# FIELDSNOTES. Barbara Lee Keyfitz: Director elect. The National Program on Complex Data Structures (NPCDS)

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2 A Monstrous Tale: The Work of John McKay THE 2003 CRM-FIELDS PRIZE was shared by Edwin Perkins of UBC and John McKay of Concordia University. Perkins spoke about his work in November 2003 (see the January 2004 Fields Notes). McKay delivered his lecture on March 8, McKay s observations over the course of more than two decades led to discoveries that climaxed in the Fields Medal of Richard Borcherds. In his talk at the Fields Institute, McKay recounted some of those observations and the myriad mathematical connections that they opened up. These connections seem to show no sign of abating and the talk presented several new themes. McKay begins with the j-function. Define the complex function E = (σ 3 (1)q + σ 3 (2)q ), where σ 3 (n) is the sum of the cubes of the positive divisors of n and q = exp 2πiz. The discriminant function = q(1-q) 24 (1-q 2 ) was studied by Ramanujan and leads to the j-function defined by j = E 3 /. Expanding j as a function of q, we find j(z) = q q +... This is a modular function in the sense that it is invariant under action by the modular group SL 2 (Z). That is, if a,b,c and d are integers such that ad bc =1, then j(az + b / cz + d) = j(z). All of this was known classically. But McKay was the first to notice the relevance of the equation in the title of his talk: = Here is a coefficient from the above expansion of j, while is the smallest non-trivial degree of a character of the monster group, the largest sporadic finite simple group. The order of the monster is , a number larger than 8 x The fact that there might be some relationship between the monster and modular functions first arose in an observation of Ogg. He pointed out that the prime divisors of the order of the monster group are exactly the same primes p for which the normalizer of the discrete group Γ 0 (p) in SL 2 (Z) gives a genus zero quotient of the upper half plane. However, no one had thought that there might be a relationship between the Fourier coefficients of the j-function and its analogues, and the character theory of the monster. Shortly after McKay made this simple but profound observation, he looked at the character table of the group E 8 and found another relationship between the degrees of the John McKay representation here and the Fourier coefficients of a cube root of the j-function. Based on some of McKay s observations, Conway and Norton formulated a general conjecture that became known as the Moonshine conjecture, possibly because it was so bizarre. It related character degrees of exotic finite simple groups to Fourier coefficients of certain modular functions. Why such a relationship should even exist was completely unknown until the work of Borcherds who introduced several new concepts including that of vertex algebras. Using the theory of these algebras, he was able to prove the Moonshine conjecture. But far from being the end of the story, it seems that many new connections and conjectures have been opened up. In his talk, McKay touched upon such diverse topics as quantum logic, replicable functions, dispersionless flow and Witten s conjectured 3-manifold in topological quantum field theory and how each of these is connected in some way to moonshine. John McKay presents his ideas with a characteristic modesty. Nevertheless, he has served as a catalyst for the development of some of the very important ideas of modern mathematics. Besides receiving the CRM-Fields prize, he was recently elected to the Royal Society of Canada in recognition of his many contributions. V. Kumar Murty (Toronto) 2 FIELDSNOTES FIELDS INSTITUTE Research in Mathematical Science

