiple mixing from weak hyperbolicity by the Hopf argument Yves Coudène, Boris Hasselblatt, Serge Troubetzkoy To cite this version: Yves Coudène, Boris Hasselblatt, Serge Troubetzkoy. Multiple mixing from weak hyperbolicity by the Hopf argument. 2014. hal-01006451v2 HAL Id: hal-01006451 https://hal.archives-ouvertes.fr/hal-01006451v2 Submitted on 16 Jun 2014 (v2), last revised 15 Sep 2015 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. MULTIPLE MIXING FROM WEAK HYPERBOLICITY BY THE HOPF ARGUMENT YVES COUDÈNE, BORIS HASSELBLATT AND SERGE TROUBETZKOY ABSTRACT. We show that using only weak hyperbolicity (no smoothness, com- pactness or exponential rates) the Hopf argument produces multiple mixing in an elementary way. While this recovers classical results with far simpler proofs, the point is the broader applicability implied by the weak hypothe- ses. Some of the results can also be viewed as establishing “mixing implies multiple mixing” outside the classical hyperbolic context. 1. INTRODUCTION The origins of hyperbolic dynamical systems are connected with the efforts by Boltzmann and Maxwell to lay a foundation under statistical mechanics. In today’s terms their ergodic hypothesis was that the mechanical system defined by molecules in a container is ergodic, and the difficulties of establishing this led to the search for any mechanical systems with this property. -
AR 96 Covers/Contents
ICTP FULL TECHNICAL REPORT 2018 INTRODUCTION This document is the full technical report of ICTP for the year 2018. For the non-technical description of 2018 highlights, please see the printed “ICTP: A Year in Review” publication. 2 ICTP Full Technical Report 2018 CONTENTS Research High Energy, Cosmology and Astroparticle Physics (HECAP) ........................................................................ 7 Director's Research Group – String Phenomenology and Cosmology ...................................... 30 Condensed Matter and Statistical Physics (CMSP)............................................................................................ 32 Sustainable Energy Synchrotron Radiation Related Theory Mathematics ........................................................................................................................................................................ 63 Earth System Physics (ESP) ......................................................................................................................................... 71 Applied Physics Multidisciplinary Laboratory (MLab) ....................................................................................................... 97 Telecommunications/ICT for Development Laboratory (T/ICT4D) ....................................... 108 Medical Physics ................................................................................................................................................. 113 Fluid Dynamics .................................................................................................................................................. -
Diagonalizable Flows on Locally Homogeneous Spaces and Number
Diagonalizable flows on locally homogeneous spaces and number theory Manfred Einsiedler and Elon Lindenstrauss∗ Abstract.We discuss dynamical properties of actions of diagonalizable groups on locally homogeneous spaces, particularly their invariant measures, and present some number theoretic and spectral applications. Entropy plays a key role in the study of theses invariant measures and in the applications. Mathematics Subject Classification (2000). 37D40, 37A45, 11J13, 81Q50 Keywords. invariant measures, locally homogeneous spaces, Littlewood’s conjecture, quantum unique ergodicity, distribution of periodic orbits, ideal classes, entropy. 1. Introduction Flows on locally homogeneous spaces are a special kind of dynamical systems. The ergodic theory and dynamics of these flows are very rich and interesting, and their study has a long and distinguished history. What is more, this study has found numerous applications throughout mathematics. The spaces we consider are of the form Γ\G where G is a locally compact group and Γ a discrete subgroup of G. Typically one takes G to be either a Lie group, a linear algebraic group over a local field, or a product of such. Any subgroup H < G acts on Γ\G and this action is precisely the type of action we will consider here. One of the most important examples which features in numerous number theoretical applications is the space PGL(n, Z)\ PGL(n, R) which can be identified with the space of lattices in Rn up to homothety. Part of the beauty of the subject is that the study of very concrete actions can have meaningful implications. For example, in the late 1980s G. -
