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Symmetric Products of Surfaces; a Unifying Theme for Topology
Symmetric products of surfaces; a unifying theme for topology and physics Pavle Blagojevi´c Mathematics Institute SANU, Belgrade Vladimir Gruji´c Faculty of Mathematics, Belgrade Rade Zivaljevi´cˇ Mathematics Institute SANU, Belgrade Abstract This is a review paper about symmetric products of spaces SP n(X) := n X /Sn. We focus our attention on the case of 2-manifolds X and make a journey through selected topics of algebraic topology, algebraic geometry, mathematical physics, theoretical mechanics etc. where these objects play an important role, demonstrating along the way the fundamental unity of diverse fields of physics and mathematics. 1 Introduction In recent years we have all witnessed a remarkable and extremely stimulating exchange of deep and sophisticated ideas between Geometry and Physics and in particular between quantum physics and topology. The student or a young scientist in one of these fields is often urged to master elements of the other field as quickly as possible, and to develop basic skills and intuition necessary for understanding the contemporary research in both disciplines. The topology and geometry of manifolds plays a central role in mathematics and likewise in physics. The understanding of duality phenomena for manifolds, mastering the calculus of characteristic classes, as well as understanding the role of fundamental invariants like the signature or Euler characteristic are just examples of what is on the beginning of the growing list of prerequisites for a student in these areas. arXiv:math/0408417v1 [math.AT] 30 Aug 2004 A graduate student of mathematics alone is often in position to take many spe- cialized courses covering different aspects of manifold theory and related areas before an unified picture emerges and she or he reaches the necessary level of maturity. -
Meetings & Conferences of The
Meetings & Conferences of the AMS IMPORTANT INFORMATION REGARDING MEETINGS PROGRAMS: AMS Sectional Meeting programs do not appear in the print version of the Notices. However, comprehensive and continually updated meeting and program information with links to the abstract for each talk can be found on the AMS website. See http://www.ams.org/ meetings/. Final programs for Sectional Meetings will be archived on the AMS website accessible from the stated URL and in an electronic issue of the Notices as noted below for each meeting. Tamar Zeigler, Technion, Israel Institute of Technology, Tel Aviv, Israel Patterns in primes and dynamics on nilmanifolds. Bar-Ilan University, Ramat-Gan and Special Sessions Tel-Aviv University, Ramat-Aviv Additive Number Theory, Melvyn B. Nathanson, City University of New York, and Yonutz V. Stanchescu, Afeka June 16–19, 2014 Tel Aviv Academic College of Engineering. Monday – Thursday Algebraic Groups, Division Algebras and Galois Co- homology, Andrei Rapinchuk, University of Virginia, and Meeting #1101 Louis H. Rowen and Uzi Vishne, Bar Ilan University. The Second Joint International Meeting between the AMS Applications of Algebra to Cryptography, David Garber, and the Israel Mathematical Union. Holon Institute of Technology, and Delaram Kahrobaei, Associate secretary: Michel L. Lapidus City University of New York Graduate Center. Announcement issue of Notices: January 2014 Asymptotic Geometric Analysis, Shiri Artstein and Boaz Program first available on AMS website: To be announced Klar’tag, Tel Aviv University, and Sasha Sodin, Princeton Program issue of electronic Notices: To be announced University. Issue of Abstracts: Not applicable Combinatorial Games, Aviezri Fraenkel, Weizmann University, Richard Nowakowski, Dalhousie University, Deadlines Canada, Thane Plambeck, Counterwave Inc., and Aaron For organizers: To be announced Siegel, Twitter. -
Discurso De Investidura Como Doctor “Honoris Causa” Del Excmo. Sr
Discurso de investidura como Doctor “Honoris Causa” del Excmo. Sr. Simon Kirwan Donaldson 20 de enero de 2017 Your Excellency Rector Andradas, Ladies and Gentlemen: It is great honour for me to receive the degree of Doctor Honoris Causa from Complutense University and I thank the University most sincerely for this and for the splendid ceremony that we are enjoying today. This University is an ancient institution and it is wonderful privilege to feel linked, through this honorary degree, to a long line of scholars reaching back seven and a half centuries. I would like to mention four names, all from comparatively recent times. First, Eduardo Caballe, a mathematician born in 1847 who was awarded a doctorate of Science by Complutense in 1873. Later, he was a professor in the University and, through his work and his students, had a profound influence in the development of mathematics in Spain, and worldwide. Second, a name that is familiar to all of us: Albert Einstein, who was awarded a doctorate Honoris Causa in 1923. Last, two great mathematicians from our era: Vladimir Arnold (Doctor Honoris Causa 1994) and Jean- Pierre Serre (2006). These are giants of the generation before my own from whom, at whose feet---metaphorically—I have learnt. Besides the huge honour of joining such a group, it is interesting to trace one grand theme running the work of all these four people named. The theme I have in mind is the interplay between notions of Geometry, Algebra and Space. Of course Geometry begins with the exploration of the space we live in—the space of everyday experience. -
