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REPORTS AND RECORDS OF SOCIETY MEETINGS Contents LMS SW&SW Regional
Meeting 2007 – records RECORDS OF PROCEEDINGS AT MEETINGS REGIONAL ORDINARY MEETING held on Wednesday 30 May 2007 at the University of Cardiff. At least 40 members and visitors were present for all or part of the meeting. The meeting began at 3 pm, with the President, Professor J.F. TOLAND, FRS, FRSE, in the Chair. Eight people were elected to Ordinary Membership: M.P. Holland, B. Klopsch, A. Laptev, J.P.T. Meyer, R. Moser, A.O. Philips, D. Schley, A. Vishik; and two were elected to Associate Membership: H.D. Burton, S.A. Munday. Four members signed the book and were admitted to the Society. PROFESSOR W.D. EVANS introduced a lecture given by Professor M. Aizenman on The curious effects of disorder on spectra of random operators. After tea, Professor Evans introduced a lecture given by Professor T. Sunada on The K4 crystal – a new crystal structure similar to the diamond lattice. Professor Toland expressed the thanks of the Society to the local organiser and the speakers for putting on such an excellent meeting. After the meeting there was a recital given by Nicole Lamartine (soprano) and Matthias Langer (piano) at Aberdare Hall followed by a dinner. LMS SW&SW REGIONAL MEETING 2007 A South West and South Wales Meeting of the LMS was held in Cardiff University on 30 May 2007. This was part of a 4-day workshop in Cardiff on Analysis on graphs and fractals which was a satellite meeting of the programme Analysis on graphs and its applications in progress at the Newton Institute between January and July 2007. There were about 40 LMS members and visitors present. The President, Professor John Toland, opened the meeting and then formally invited Society members to come forward to sign the precious Members' Book dating back to the Society's foundation in 1865. This was followed by lectures given by two very distinguished speakers who were introduced by Professor Des Evans, one of the organisers of the meeting.
The first talk was given by Professor Michael Aizenman (Princeton) and had the intriguing title The curious effects of disorder on spectra of random operators. Michael started with an introduction on the Anderson model of mathematical physics and went on to survey the current state of knowledge on the rigorous justification of Anderson localisation for random Schrödinger operators and the related area of spectral level statistics. While a number of results are known concerning the appearance of pure point spectra and localised states, there are few results on the survival of delocalised states and the existence of the mobility edge, except in the case of operators on trees. Related conjectures were formulated and a surprising recent result for Bethe lattices given. During the break for tea and coffee between the two lectures, a display of LMS books and periodicals was organised by Susan Oakes. Some non-members were persuaded of the great benefits of joining the Society! Professor Toshikazu Sunada (Meiji University, Tokyo) then gave his talk on The K4 crystal – a new crystal structure similar to the diamond lattice. Toshi initially described the abstract notion of a crystal lattice as a graph with a free action of a free abelian finitely generated group. The so-called maximal abelian coverings of finite regular graphs were presented: among these, the diamond lattice and K4 lattice are particularly interesting. These share many important properties: they have extensive symmetry groups; have isometric realisations in 3-dimensional space such that their automorphisms can be realised as rigid bodies; and they minimise some natural variational functional. They are in fact the only 3-dimensional lattices with such properties. The natural question raised was whether carbon atoms could form the K4 lattice, and if so, what would be its properties? The very successful meeting ended with a reception and a delightful recital by Nicole Sheldon (soprano) accompanied by Matthias Langer. This was followed by a dinner in the University's Aberdare Hall. Peter Kuchment LMS INVITED LECTURES 2007
The 2007 LMS Invited Lectures were held in Oxford between 10–14 April, consisting of 10 lectures given by Professor David Ben-Zvi of the University of Texas at Austin on the Geometric Langlands correspondence. The topic attracted a large audience, consisting of about 40 UK-based mathematicians and physicists including 15 graduate students, a further 20 Europeans, as well as participants from as far afield as Jerusalem, Seoul and Seattle. The speaker broke down the subject into five parts, respectively on geometric function theory, moduli of bundles on algebraic curves and the Hitchin fibration, geometric Hecke operators, the topological field theory approach of Kapustin and Witten, and applications to representation theory following Bezrukhavnikov's work on the tamely ramified case and his own recent work with Nadler on representations of real algebraic groups. Highlights