6 Weekend Workshops on Arithmetic and Geometry of Algebraic Varieties THE WEEKEND WORKSHOPS ON CALABI-YAU varieties and mirror symmetry at the Fields Instite were very successful. In , we broadened the subjects to Arithmetic and Geometry of Algebraic Varieties (in particular, higher dimensional varieties) with special emphasis on Calabi-Yau varieties and mirror symmetry. These weekend workshops organized by myself and James Lewis (Alberta) have been getting to be known in the community. Besides local Canadian regular participants, a number of participants came from various parts of the United States. Due to high interest, we organized three workshops in the academic year (October 4-5 and November 8-9, 2003; March 27-28, 2004). The three workshops were a huge success. Postdoctoral fellows and graduate students were especially encouraged to take part in the activities. All participants and also several colleagues who could not be present at this year's workshops have already asked for schedules for the workshops in the coming year. We are planning the first workshop in the academic year in October. For a listing of speakers and titles see the Fields website. Noriko Yui (Queen's) Workshop on Nonlinear Wave Equations DURING THE WEEK OF MARCH 15-19, the Fields Institute hosted a workshop on nonlinear wave equations, as a component of the Thematic Program Year on Partial Differential Equations. This meeting was organized by Claude Bardos (Université de Paris 7), Jim Colliander (Toronto), Walter Craig (McMaster), Nick Ercolani (Arizona), and Catherine Sulem (Toronto). This was one of a pair of workshops on related material, the second being the workshop on kinetic theory, which was held one week later. The design of holding major workshops as pairs, in closely aligned weeks was to allow mathematicians who are working in both areas to attend as one extended event. Included in this time frame was the short-course on the theory of collisionless plasmas, which took place in the intervening week. The focus of the week on nonlinear wave equations included nonlinear microlocal analysis of evolution equations, results and ideas from Hamiltonian PDE, existence theory for the initial value problem (both local and global in time, and in optimal function spaces), and applications of this theory to a number of physically relevant partial differential Walter Craig, Claude Bardos, Nick Ercolani, Catherine Sulem and Jim Colliander equations. The setting of this work ranged from Einstein's equations of general relativity, to the nonlinear equations of fluid dynamics, to the nonlinear Schroedinger equations of plasma physics, including nonlinear optics and multi-particle quantum mechanics. Participants came from South America, numerous European countries, as well as Canada and the US. The number of postdoctoral speakers and graduate student participants was notably very high. There are numerous nonlinear evolution equations that have been derived in a context relevant to mathematical physics, for which it is a challenging mathematical problem to analyze the properties of the solution map for time evolution of the system. The theme of this workshop was on recent results on precise properties of solutions of the initial value problem, involving new techniques of harmonic and microlocal analysis for this purpose. Principal questions include the well-posedness of the nonlinear equations, and in which function spaces, the precise regularity of solutions, and the phenomenon of the formation of singularities as compared with the possibility of globally defined evolution in time. One of the main themes was the close analogy between microlocal techniques for linear PDE and the analysis of nonlinear kinetic equations. In fact the timing of the meeting to be close to both the workshop on kinetic theory and the short-course on collisionless plasmas was useful in this context. The schedule of 25 talks kept the schedule full but not too busy, and left the participants time to work and to discuss mathematics together with each other. I personally found the talks very informative, the nature of the audience very receptive, and all in all it was a very positive event in the active life of our program year. Walter Craig (McMaster) 6 FIELDSNOTES FIELDS INSTITUTE Research in Mathematical Science

9 The high point of the social events was undoubtedly the conference banquet on Sunday, Jan 18, at Waterloo Inn. A superb meal was followed by an inspiring talk by Mike Lazaridis, co-ceo, Research in Motion (RIM), and the visionary behind the Perimeter Institute (also a major benefactor of the Institute for Quantum Information). Lazaridis shared his passion for fundamental physics, and spoke at length on how it has resulted in significant technological advances. He stressed the importance of a scientifically aware society and government, and the role of researchers in achieving this. The evening was capped by a highly entertaining, semi-serious talk by Charles Bennett, IBM Research, Yorktown Heights, a pioneer in quantum information processing. Bennett regaled the audience with amusing snippets from the scientific talks at the workshop. Then, inspired by the exhibits at the old the Seagram museum (the conference venue), he went on to the physics of distillation and its parallels in entanglement purification in quantum information theory. The meeting was from all accounts very successful, thanks in no small part due to the months of preparation by Daniel Gottesman (Perimeter) and Michele Mosca (U. Waterloo & Perimeter), with whom I co-chaired the event, and Wendy Reibel (IQC) who most efficiently lead the local organization and meticulously ensured that everything went like clockwork. Thanks also to Fields, and all our sponsors, for the support and help that made it all possible! For more details on QIP 2004, please visit the conference website Ashwin Nayak (Waterloo and Perimeter) Young Mathematicians Conference: Partial Differential Equations and Dynamical Systems A SATELLITE OF THE THEMATIC PROGRAM on Partial Differential Equations at the Fields Institute, the two day conference provided an opportunity for promising young Canadian mathematicians to present their work in a very friendly and active atmosphere, to meet each other, and to interact with more senior mathematicians. There were twelve 30 minute talks delivered by junior scientists. There was also a poster session presented by the lively joint effort of nine invited junior scientists. The participants were also asked to give a five minute oral summary, during the regular seminar time, of the results described in the posters (for a complete list of speaker names and titles see the Fields website). Two 45 minute talks were delivered by senior scientists. N. Kamran (McGill) spoke on solutions of the wave equation problems in the context of general relativity and in particular on his results on wave propagation with a Kerr black hole background Lorentzian metric. A. Nachman (Toronto) presented an overview of inverse problems in partial differential equations. The success of the conference was the result of a team effort. The organizers, Stefanella Boatto (McMaster), Walter Craig (McMaster) and Catherine Sulem (Toronto) would like to extend their particular thanks to the chairs of the sessions, Robert McCann (Toronto), Matheus Grasselli (McMaster), Bill Langford (Guelph), Dmitry Pelinovsky (McMaster,) and Jianhong Wu (York). They participated actively by promoting and taking part in the mathematical discussions. The Fields Institute provided an excellent venue for presentations, discussions, and collaborative activities. We extend our thanks in particular to the Institute staff, Alexa Brand and Alison Conway. The conference was supported by the Fields Institute, the NSERC Leadership Support Initiative, and the Canada Research Chairs Program. Stefanella Boatto (McMaster) Paul Tupper and Stefanella Boatto FIELDS INSTITUTE Research in Mathematical Science FIELDSNOTES 9

