On the Move Our Institute in Oxford Houses a Large and Brilliant Community of Mathematicians
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Oxford Mathematical Institute March 2003 Issue 1 Newsletter Inside this issue Meeting of minds, 2 International Mathematical Congress 2002, Beijing Time to celebrate, 3 Special symposium on Topology, Geometry, and Quantum Field Theory On the move Our Institute in Oxford houses a large and brilliant community of mathematicians. Their achievements do not often attract the attention of the media since it is not in the nature of mathematicians to practise immodesty and it is not in the nature of journalists to inquire too deeply into mathematics. But over the decades, and indeed the centuries, the work of Oxford mathematicians has had a profound impact on our society. Talking with scientists, 4 This Newsletter tells our friends, and our former students and colleagues, something of About climate, cancer and superconductors the successes and achievements of the past year. Success, however,brings its problems. Our current building, which seemed so spacious The business of problem when it opened in 1966, is now hopelessly overcrowded. Our colony in Little Clarendon solving, 6 In finance, and in industry Street gradually expands as we take over more floors of Dartington House, but we desperately need a new building in which the whole community – professors, lecturers, post-docs, and graduate students – can work together in comfort.We also need facilities to draw undergraduates more fully into the life of the department. Informal interaction is the lifeblood of mathematics, and it is greatly inhibited by scattering people over different colleges and University buildings. The world of books, 7 In January of 2002, our External Advisory Panel set us down the road of planning a New books from Roger Penrose (p.2), new building for the whole department. It is early days yet, but we hope to see these David Acheson, Marcus du Sautoy, and tentative first steps come to fruition over the coming years. Subsequent issues of the Robin Wilson Newsletter will report progress. Congratulations, 8 N. M. J. Woodhouse Prizes and awards Chairman, Mathematical Institute Meeting of minds Mathematics, like music, is a language all its own and one that crosses national boundaries. Nowhere is this seen more clearly than at international conferences and this year in Beijing was no exception when China played host to 3000 mathematicians from around the world for the four-yearly International Congress, ICM 2002.The Opening Ceremony brought the mathematicians – in a fleet of buses driven through streets cleared of traffic – to the Great Hall of the People where in the presence of Jiang Zemin, President of the People’s Republic of China, the conference was declared open. Oxford was well represented: Professor Frances Kirwan was invited to be one of the 20 Plenary speakers, Professor Ulrike Tillmann was also invited to speak. At the Congress Professor John Ball was elected to the four-year Presidency of the International Mathematical Union. Moduli Spaces of Riemann Surfaces Frances Kirwan writes about her Beijing Riemann realised that a topological surface An important new book from Roger Penrose paper on Moduli Spaces of Riemann Surfaces. can usually be made into a Riemann surface is scheduled for publication in spring 2003. in lots of different ways (although this is not Ambitious in scope and scale, The Road to true for a sphere, which can be made into a Reality is Penrose’s picture of the physical Riemann surface in essentially only one way). universe complete with a masterly survey of In fact he observed that if S is a ‘sphere with g MARC ATKINS the mathematical underpinnings. handles’ where g ≥ 2, then there is a (3g - 3)-dimensional family of complex ‘I want to describe as clearly as possible our parameters which describe the possible present understanding of the universe and to Riemann surfaces which are topologically convey a feeling for its deep beauty and philo- the same as S.In modern mathematical sophical implications as well as its intricate terminology, Riemann’s observation amounts interconnections.’ to the fact that the moduli space of compact Penrose, Roger The Road to Reality Jonathan Riemann surfaces of genus g has complex ‘A surface in Euclidean 3-space R3 is easy Cape (2003) 0-224-04447-8 dimension 3g - 3 when g ≥ 2. to visualise; it is typically defined by one equation in three variables x, y and z,just Moduli spaces are very beautiful geometric as the unit sphere is defined by the familiar spaces which arise in classification problems equation x 2 + y 2 + z 2 = 1. The crucial in geometry: a moduli space for a given property which a surface must have is that classification problem is the set of equivalence locally it ‘looks like’ Euclidean 2-space R 2. classes of the geometric objects to be classi- We get different kinds of surface depending fied, itself endowed with a