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REPORTS AND RECORDS OF SOCIETY MEEETINGS Contents LMS Ordinary Meeting on Tuesday 22 July: record RECORDS OF PROCEEDINGS AT MEETINGS ORDINARY MEETING held on Tuesday 22 July 2003, jointly with the Edinburgh Mathematical Society, as part of the International Centre for Mathematical Sciences ‘Hodge Centenary Meeting’, held at the University of Edinburgh. Over 100 members and visitors were present for all or part of the meeting. The meeting was opened at 2.00 pm by the Edinburgh Mathematical Society President, Professor A. GILLESPIE, who chaired a business meeting of that Society. The President, Professor P. GODDARD, FRS, then assumed the Chair. Five people were elected to Ordinary Membership of the London Mathematical Society: M. Carr, M. de Visscher, D.J. Hoyt, M.E. Jimenez Contreras and J. Virtanen; five people were elected to Associate Membership: C.A. Hoenselaers, E. Katirtzoglou, A.A. Miller, M. Pistoriu and M.J. Thompson; and one person was elected to Reciprocity Membership: J.C. Goodwin (Australian Math. Soc.). The Records of the Proceedings of the Society Meetings held on 11 March and 14 May 2003 were signed as a correct record. The President announced the award of the Joint LMS-IMA David Crighton Medal for 2003 to Professor J.M. Ball, FRS. Professor Goddard introduced a lecture given by Sir Michael Atiyah, FRS, on ‘Sir William Hodge – The Man and the Mathematician’. Professor Gillespie introduced a lecture given by Sir Roger Penrose, FRS, on ‘Mathematical Experiences as a Cambridge Research Student under William Hodge’. After tea, Professor P. GRIFFITHS, Institute of Advanced Studies, Princeton, introduced a lecture given by Professor F. Hirzebruch, For Mem RS, on ‘Hodge Numbers, Chern Numbers, Catalan Numbers’. Professor Goddard closed the meeting and expressed the thanks of the Society to the Edinburgh Mathematical Society, the ICMS, University of Edinburgh and the speakers for putting on such an excellent meeting. After the meeting a reception and dinner were held at the Playfair Library, University of Edinburgh.
Hodge was born in Edinburgh and spent most of his academic career in Cambridge. He pioneered the use of differential geometry (in particular Kähler geometry) and functional analysis (harmonic theory) in algebraic geometry. This was revolutionary at the time and produced deep and surprising results that have now become standard tools in geometry, and even in areas of physics and number theory. His work also led naturally to a question of whether some of his analytically defined topological invariants (the Hodge classes) of an algebraic variety could be described algebraically (as algebraic cycles). This is the famous Hodge conjecture, one of the Clay millennium prizes for which $1 million is offered.
Even BBC Scotland was there, filming their ‘and finally’ piece for the 6 O’clock News. Interspersed with clips of lectures and interviews with Atiyah, Griffiths and Elmer Rees was 1970s BBC archive footage presumably meant to illustrate the Hodge conjecture: an Open University lecturer (complete with bad beard and flares) in front of a graph of y=x, and a young man (worse beard, couldn’t see his trousers) solving the Rubik’s cube in record time. We then cut back to the studio where the newsreader added 7 and 9 and wondered about a sum so hard that it was worth $1 million. The whole thing was mostly hilarious rather than patronising, describing mathematics with awe as well as the usual bemusement; most people cried with laughter as it was played back to the participants. The conference also featured problem sessions for people to ask questions (‘What was Beilinson talking about ?’) and so most people got a good deal out of the week, as well as getting to see the most stellar collection of mathematicians outside of the ICM. Richard Thomas
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The
International Centre for Mathematical Sciences, Edinburgh, held a conference
from 20-26 July, to commemorate the centenary of Sir William Hodge,
the originator of Hodge theory – ‘one of the landmarks in
the history of mathematics in the 20th century’ according to Hermann
Weyl. There was a quite spectacular array of speakers, including four
Fields Medallists, the world’s leading experts in Hodge theory,
former colleagues and students of Hodge such as Fritz Hirzebruch and
Sir Roger Penrose, and members of the Hodge family.
A
joint meeting of the London Mathematical Society and Edinburgh Mathematical
Society was held on the Tuesday of the conference. Sir Michael Atiyah
gave a general talk about Hodge’s life, full of photographs, family
archives, funny stories and his personal reminiscences as a former student
of Hodge. Professor Hirzebruch described more of the history of the
mathematics around at that time and gave an interesting lecture on characteristic
classes. Sir Roger Penrose, who spent a year as Hodge’s student,
described his very non-Hodge-like results from that year which preceded
his celebrated diagramatic methods in tensorial and twistorial calculus.
In the conference itself Maxim Kontsevich gave a talk about his noncommutative
Hodge theory, using similar Feynman diagram methods to organise homological
algebraic information, and solved the problem of defining (or renormalising)
the physical theory of ‘Kodaira-Spencer gravity’ (at least
according to those who understood it). Edward Witten gave a fascinating
talk about the physics of gauge theories (which can be thought of as
non-abelian Hodge theory) leading to a series of mathematical conjectures.
These and David Morrison’s excellent talk (with colour graphics,
no less, a rarity in pure mathematics) on birational geometry and Hodge
theory, illustrated the prominent role of Hodge theory in the interaction
between geometry and string theory.
There
were talks about pure Hodge theory and the Hodge conjecture by the giants
of the field: Griffiths, Beilinson, Green, Soulé, Schmid, Voisin
and Bloch (in chronological order). There were also talks by Cheeger,
McDuff and Donaldson illustrating how some parts of Hodge’s Kähler
methods can now be extended to other settings, in particular symplectic
geometry. One of Simon Donaldson’s talks in fact announced an extension
of these methods to almost all smooth compact 4-manifolds (those with
b+>1) to give an astounding structure theorem for 4-manifolds in
terms of a generalisation of Lefschetz pencils. Having polished off
4-dimensional differential topology he turned to three dimensions, giving
an enlightening account of Perelman’s work on the Poincaré
conjecture (another Clay $1 million prize) and Thurston’s geometrisation
conjecture, as described by Michael Singer in last month’s Newsletter.
This lead into a half-hour discussion session on the topic, in particular
giving us a chance to hear Kontsevich and Witten discuss a few of the
physical ideas motivating the work.