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REPORTS AND RECORDS OF SOCIETY MEETINGS Contents LMS Ordinary Meeting 10 June 2005 - record RECORDS OF PROCEEDINGS AT MEETINGS ORDINARY MEETING held on Friday 10 June 2005 at the Mathematical Institute, Oxford, during a Colloquium. About 100 members and visitors were present for all or part of the meeting. The meeting began at 4.25 pm, after the Colloquium had been opened by Professor T.J. LYONS FRS, and was Chaired by the LMS President Professor F.C. KIRWAN FRS. Three people signed the book and were admitted to the Society. The President, on Council’s behalf, presented a certificate to the 2004 Honorary Member of the Society, Professor I.M. Singer. The Colloquium then resumed with Professor Lyons in the Chair. Professor I.M. Singer gave a lecture on The projective Dirac operator and its fractional analytic index. The meeting was followed by an informal reception. RECORDS OF PROCEEDINGS AT MEETINGS ORDINARY MEETING held on Friday 17 June 2005 at University College London. About 60 members and visitors were present for all or part of the meeting. The meeting began at 3.00 pm, with the President, Professor F.C. KIRWAN FRS, in the Chair. Four people were elected to Ordinary Membership: J. Fliege, J.E. Gough, H. Koohy, D. Martin; two people were elected to Associate Membership: P.N.J. Eagle, J.E. Hinchcliffe; and two people were elected to Reciprocity Membership: J.P. Howard, T.M. Rassias. One person signed the book and was admitted to the Society. The President, on Council’s behalf, proposed that Professor J-P. Bourguignon, of the Institut des Hautes Études Scientifiques, be elected to Honorary Membership of the Society. This was approved by acclaim. The President then announced the awards of the prizes for 2005: Pólya Prize – Professor Sir Michael Berry FRS (University
of Bristol); The President read short versions of the citations, to be published in full in the Bulletin. The President introduced a lecture given by Professor J. Barrow-Green on An indulgent freedom: 100 years of presidential addresses, on the occasion of the launch of The Book of Presidents: 1865-1965. The President then introduced the 2004 Naylor Prize Lecture given by Professor R. Jozsa on An invitation to quantum computation and recent theoretical developments. After tea, the President introduced a lecture given by Sir Roger Penrose on Quanglement, spin-networks, and twistor theory. After the meeting, a reception was held at De Morgan House, at which the President opened the Faces of Mathematics exhibition. The reception was followed by a dinner at the Il Fornello Restaurant. LMS MEETING 17 June 2005 The meeting opened with An indulgent freedom: 100 years of Presidential addresses by June Barrow-Green, given on the occasion of the launch of The Book of Presidents. Many of the older accompanying photographs were taken from the collection assembled by Robert Tucker who had been Secretary from 1866-1902. This had been on loan to the Science Museum in 1981 but was repatriated, catalogued and completed by the Society’s Administrator, Susan Oakes. Presidential addresses have been very varied, from major reviews to the highly specialized. Pure mathematics had increasingly dominated. Among the more important were those of Smith, Greenhill and Jeffrey. Richard Jozsa (Bristol), the 2005 Naylor Prize winner, then spoke on An invitation to quantum computation and recent theoretical developments. Bits in a quantum computer (qubits) allow linear superpositions of 0 and 1 states, unlike classical bits, but the really crucial new features in quantum computing are quantum measurement and quantum entanglement. The first of these implies irreversible loss of information when a measurement is made. Josza had strengthened the known impossibility theorem for duplication of a quantum state to show that one needed at least the full information of the second copy. Quantum entanglement depends on the fact that the state space of n
qubits is the tensor product of the individual qubit states, rather
than the Cartesian product as in the classical case. An n-qubit
state need not be the product of 1-qubit states. Quantum gates are defined
by unitary operations on the state: these cannot be efficiently simulated
on a classical computer. They can be modelled using three basic 2-qubit
gates: an algorithm involves applying them to pairs in n qubits,
followed by measurement of the leftmost bits. Examples of the improved
efficiency possible were given, some being solvable in a single step.
However, the classic problem of satisfiability, ‘given a Boolean
formula in n variables, are there values for which it is true?’,
to which any NP (nondeterministic polynomial time) problem can be reduced,
requires O( This raises the issue of what information an n-qubit state can give. One key possibility is ‘pattern recognition’ of periodicity. This is involved in Shor’s famous quantum factoring algorithm. Further examples are the hidden subgroup problem for symmetry of functions on groups, and Pell’s equation. It is not clear if all algorithms are of this type, but Jozsa suggested that quantum algorithms are classical algorithms ‘punctuated by quantum effects’. This led on to consideration of models of computing, in particular the methods based solely on measurements relative to carefully-chosen bases at each step. The conclusion that the power of entanglement is offset by the loss of information in measurement may imply that quantum methods cannot efficiently solve NP complete problems. This limitation on computing power may be a fundamental principle for physical theories. After tea, Sir Roger Penrose (Oxford, 2004 De Morgan medallist) spoke on Quanglement, spin networks, and twistor theory. He noted that (i) even simple quantum particle detection implied some superluminal connection between points on the wavefront, and (ii) accounting for entangled state measurements at separated points required information to travel into the past. He then described two ways of encoding quantum geometry which have appropriate features to describe entanglement. Spin networks, which are used in loop quantum gravity, are based on the idea of labelled graphs whose vertices obey the usual spin rules. One can assign norms to such graphs combinatorially. It had been shown that in the large N limit one can reconstruct usual Euclidean space. Although these networks are static and give an incomplete explanation of phenomena, they were a motivation for twistor theory. Twistors can be introduced by considering the space PN of null rays in flat Minkowski spacetime. The transformation of the celestial spheres of relatively moving observers is conformal, but to give this an adequate setting one has to complexify PN. Doing so leads to the complex manifold PT, where helicities are attached to the rays. The correspondence with Minkowski space is related to the Klein correspondence and to classical results of Lie. Considering the SL(2,C)-spinor description of a massless spinning particle leads to the representation of twistors as spinor pairs. Quantization by making twistors and dual twistors non-commuting leads
to helicity eigenstates given by homogeneous holomorphic functions on
PT, the wave functions in Minkowski space being their contour integrals.