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Arxiv:1807.04136V2 [Math.AG] 25 Jul 2018
HITCHIN CONNECTION ON THE VEECH CURVE SHEHRYAR SIKANDER Abstract. We give an expression for the pull back of the Hitchin connection from the moduli space of genus two curves to a ten-fold covering of a Teichm¨ullercurve discovered by Veech. We then give an expression, in terms of iterated integrals, for the monodromy representation of this connection. As a corollary we obtain quantum representations of infinitely many pseudo-Anosov elements in the genus two mapping class group. Contents 1. Introduction 2 1.1. Acknowledgements 6 2. Moduli spaces of vector bundles and Hitchin connection in genus two 6 2.1. The Heisenberg group 8 2.2. The Hitchin connection 10 2.2.1. Riemann surfaces with theta structure 11 2.2.2. Projectively flat connections 12 3. Teichm¨ullercurves and pseudo-Anosov mapping classes 16 3.1. Hitchin connection and Hyperlogarithms on the Veech curve 20 4. Generators of the (orbifold) fundamental group 25 4.1. Computing Monodromy 26 References 31 arXiv:1807.04136v2 [math.AG] 25 Jul 2018 This is author's thesis supported in part by the center of excellence grant 'Center for Quantum Geometry of Moduli Spaces' from the Danish National Research Foundation (DNRF95). 1 HITCHIN CONNECTION ON THE VEECH CURVE 2 1. Introduction Let Sg be a closed connected and oriented surface of genus g ¥ 2, and consider its mapping class group Γg of orientation-preserving diffeomorphisms up to isotopy. More precisely, ` ` Γg :“ Diffeo pSgq{Diffeo0 pSgq; (1) ` where Diffeo pSgq is the group of orientation-preserving diffeomorphisms of Sg, and ` Diffeo0 pSgq denotes the connected component of the identity. -
Comments on the 2011 Shaw Prize in Mathematical Sciences - - an Analysis of Collectively Formed Errors in Physics by C
Global Journal of Science Frontier Research Physics and Space Science Volume 12 Issue 4 Version 1.0 June 2012 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4626 & Print ISSN: 0975-5896 Comments on the 2011 Shaw Prize in Mathematical Sciences - - An Analysis of Collectively Formed Errors in Physics By C. Y. Lo Applied and Pure Research Institute, Nashua, NH Abstract - The 2011 Shaw Prize in mathematical sciences is shared by Richard S. Hamilton and D. Christodoulou. However, the work of Christodoulou on general relativity is based on obscure errors that implicitly assumed essentially what is to be proved, and thus gives misleading results. The problem of Einstein’s equation was discovered by Gullstrand of the 1921 Nobel Committee. In 1955, Gullstrand is proven correct. The fundamental errors of Christodoulou were due to his failure to distinguish the difference between mathematics and physics. His subsequent errors in mathematics and physics were accepted since judgments were based not on scientific evidence as Galileo advocates, but on earlier incorrect speculations. Nevertheless, the Committee for the Nobel Prize in Physics was also misled as shown in their 1993 press release. Here, his errors are identified as related to accumulated mistakes in the field, and are illustrated with examples understandable at the undergraduate level. Another main problem is that many theorists failed to understand the principle of causality adequately. It is unprecedented to award a prize for mathematical errors. Keywords : Nobel Prize; general relativity; Einstein equation, Riemannian Space; the non- existence of dynamic solution; Galileo. GJSFR-A Classification : 04.20.-q, 04.20.Cv Comments on the 2011 Shaw Prize in Mathematical Sciences -- An Analysis of Collectively Formed Errors in Physics Strictly as per the compliance and regulations of : © 2012. -
The Dark Enlightenment
The Dark Enlightenment Nick Land Part 1: Neo-reactionaries head for the exit March 2, 2012 Enlightenment is not only a state, but an event, and a process. As the designation for an historical episode, concentrated in northern Europe during the 18th century, it is a leading candidate for the ‘true name’ of modernity, capturing its origin and essence (‘Renaissance’ and ‘Industrial Revolution’ are others). Between ‘enlightenment’ and ‘progressive enlightenment’ there is only an elusive difference, because illumination takes time – and feeds on itself, because enlightenment is self-confirming, its revelations ‘self-evident’, and because a retrograde, or reactionary, ‘dark enlightenment’ amounts almost to intrinsic contradiction. To become enlightened, in this historical sense, is to recognize, and then to pursue, a guiding light. There were ages