Memories of Vladimir Arnold Boris Khesin and Serge Tabachnikov, Coordinating Editors
Memories of Vladimir Arnold Boris Khesin and Serge Tabachnikov, Coordinating Editors Vladimir Arnold, an eminent mathematician of functions of two variables. Dima presented a two- our time, passed away on June 3, 2010, nine days hour talk at a weekly meeting of the Moscow before his seventy-third birthday. This article, Mathematical Society; it was very uncommon along with one in the previous issue of the Notices, for the society to have such a young speaker. touches on his outstanding personality and his Everybody ad- great contribution to mathematics. mired him, and he certainly de- Dmitry Fuchs served that. Still there was some- thing that kept Dima Arnold in My Life me at a distance Unfortunately, I have never been Arnold’s student, from him. although as a mathematician, I owe him a lot. He I belonged to was just two years older than I, and according to a tiny group of the University records, the time distance between students, led by us was still less: when I was admitted to the Sergei Novikov, Moscow State university as a freshman, he was a which studied al- sophomore. We knew each other but did not com- gebraic topology. municate much. Once, I invited him to participate Just a decade in a ski hiking trip (we used to travel during the before, Pontrya- winter breaks in the almost unpopulated northern gin’s seminar in Russia), but he said that Kolmogorov wanted him Moscow was a to stay in Moscow during the break: they were true center of the going to work together. -
1914 Martin Gardner
ΠME Journal, Vol. 13, No. 10, pp 577–609, 2014. 577 THE PI MU EPSILON 100TH ANNIVERSARY PROBLEMS: PART II STEVEN J. MILLER∗, JAMES M. ANDREWS†, AND AVERY T. CARR‡ As 2014 marks the 100th anniversary of Pi Mu Epsilon, we thought it would be fun to celebrate with 100 problems related to important mathematics milestones of the past century. The problems and notes below are meant to provide a brief tour through some of the most exciting and influential moments in recent mathematics. No list can be complete, and of course there are far too many items to celebrate. This list must painfully miss many people’s favorites. As the goal is to introduce students to some of the history of mathematics, ac- cessibility counted far more than importance in breaking ties, and thus the list below is populated with many problems that are more recreational. Many others are well known and extensively studied in the literature; however, as our goal is to introduce people to what can be done in and with mathematics, we’ve decided to include many of these as exercises since attacking them is a great way to learn. We have tried to include some background text before each problem framing it, and references for further reading. This has led to a very long document, so for space issues we split it into four parts (based on the congruence of the year modulo 4). That said: Enjoy! 1914 Martin Gardner Few twentieth-century mathematical authors have written on such diverse sub- jects as Martin Gardner (1914–2010), whose books, numbering over seventy, cover not only numerous fields of mathematics but also literature, philosophy, pseudoscience, religion, and magic. -
Mathematisches Forschungsinstitut Oberwolfach Emigration Of
Mathematisches Forschungsinstitut Oberwolfach Report No. 51/2011 DOI: 10.4171/OWR/2011/51 Emigration of Mathematicians and Transmission of Mathematics: Historical Lessons and Consequences of the Third Reich Organised by June Barrow-Green, Milton-Keynes Della Fenster, Richmond Joachim Schwermer, Wien Reinhard Siegmund-Schultze, Kristiansand October 30th – November 5th, 2011 Abstract. This conference provided a focused venue to explore the intellec- tual migration of mathematicians and mathematics spurred by the Nazis and still influential today. The week of talks and discussions (both formal and informal) created a rich opportunity for the cross-fertilization of ideas among almost 50 mathematicians, historians of mathematics, general historians, and curators. Mathematics Subject Classification (2000): 01A60. Introduction by the Organisers The talks at this conference tended to fall into the two categories of lists of sources and historical arguments built from collections of sources. This combi- nation yielded an unexpected richness as new archival materials and new angles of investigation of those archival materials came together to forge a deeper un- derstanding of the migration of mathematicians and mathematics during the Nazi era. The idea of measurement, for example, emerged as a critical idea of the confer- ence. The conference called attention to and, in fact, relied on, the seemingly stan- dard approach to measuring emigration and immigration by counting emigrants and/or immigrants and their host or departing countries. Looking further than this numerical approach, however, the conference participants learned the value of measuring emigration/immigration via other less obvious forms of measurement. 2892 Oberwolfach Report 51/2011 Forms completed by individuals on religious beliefs and other personal attributes provided an interesting cartography of Italian society in the 1930s and early 1940s. -