included a formulation of the geometric Langlands story, including the action of Hecke operators, as a three-tier quantum field theory (though time luckily ran out just before we got to the headache involved in contemplating 4-tier quantum field theories), and an explanation how the Springer resolution arises as a derived loop space. The lectures were generally thought to be very well structured, clear and inspriring; they were recorded on video, and will soon be available for download from David's NSF-funded resource webpage GRASP. There were additional lectures by Dmitriy Rumynin on the finite characteristic case; by Constantin Teleman on some constructions and computations supporting and fine-tuning the expected Langlands equivalence; and by Alexander Schmitt and Luis Alvarez-Consul on aspects of moduli spaces of sheaves on algebraic varieties. The Oxford weather was at its best, contributing in its small way to the success of an enjoyable and thought-provoking week. Balázs Szendröi GRESHAM COLLEGE LMS LECTURE 2007
Delivered at Gresham College on 22 May, this lecture, a free public lecture in the tradition of the college, was the second in a series of joint lectures of the College and the London Mathematical Society. Professor Timothy Gowers, FRS, began his lecture on Multiplying and dividing whole numbers: why it is more difficult than you might think with some general comments on the difficulty of describing to non-mathematicians what mathematicians do, particularly explaining that this does not simply consist of such things as multiplying larger and larger numbers together. He then continued by saying 'I want to give some idea of what research in maths is like, so I have decided to talk about multiplying larger and larger numbers together', explaining that the emphasis would be on how to go about such a task. Leading into this he carried out long multiplication of two four digit numbers, showing that essentially this involved 16 multiplications of single digit numbers, together with some addition. He then explained that when multiplying very much larger numbers (as is necessary in some encryption methods) an approach which reduces the enormous number of multiplications involved is worthwhile, and began by describing a method giving a 25% reduction. This was achieved by splitting each of the two strings of digits into two parts, so that for instance two thousand-digit numbers would each consist of a pair of 500 digit strings. Labelling these strings as A and B for the first number, C and D for the second, the trick was then to notice that the final answer could be obtained from the three numbers AC, CD and AD + BC, and this could be done by three rather than four multiplications of 500 digit numbers (since AD + BC = (A + B)(C + D) – AC – BD), so that 750,000 rather than a million individual single digit multiplications were required to obtain the final product. This process of subdivision could then be repeated, with further savings. This approach was then used to suggest the idea of seeking a method
where the convolution aspect of multiplying two sums The second part of the talk concentrated on factorisation, beginning with a specific example, expressing the number 437 as a product of prime numbers by the direct method of explicitly testing each possible prime factor. It was then shown that using this method to factorise a thousand digit number could be estimated to take 10485 years. Having thus established that efficient factorisation of large numbers is a difficult problem, and remarked that the fact that there is no fast algorithm for factorising a large number which is the product of two large primes is a basis for everyday digital security, Professor Gowers ended by describing a method for showing that an integer is not prime without actually finding any non-trivial factors. The method uses Fermat's little theorem, which states that if a number p is a prime number then (for any natural number a) a p–1 is always equal to 1 mod p. He illustrated the method by calculating 290 mod 91, by hand on the board, achieving the answer 64, and thus showing that 91 is not prime. Professor Gowers engaged his audience throughout, carrying out many calculations on the board as he proceeded and developing his arguments in a clear and entertaining way. He conveyed complex ideas using carefully constructed examples and considerable expository skill. The lecture hall was packed, and a further room with video link also overflowing. This second lecture in the series again shows that there is a public appetite for mathematics, particularly when delivered with the verve and accomplishment shown by this lecturer. The lecture may be found in video and audio format on the Gresham College web pages at www.gresham.ac.uk. F.A. Rogers
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. While it is not known if this is the fastest possible technique for
multiplication, an open and difficult problem in mathematics, fast Fourier
transformations have certainly revolutionised algorithms.