geometric struc- on our interpretation of ‘looks like’: a ture which reflects the way the objects can A feature of science today is the topological surface if our interpretation vary geometrically. gradual erosion of boundaries within uses continuous functions, a smooth surface Understanding the topology of the moduli and between subjects.This is seen in if we use differentiable functions and a spaces of compact Riemann surfaces of genus mathematics too and some of the most Riemann surface (called after the great g has for a long time been an important goal 19th century German geometer Bernhard in algebraic geometry, and very recently a sig- exciting of recent research has resulted 2 Riemann) if we replace R with the complex nificant advance has been made concerning from the imaginative use of ideas and plane C and use functions which are differen- the ‘limiting case’ when g tends to infinity. tools from one branch of the subject in tiable in the sense of complex numbers (i.e., In fact this progress has been made via a re- the elucidation of another,apparently holomorphic functions). interpretation of the problem in purely topo- logical terms, and it is not algebraic geometers unrelated area. According to a very old joke, a topologist is someone who cannot tell a teacup from a but a group of topologists, including Ulrike doughnut, the point being that the surface Tillmann at Oxford, who deserve the credit of a teacup is topologically the same as the for this particular advance.’ surface of a (ring) doughnut. In fact, each is Unfortunately Frances was unable to present topologically a torus, sometimes called a her paper in person as a result of illness. ‘sphere with one handle’. 2 OXFORD MATHEMATICAL INSTITUTE NEWSLETTER MARCH 2003 ISSUE 1 Time to celebrate Mathematics and theoretical physics have always been intertwined. It used to be said that advances in physics sometimes made use of pre-existing mathe- matics, and in turn could influence new mathematical research. Over the past twenty years, however,it has become apparent that ideas from particle physics can provide new techniques for studying traditional problems in mathematics. One such topic is string theory with its links to geometry and topology. NATHALIE WAHL NATHALIE String Topology, Geometry Theory & MARC ATKINS & Quantum Field Theory Oxford has been very much at the centre of this new interaction Topology of physics and mathematics. The perfect opportunity for Oxford to hold an interdisciplinary symposium was presented by Graeme Ulrike Tillmann on the Segal’s 60th birthday. Formerly a Fellow of St. Catherine’s and now a background to her Beijing Fellow of All Souls, Graeme Segal has spent most of his working life lecture String Theory and in Oxford. He has been central to the mathematics-physics dialogue Topology during the past two decades and many distinguished scientists from ‘In classical mechanics a particle moves along a trajectory which both camps were delighted to honour a colleague, teacher, and minimizes its ‘action’ – a certain functional of its paths. In Feynman’s friend and to attend the symposium and celebration banquet. path-integral approach to quantum mechanics the basic idea is the Ed Witten (Princeton) gave the first Astor lecture and this was ‘amplitude’ for the particle to get from one place to another in a given open to all members of the University. Mini-lecture courses by time, and is calculated as an average over all possible paths from one Mike Hopkins (MIT) and Robbert Dijkgraaf (Amsterdam) produced position to the other, weighted with the exponential of the classical stimulating dialogue; these and all the lectures were of the highest action. quality. Much was done in coffee breaks, over lunch, and in the Particle theory only works well when gravity is ignored. In searching evenings in never-ending discussions. Interdisciplinary dialogue was for a theory that unifies quantum mechanics and gravity theoretical the goal of the symposium and as such, it was a great success. physicists have invented string theory. The point particle is replaced The Symposium was organised by Ulrike Tillmann assisted by conference secre- by a ‘string’,a little loop. As time evolves the string moves and sweeps taries Brenda Willoughby and Laura Mildenhall. The Symposium was supported out a surface, and while it does this it occasionally splits or interacts by the London Mathematical Society, the EPSRC, and Oxford University. with other strings. To calculate the ‘amplitude’ of a string it is necessary to take an average – called the path-integral – over all possible surfaces in space- time. Unfortunately the space of all possible surfaces is too large and no rigorous mathematical way of taking such an integral is known. If the theory is conformally invariant, however, it is possible to replace MARCO GUALTIERI the space of all surfaces by the moduli space of compact Riemann surfaces, as described by Frances Kirwan in her article.