of darkness, and then enlightenment came. Clearly, advance has demonstrated itself, offering not only improvement, but also a model. Furthermore, unlike a renaissance, there is no need for an enlightenment to recall what was lost, or to emphasize the attractions of return. The elementary acknowledgement of enlightenment is already Whig history in miniature. Once certain enlightened truths have been found self-evident, there can be no turning back, and conservatism is pre-emptively condemned – predestined — to paradox. F. A. Hayek, who refused to describe himself as a conservative, famously settled instead upon the term ‘Old Whig’, which – like ‘classical liberal’ (or the still more melancholy ‘remnant’) – accepts that progress isn’t what it used to be. What could an Old Whig be, if not a reactionary progressive? And what on earth is that? Of course, plenty of people already think they know what reactionary modernism looks like, and amidst the current collapse back into the 1930s their concerns are only likely to grow. -
Herman Heine Goldstine
Herman Heine Goldstine Born September 13, 1913, Chicago, Ill.; Army representative to the ENIAC Project, who later worked with John von Neumann on the logical design of the JAS computer which became the prototype for many early computers-ILLIAC, JOHNNIAC, MANIAC author of The Computer from Pascal to von Neumann, one of the earliest textbooks on the history of computing. Education: BS, mathematics, University of Chicago, 1933; MS, mathematics, University of Chicago, 1934; PhD, mathematics, University of Chicago, 1936. Professional Experience: University of Chicago: research assistant, 1936-1937, instructor, 1937-1939; assistant professor, University of Michigan, 1939-1941; US Army, Ballistic Research Laboratory, Aberdeen, Md., 1941-1946; Institute for Advanced Study, Princeton University, 1946-1957; IBM: director, Mathematics Sciences Department, 1958-1965, IBM fellow, 1969. Honors and Awards: IEEE Computer Society Pioneer Award, 1980; National Medal of Science, 1985; member, Information Processing Hall of Fame, Infornart, Dallas, Texas, 1985. Herman H. Goldstine began his scientific career as a mathematician and had a life-long interest in the interaction of mathematical ideas and technology. He received his PhD in mathematics from the University of Chicago in 1936 and was an assistant professor at the University of Michigan when he entered the Army in 1941. After participating in the development of the first electronic computer (ENIAC), he left the Army in 1945, and from 1946 to 1957 he was a member of the Institute for Advanced Study (IAS), where he collaborated with John von Neumann in a series of scientific papers on subjects related to their work on the Institute computer. In 1958 he joined IBM Corporation as a member of the research planning staff. -
The Road to Afghanistan
Introduction Hundreds of books—memoirs, histories, fiction, poetry, chronicles of military units, and journalistic essays—have been written about the Soviet war in Afghanistan. If the topic has not yet been entirely exhausted, it certainly has been very well documented. But what led up to the invasion? How was the decision to bring troops into Afghanistan made? What was the basis for the decision? Who opposed the intervention and who had the final word? And what kind of mystical country is this that lures, with an almost maniacal insistence, the most powerful world states into its snares? In the nineteenth and early twentieth century it was the British, in the 1980s it was the Soviet Union, and now America and its allies continue the legacy. Impoverished and incredibly backward Afghanistan, strange as it may seem, is not just a normal country. Due to its strategically important location in the center of Asia, the mountainous country has long been in the sights of more than its immediate neighbors. But woe to anyone who arrives there with weapon in hand, hoping for an easy gain—the barefoot and illiterate Afghans consistently bury the hopes of the strange foreign soldiers who arrive along with battalions of tanks and strategic bombers. To understand Afghanistan is to see into your own future. To comprehend what happened there, what happens there continually, is to avoid great tragedy. One of the critical moments in the modern history of Afghanistan is the period from April 27, 1978, when the “April Revolution” took place in Kabul and the leftist People’s Democratic Party seized control of the country, until December 27, 1979, when Soviet special forces, obeying their “international duty,” eliminated the ruling leader and installed 1 another leader of the same party in his place.