15 Fields Activities Chalk it up to Mathematics MAY TO SEPTEMBER 2004 Detailed information: Thematic Programs FIELDS PARTIAL DIFFERENTIAL EQUATIONS, MAY 3-5, 2004 Coxeter Lecture Series, Sergei Kuksin MAY 10-13, 2004 Distinguished Lecture Series, Jean Bourgain MAY 12 15, 2004 Short Course on Hamiltonian Partial Differential Equations MAY 17 21, 2004 Workshop on Integrable and Near-integrable Hamiltonian PDE (joint with CRM) MAY 24 28, 2004 Workshop on Hamiltonian Dynamical Systems (joint with CRM), at CRM JUNE 1 11, 2004 Workshop on Semi-classical Analysis (joint with CRM) June 1 5, at CRM; June 7 11, at Fields JUNE 14 18, 2004 Conference on Free Surface Water Waves GEOMETRY OF STRING THEORY, General Scientific Activities at Fields unless otherwise indicated MAY 5-9, 2004 Representation Theory of p-adic Groups, at University of Ottawa MAY 6-8, 2004 Mathematics of Computer Algebra and Analysis, at University of Waterloo MAY 8-18, 2004 Quantum Information Geometry and Quantum Computing May 7-17, at McMaster University; May 18, at Fields MAY 12 14, 2004 Shape Optimization and Applications, at University of Ottawa MAY 12 15, 2004 Large-Scale Nonlinear and Semi-definite Programming, at University of Waterloo MAY 12 16, 2004 The Coxeter Legacy: Reflections and Projections, at University of Toronto MAY 14 18, 2004 Geometry and Topology of Manifolds, at McMaster University MAY 17-18, Canadian Symposium on Abstract Harmonic Analysis, at University of Western Ontario MAY 18 19, 2004 Southern Ontario Statistical Graduate Student Seminar, at University of Waterloo MAY 19 23, Canadian Operator Theory and Operator Algebra Symposium, at University of Waterloo MAY 25-27, 2004 Model Theory Workshop JUNE 7 26, 2004 Summer School on Probability Models and Statistical Analyses for Ranking Data, at University of Ottawa JUNE 11-13, 2004 Mathematics of Learning Objects Symposium, at University of Western Ontario JUNE 17 19, 2004 Conference in Honour of D. Brownawell, at University of Waterloo JUNE 20 25, 2004 Canadian Number Theory Association VIII Meeting, at University of Toronto JUNE 21-25, 2004 Fourth Canadian Summer School on Quantum Information, at University of Waterloo JUNE 22, 2004 Public Lecture by Hendrik Lenstra JULY 16-20, 2004 Algebraic Topological Methods in Computer Science II, at University of Western Ontario JULY 19-23, 2004 Quantum Information and Quantum Control JULY 28-30, 2004 MOPTA 2004 McMaster Optimization Conference: Theory and Applications, at McMaster University AUG. 4-7, 2004 IMS, New Researchers Conference, at York University AUG. 5-6, 2004 Workshop on Missing Data Problems AUG. 6-7, 2004 New Directions in Probability Theory SEPT , 2004 Distinguished Lecture Series in Statistical Science, Sir David Cox FIELDS INSTITUTE Research in Mathematical Science FIELDSNOTES 15

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