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Meeting Reports and Records Contents LMS Ordinary Meeting on Friday 20 June: record RECORDS OF PROCEEDINGS AT MEETINGS ORDINARY MEETING held on Friday 20 June 2003 at University College London. At least 47 members and visitors were present for all or part of the meeting. The meeting began at 3.30 pm, with the President, Professor P. GODDARD, FRS, in the Chair. The President announced the awards of the Polya Prize to Professor A.J. Macintyre, FRS, of the University of Edinburgh, the Berwick Prize to Dr T. Bridgeland of the University of Edinburgh, the Senior Whitehead Prize to Dr P. Neumann of Oxford University, and Whitehead Prizes to Dr N. Dorey of University of Wales, Swansea, to Dr T. Hall of Liverpool University, to Dr M. Lackenby of St. Catherine’s College and the University of Oxford, and to Dr M. Nazarov of the University of York. The President read short versions of the citations, which would be published in full in the Bulletin. The President, on Council’s behalf, proposed that Professor Pierre Deligne be elected to Honorary Membership of the Society. This was approved by acclaim. The President read a short version of the citation, to be published in full in the Bulletin. Seven people were elected to Ordinary Membership: M.G. Blyth, G. El, J. Mao, S.E. Mikhailov, J. Talabany, C. Yastremiz, A.S.I. Zinober; and three people were elected to Associate Membership: S. Hendren, K.J. Shackleton, I. Shah. Three members signed the book and were admitted to the Society. GENERAL MEETING With Professor P. Goddard, FRS, in the Chair. On a recommendation from Council it was agreed to elect Dr D.J. Collins and Dr A.R. Camina to be appointed scrutineers in the forthcoming Council elections 2004. The Ordinary Meeting then resumed. The President announced, with regret, that Professor J.C. Rickard, of the University of Bristol, Senior Berwick Prizewinner for 2002, who was to have given the first lecture, had been unavoidably delayed. It was intended to reschedule his lecture for a later date. The Fröhlich Lecture was then given by Professor M.J. Taylor, FRS, on ‘Die Fröhliche Wissenschaft’. A meeting of the Society was held on Friday 20 June at University College London. It was chaired by the President, Professor Peter Goddard, who began by inviting new members to add their names to the original Membership Book which dates from the foundation of the Society in 1865. D. Cariolaro, P. Fleischmann and R.J. Shank duly signed, and were welcomed into the LMS. The President then read aloud citations for the 2003 LMS prizewinners, some of whom were present, and the audience acknowledged their fine achievements with enthusiastic clapping. The first lecture, on ‘The stable module category of a finite group algebra’ was to have been delivered by Professor Jeremy Rickard, Senior Berwick prizewinner for 2002. Unfortunately, though, he fell victim to our endemic problems on the railways and was seriously delayed. Apparently a fire close to the line at Burnham had affected trains into Paddington. All was not lost, however, and Professor Martin Taylor stepped into the breach and gave an impromptu update on preparations for the forthcoming International Review of British Mathematics and Statistics. Following tea, we had the first Fröhlich Lecture, in memory of Professor Albrecht Fröhlich, appropriately delivered by Martin Taylor who had been a research student of his at King’s College and with whom he had subsequently produced several important number theoretic publications. Ali’s widow, Dr Ruth Fröhlich, and their daughter, Sorrel, were present for the lecture, entitled ‘Die Fröhliche Wissenschaft’, but their son, Shaun, was able to come for the reception and dinner only. Martin Taylor’s lecture was well received and enthusiastically applauded. At this stage Jeremy Rickard arrived and the President made a presentation of the Senior Berwick Prize certificate. We were assured that the lecture would take place at a later date. Following the lecture, there was a well-attended reception at De Morgan House and those attending took advantage of a balmy summer’s evening to spill out into the attractive garden. More than thirty people then proceeded to Poon’s Restaurant on Woburn Place to enjoy a pleasant Chinese meal and good company. R.T. Curtis On 23 July, the participants at the Hodge Centenary meeting (LMS Newsletter 310, pp 25) run by the International Centre for Mathematical Sciences at Edinburgh took time off from Hodge theory to hear about recent progress on the Poincaré conjecture. The focus of attention was the remarkable work of Grisha Perelman, which has appeared in a series of preprints dating from November 2002. Professor Simon Donaldson, FRS, gave a lecture describing the Poincaré conjecture, Thurston’s geometrization conjecture, and Richard Hamilton’s approach via Ricci flow to these conjectures. He then explained, in outline, how Grisha Perelman appears to have overcome a major stumbling block in Hamilton’s programme. The geometrization conjecture is motivated by the well known fact that every closed surface admits Riemannian metrics of constant Gauss curvature. (This is closely related to the uniformization theorem of Riemann surface theory.) Two-dimensional Poincaré, the statement that every closed simply connected surface is the standard round sphere follows rather easily from this. It is not the case that every closed 3-manifold admits metrics of constant curvature: but Thurston’s geometrization conjecture asserts that every such manifold can be split into geometric pieces. These geometric pieces include spaces of constant curvature as well as 5 other standard types. As in the 2-dimensional case, the Poincaré conjecture would follow easily if the geometrization conjecture were true. Perelman’s work, which is aimed at proving the full geometrization conjecture, is based on the Ricci flow equation which was introduced by Hamilton in the early 1980s. This equation, dg/dt = - 