Inspection of the Malcolm MacCallum LMS RECEPTION The Society Meeting was then followed by a reception at De Morgan House at which members and guests had the opportunity to view the Faces of Mathematics exhibition and purchase copies of The Book of Presidents. This must have been a record for the largest number of past, present and possibly future Presidents in one room at the same time: at least seven were counted. Several members took advantage of the unique opportunity to have their copy of the book multiply signed. The Faces of Mathematics exhibition focuses on the personalities of twenty influential mathematicians, in the form of large black and white portraits. It aims to penetrate the world of mathematics research and presents the human side of this often austere and challenging area of modern science. Alongside each of the portraits, a text-based display panel conveys the subject's research interests and personal viewpoint on mathematics. The project was funded by a Partnership in Public Understanding Award from the Engineering and Physical Sciences Research Council. Nick Gilbert (Project coordinator) is a Senior Lecturer in Mathematics at Heriot-Watt University, Edinburgh. He regularly publishes research work in group theory and topology, and has a particular interest in the public understanding of mathematics. He is a London Mathematical Society Holgate Lecturer. The photographs were taken by Marc Atkins who has lived and worked in Europe and North America and has exhibited extensively including London, Paris, Rome and New York. His work has been published in books and magazines worldwide, and features in both private and public collections, including the National Portrait Gallery, London. Nick Gilbert has kindly agreed that the set of ‘Faces’ should be housed at De Morgan House. Once the current modification work on DMH is complete it is planned to find appropriate places throughout the building to display as many as possible.
POPULAR LECTURES REPORT The Popular Lectures on Friday 15 July attracted a large, suitably mixed audience to the Royal Institution, with all ages from teenagers upwards well represented. The lecture theatre is of course remarkably beautiful, but it can feel cramped and, on a very hot evening, I was a little apprehensive as to my comfort. I need not have worried: the two lectures were both so enthralling that neither the heat nor the lack of legroom intruded on my consciousness at all during the evening. The two talks were very different. Alan Slomson told us What Computers Cannot Do. Eschewing digital display equipment in favour of the overhead projector, he took us through the logic of Unlimited Register Machines (which are similar to Turing Machines), proving (in full) that there is no algorithm that can predict in advance whether a given programme will eventually terminate or not. I would not have imagined in advance that it was possible to demonstrate this result to a lay audience in the time available, and Alan's achievement in doing so was remarkable. Alan finished his witty and entertaining talk by mentioning two other problems for which it has been proved that no algorithm can exist: whether or not a given Diophantine equation has a solution and whether or not an expression can be found for the integral of a given expression. This talk was a wonderful demonstration of how to present pure mathematics to a general audience. After the interval, Joan Lasenby gave a more practical demonstration of the use of mathematics in her talk on The Mathematics of Shrek. Switching more comfortably between a variety of software applications than I have ever been able to do, she showed us extracts from recent big-budget animations and (presumably smaller-budget) work by her students and her son, and gave us an idea of the mathematics on which such 3-D transformations depend, from complex numbers to fractals. She also showed what can be done quickly and easily with readily available, cheap software, and offered inside information into how the teams of animators in Hollywood studios work. This is a fast-moving area which draws on many areas of science, from mathematics and engineering to the neuroscience of vision and gait analysis, and it was fascinating to see how necessary all these diverse disciplines are if an animated figure, however convincing as a still image, is to remain realistic when it moves. Again, the audience was fully engaged, and a series of questions drew further insights from the speaker. I thought this was a remarkably successful evening. The organisation seemed very smooth: the talks complemented each other beautifully, offering a balance of pure mathematics and applications, rigorous proof and overview, the pleasures of symbolic manipulation and purely visual enjoyment. Such an audience, containing specialists as well as the general public, must be particularly difficult for a speaker. I went with a friend, a visual artist with an interest in film but no mathematics background: she had no trouble at any stage in following both talks, and we were both thoroughly entertained. The lessons for any sixth-former considering a future in mathematics were very positive: we were shown that mathematics is fun and that it offers exciting career possibilities! The programme is to be repeated in Manchester on Wednesday 28 September and will be recorded for subsequent release on DVD (which will be available from the LMS: a useful purchase for school and university maths departments seeking resources that will stimulate their students). Tony Mann
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) time.
ech cohomology of the non-singular regions shows the
wave functions really correspond to cohomology classes. Cohomology is
non-local, a point illustrated, literally, by the famous tribar invented
by Penrose and his father and used in Escher’s art, but measurement
of the wave function is local: this may capture point (i). Similarly
representing systems of n particles in the cohomology of a tensor
product of twistor spaces may provide a route to description of (ii).