September 2014
LONDONLONDON MATHEMATICALMATHEMATICAL SOCIETYSOCIETY NEWSLETTER No. 439 September 2014 Society Meetings HIGHEST HONOUR FOR UK and Events MATHEMATICAN Professor Martin Hairer, FRS, 2014 University of Warwick, has become the ninth UK based Saturday mathematician to win the 6 September prestigious Fields Medal over Mathematics and the its 80 year history. The medal First World War recipients were announced Meeting, London on Wednesday 13 August in page 15 a ceremony at the four-year- ly International Congress for 1 Wednesday Mathematicians, which on this 24 September occasion was held in Seoul, South Korea. LMS Popular Lectures See page 4 for the full report. Birmingham page 12 Friday LMS ANNOUNCES SIMON TAVARÉ 14 November AS PRESIDENT-DESIGNATE LMS AGM © The University of Cambridge take over from the London current President, Professor Terry Wednesday Lyons, FRS, in 17 December November 2015. SW & South Wales Professor Tavaré is Meeting a versatile math- Plymouth ematician who has established a distinguished in- ternational career culminating in his current role as The London Mathematical Director of the Cancer Research Society is pleased to announce UK Cambridge Institute and Professor Simon Tavaré, Professor in DAMTP, where he NEWSLETTER FRS, FMedSci, University of brings his understanding of sto- ONLINE: Cambridge, as President-Des- chastic processes and expertise newsletter.lms.ac.uk ignate. Professor Tavaré will in the data science of DNA se- (Cont'd on page 3) LMS NEWSLETTER http://newsletter.lms.ac.uk Contents No. 439 September 2014 15 44 Awards Partial Differential Equations..........................37 Collingwood Memorial Prize..........................11 Valediction to Jeremy Gray..............................33 Calendar of Events.......................................50 News LMS Items European News.................................................16 HEA STEM Strategic Project........................... -
Richard Von Mises's Philosophy of Probability and Mathematics
“A terrible piece of bad metaphysics”? Towards a history of abstraction in nineteenth- and early twentieth-century probability theory, mathematics and logic Lukas M. Verburgt If the true is what is grounded, then the ground is neither true nor false LUDWIG WITTGENSTEIN Whether all grow black, or all grow bright, or all remain grey, it is grey we need, to begin with, because of what it is, and of what it can do, made of bright and black, able to shed the former , or the latter, and be the latter or the former alone. But perhaps I am the prey, on the subject of grey, in the grey, to delusions SAMUEL BECKETT “A terrible piece of bad metaphysics”? Towards a history of abstraction in nineteenth- and early twentieth-century probability theory, mathematics and logic ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof. dr. D.C. van den Boom ten overstaan van een door het College voor Promoties ingestelde commissie in het openbaar te verdedigen in de Agnietenkapel op donderdag 1 oktober 2015, te 10:00 uur door Lukas Mauve Verburgt geboren te Amersfoort Promotiecommissie Promotor: Prof. dr. ir. G.H. de Vries Universiteit van Amsterdam Overige leden: Prof. dr. M. Fisch Universitat Tel Aviv Dr. C.L. Kwa Universiteit van Amsterdam Dr. F. Russo Universiteit van Amsterdam Prof. dr. M.J.B. Stokhof Universiteit van Amsterdam Prof. dr. A. Vogt Humboldt-Universität zu Berlin Faculteit der Geesteswetenschappen © 2015 Lukas M. Verburgt Graphic design Aad van Dommelen (Witvorm) -
The Catalan Mathematical Society EMS June 2000 3 EDITORIAL
CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics The Open University Milton Keynes MK7 6AA, UK e-mail: [email protected] ASSOCIATE EDITORS STEEN MARKVORSEN Department of Mathematics Technical University of Denmark Building 303 NEWSLETTER No. 36 DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] KRZYSZTOF CIESIELSKI June 2000 Mathematics Institute Jagiellonian University Reymonta 4 30-059 Kraków, Poland EMS News : Agenda, Editorial, 3ecm, Bedlewo Meeting, Limes Project ........... 2 e-mail: [email protected] KATHLEEN QUINN Open University [address as above] Catalan Mathematical Society ........................................................................... 3 e-mail: [email protected] SPECIALIST EDITORS The Hilbert Problems ....................................................................................... 10 INTERVIEWS Steen Markvorsen [address as above] SOCIETIES Interview with Peter Deuflhard ....................................................................... 14 Krzysztof Ciesielski [address as above] EDUCATION Vinicio Villani Interview with Jaroslav Kurzweil ..................................................................... 16 Dipartimento di Matematica Via Bounarotti, 2 56127 Pisa, Italy A Major Challenge for Mathematicians ........................................................... 20 e-mail: [email protected] MATHEMATICAL PROBLEMS Paul Jainta EMS Position Paper: Towards a European Research Area ............................. 24 -
October 2008