2r, g(0) = g0 (where r is the Ricci curvature of g) defines a one-parameter family of Riemannian metrics g(t) starting from an arbitrary initial metric g0. In general terms, one hopes that this flow will improve the initial metric g0: indeed Hamilton proved that if g0 is a metric of positive Ricci curvature on a closed 3-manifold, then g(t) tends to a metric of constant positive curvature as t increases. Since then, Hamilton and others have proved many further results which point towards a possible proof of the geometrization conjecture by a detailed analysis of the Ricci flow. Before Perelman’s work, the main problem in this approach can be explained, roughly, as follows. If one has no information about the initial metric g0, the Ricci flow can usually not be extended beyond some critical finite time T: the solution blows up because of the nonlinearities in the equation. In order to understand the limiting behaviour of g(t) as t approaches T, it is necessary to control the injectivity radius of g(t). Perelman has solved this problem through the introduction of a new notion of ‘entropy’. This behaves monotonically for a family of metrics evolving by the Ricci flow, and at the same time controls the injectivity radius. Taken together, these two properties of the entropy give information about the injectivity radius of g(t) as t approaches T, just as required. While it will take months or perhaps years for the mathematical commnity to agree on whether the Poincaré conjecture has finally been settled, it is clear that Perelman’s work will have a great impact on future work in geometric flow equations, and will lead to a deeper understanding of the geometric meaning and significance of the Ricci flow itself. Michael Singer The ICMS is aware of the need to bring the latest and best developments in mathematics worldwide rapidly to the attention of the UK community and sees this as one of its roles. The initiative to hold this Poincaré Conjecture afternoon, which arose at very short notice, is a case in point. John Toland Well, follow that! In the last couple of years we have had double pendulums hanging vertically down, and performing tricks that no decent-minded pendulum ought to know about, and quadruple pendulums hanging upwards (?) and staying there, in spite of all our instincts that they should collapse and behave properly. So what will we get next year? The LMS Popular Lectures have been running for 21 years, having been started in 1982 as a way of attempting to share with a large audience some of the delights and unravel for them some of the mysteries of our subject. I suspect, too, that many professionals have been pleased to have an insight into what some of their fellow mathematicians working in totally unrelated fields have been getting up to. I first became aware of the lectures some fifteen years ago, and regularly turned up at Imperial College before they migrated to Bloomsbury, a move which came with that of the Society from Piccadilly to De Morgan House in Russell Square. Over the years we have tied ourselves in knots; floated, spun and tumbled; married, voted and chosen; stamped (through mathematics); hopped (mad, with probability); and juggled. When I first started to attend I sometimes attempted to make notes. This was usually in vain. Somehow the lectures did not lend themselves to that sort of attention. Much better to let it happen and enjoy them as a theatrical experience (which many of them were!) and then later hire the video to refresh one’s memory of the trickier bits.
And so to this year. Marcus du Sautoy investigated with us The Music of the Primes. There was some intriguing music being played as we entered the lecture hall, but no reference was made to it later. Was it relevant? Was it some strange piece in which the primes had been converted into musical notes? I know not. Marcus took us on a tour of the primes, reminded us that they were infinite (with proof), and that, perhaps, there was an infinite number of prime pairs. We investigated how often they occurred, and puzzled over trying to work out the next number in a sequence – only to dissolve in rather embarrassed laughter when it turned out to be the Lottery numbers from, was it, last September? We even had a dab at Fermat’s Last Theorem. And then David Acheson: Mathematics, Music and the Electric Guitar. David’s recently published book, 1089 & all that, furnished some of the material for this lecture, and highly entertaining it was. From the ‘trick’ that involves 1089, the divergence of the harmonic series and the intriguing convergence of
And to top it all off we had a duet between our two lecturers on trumpet and guitar – Autumn Leaves. So what will we get next year? Martin Perkins
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I
have usually found the lectures to be pitched at what I thought was
the right level – they should be accessible to someone at the end
of the lower sixth (Year 12 in current jargon) but there should also
be something further to puzzle them and to keep those with greater knowledge
on their toes. The art of constructing such a lecture is not easily
acquired, and the Society has been very fortunate (and skilled) in its
selection of its lecturers. Inevitably one’s visual memory is what
dominates in thinking back to previous lectures. Colin Wright explaining
the mathematics of juggling is a case in point (and how anyone can juggle
while riding a unicycle is still beyond me – but I know he can
because I’ve seen it).
via
some famous errors, such as Malfatti’s problem, and finally to
the oscillations of a six-legged spider (sic) and some weird simulations
of the three-body problem we came to ‘not the Indian Rope Trick’:
the stable oscillations of a series of upside-down pendulums. Common
sense tells us that of course this is quite impossible, so there, ably
assisted by the man who stood on the vibrating machine to stop it walking
off the platform, he showed it happening. What was it Victor Meldrew
used to say?