THE LONDON MATHEMATICAL SOCIETY NEWSLETTER No. 374 October 2008 Society THE PROPOSAL FOR A NEW SOCIETY Meetings In all likelihood you will now have present form fulfil many of the and Events received a copy of the proposal hopes and expectations of their for a new society, combining the members, times are changing and 2008 present London Mathematical the need for mathematics as a uni- Friday 21 November Society and Institute of Mathe- fied activity to hold and defend AGM, London matics and its Applications. For its position in the public sphere [page 3] a new society to be formed, the grows constantly greater. IMA and the LMS must both vote As the Presidents’ letter which 12–13 December separately in favour of the accompanies the report makes Joint Meeting with proposal. clear, there is a pressing need to the Edinburgh There has been debate about engage effectively with govern- Mathematical Society this for several years but mem- ment, with external bodies, with Edinburgh [page 7] bers could be forgiven for think- the media and with the public. ing that, despite progress reports A society that represents the 2009 appearing in Mathematics Today broad spectrum of the mathemat- Friday 27 February and the Newsletter, things had ical community and has a larger Mary Cartwright ‘gone quiet’. The process leading membership must inevitably carry Lecture, London up to the present proposal has greater weight. been protracted not because the Your view is important and you 31 March – 4 April two societies disagree with one will soon have an opportunity to LMS Invited Lectures another, which they do not, but take part in this important deci- Edinburgh because those developing the new sion. -
Leo Corry HILBERT's 6TH PROBLEM: BETWEEN THE
Leo Corry HILBERT’S 6TH PROBLEM: BETWEEN THE FOUNDATIONS OF GEOMETRY AND THE AXIOMATIZATION OF PHYSICS LEO CORRY1 Abstract. The sixth of Hilbert’s famous 1900 list of twenty-three problems was a programmatic call for the axiomatization of the physical sciences. It was naturally and organically rooted at the core of Hilbert’s conception of what axiomatization is all about. In fact, the axiomatic method which he applied at the turn of the twentieth-century in his famous work on the foundations of geometry originated in a preoccupation with foundational questions related with empirical science in general. Indeed, far from a purely formal conception, Hilbert counted geometry among the sciences with strong empirical content, closely related to other branches of physics and deserving a treatment similar to that reserved for the latter. In this treatment, the axiomatization project was meant to play, in his view, a crucial role. Curiously, and contrary to a once-prevalent view, from all the problems in the list, the sixth is the only one that continually engaged Hilbet’s efforts over a very long period of time, at least between 1894 and 1932. (TO APPEAR IN: Philosophical Transactions of the Royal Society (A) - DOI: 10.1098/rsta.2017.0221) 1. Introduction Of the many important and brilliant talks delivered in International Congresses of Mathematicians (ICM) ever since the inception of this institution in 1897 in Zurich, none has so frequently been quoted and, arguably, none has had the kind of pervasive influence, that had the one delivered by David Hilbert in 1900 at the second ICM in Paris, under the title of “Mathematical Problems”. -
Fermat Meets SWAC: Vandiver, the Lehmers, Computers, and Number Theory
Fermat Meets SWAC: Vandiver, the Lehmers, Computers, and Number Theory Leo Corry Tel Aviv University This article describes the work of Harry Schultz Vandiver, Derrick Henry Lehmer, and Emma Lehmer on calculations related with proofs of Fermat’s last theorem. This story sheds light on ideological and institutional aspects of activity in number theory in the US during the 20th century, and on the incursion of computer-assisted methods into pure fields of mathematical research. The advent of the electronic digital computer than 100. Extending this result beyond 100 opened a new era of unprecedented possibil- involved, above all, straightforward (if tedious) ities for large-scale number crunching. Begin- computations. Yet, little work was devoted to ning in the late 1940s, these gradually increas- such computations before 1920, and even ing possibilities were duly pursued in many then, this remained an essentially marginal branches of science. Some of them, like trend within number theory. meteorology, geophysics, or engineering sci- Thus, when electronic computers started to ence, underwent deep and quick transforma- became available in the late 1940s, few tions. Pure mathematical disciplines such as mathematicians working in the core, ‘‘pure’’ number theory can be counted among the less fields of the discipline incorporated them in receptive audiences for these newly opened their research agendas. Even fewer did so for possibilities. One way to account for this Fermat’s last theorem. Harry Schultz Vandiver somewhat ironic situation is to examine the (1882–1973) was one of the few to do so. He main research trends that shaped progress in joined forces with the couple Derrick Henry the algebraic theory of numbers from the Lehmer (1905–1991), and Emma Lehmer second half of the 19th century on.