Topic #3 ------------- OP-SF NET 8.5 ------------ September 15, 2001 ~~~~~~~~~~~~~ From: OP-SF NET Editor (muldoon@yorku.ca) Subject: Reports on OPSFA, Rome, 2001 The Sixth International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA) took place at Ostia, near Roma, Italy from 18 to 22 June, 2001. Here are reports on the symposium from Tom Koornwinder, Bill Connett and Peter Clarkson. Some impressions by Tom Koornwinder (thk@science.uva.nl) This recent conference was the sixth (or by a different counting the ninth) in a series of European meetings which started in Bar-Le-Duc, France, 1984. The regular participants of these conferences are like relatives from a large family, spread over Europe (or even the world), which come together every two or three years for a joyful reunion. Serious family matters certainly have to be discussed, but enough time should remain available for lighter talk, for good eating and drinking and for having a lot of fun. The cousins from Southern Europe, who are also most numerous, have in particular excelled during this long period in being hosts to their family. The site of the 2001 meeting was Rome, or rather Lido di Ostia, which is Rome on the sea. Ostia is certainly less exciting than Rome (a good thing for keeping participants at the lectures), but Rome is still close enough (a 30 minute train ride for only 1500 Lire) to make evening visits to the city by participants or daytime visits by accompanying persons a good option. Everything, lectures, meals and accommodation, took place in an excellent and pleasant hotel in the middle of a large neighbourhood packed with modern apartment buildings of moderate height. Town planners have given this neighbourhood a human aspect by spreading shops (including many bars) all over the neighbourhood instead of concentrating them in a shopping mall. The seaside was within 10 minutes walking distance. Those who were in, or passed through, Rome on the Sunday afternoon and evening before the meeting, could give testimony of some one million people in the streets celebrating the Italian championship of their local soccer team Roma, after a decisive match against Parma held in the city that afternoon. No hooligans here, no riots, no plundering, but young men with their girl friends, and fathers and mothers with their children all happy together about the success of their favourite club for which they had to wait so many years. (Later I read in a Dutch newspaper that there were still some disturbances and casualties.) The conference had two plenary lectures every morning. Afterwards, at least on a generic day, there were 7 contributed lectures in four parallel sessions. The plenary lectures lasted 60 minutes including discussion, the contributed lectures 30 minutes including discussion and possible change of room. The plenary lectures were the following: - A. Laforgia, M. Muldoon and P.D. Siafarikas, Commemoration of A. Elbert - R. Askey, Solutions of some q-difference equations - C. Dunkl, Special functions and generating functions associated with reflection groups - D. Sattinger, Multipeakons and the classical moment problem - D. Stanton, Orthogonal polynomials and identities of Rogers-Ramanujan type - S.K. Suslov, On Askey's conjecture - N.M. Temme, Large parameter cases of the Gauss hypergeometric function, in particular in connection with orthogonal polynomials - W. Van Assche, Multiple orthogonal polynomials Speakers in all these lectures gave excellent presentations. In the commemoration of A. Elbert the three speakers gave a very worthy and impressive account of this Hungarian mathematician as a person and as a scientist. As one of the speakers said and made evident, his work was underestimated by the mathematical community. I am regretting now that I have never been in personal contact with this interesting mathematician, who died much too young. While all plenary lectures were very interesting for me, I was in particular impressed by the lectures by Sattinger and by Van Assche. David Sattinger, coming from nonlinear pde's and integrable systems, talked about a surprising application of the classical Stieltjes moment problem and the related continued fraction expansion to peakon and antipeakon solutions of the Camassa-Holm equation. The Camassa-Holm equation is a nonlinear pde refining the KdV equation, more suitable for modelling fluid flows in thin domains. It supports solutions, so-called peakons, that are continuous but only piecewise analytic. Solutions with the peak downwards are called antipeakons. During a peakon-antipeakon collision the slope becomes infinite. Closed form of peakon-antipeakon solutions, asymptotic behaviour and scattering shift can be obtained from the continued fraction expansion and the corresponding orthogonal polynomials. A good reference is R. Beals, D.H. Sattinger and J. Szmigielski, Multipeakons and the classical moment problem, Advances in Math. 154 (2000), 229-257. Walter Van Assche talked about multiple orthogonal polynomials. This notion has its roots in the nineteenth century, from simultaneous rational approximation, in particular Hermite-Pade approximation. The theory of multiple orthogonal polynomials came up in the Eastern European literature during the past ten or twenty years. Recently it has got a further impulse by work of Van Assche and his collaborators. These polynomials occur in two variants, type I and type II. Type II means for instance that a polynomial P(x) of degree n_1+n_2+...+n_r is orthogonal to all polynomials of degree less than n_j with respect to an weight function w_j(x) on an interval Delta_j (j=1,...,r). One can define multiple analogues of the classical orthogonal polynomials. In a remarkable result of Van Assche, Geronimo and Kuijlaars the Fokas-Its-Kitaev Riemann-Hilbert problem associated with a system of orthogonal polynomials has a generalization to the multiple case. See "Riemann-Hilbert problems for multiple orthogonal polynomials", to appear in "Special functions 2000: Current perspectives and future directions", Kluwer, 2001; also downloadable from W. Van Assche's homepage. The contributed talks were of great variety, such that something could be found to everybody's taste. Two contributed lectures struck me as having deserved more emphasis by the organizers and a larger time slot. In a brilliant 25-minute lecture Peter Clarkson gave a survey of properties of the Painleve equations, restricting to Painleve II for the sake of exposition. The Painleve equations may be seen as nonlinear analogues of the classical special functions. Peter Clarkson is writing the chapter on Painleve equations in the forthcoming NIST Digital Library of Mathematical Functions (successor to the Handbook of Mathematical Functions by Abramowitz and Stegun). Dan Lozier, managing editor of this DLMF project, gave a very informative contributed lecture about the present status of this large-scale enterprise, which will be of enormous importance for the future of special functions usage. A very remarkable social event was Music for Friends on Tuesday evening, where Gino Palumbo of Universita Roma Tre, one of the conference organizers, played piano works, partly joint with Enrico Tronci, composed by himself during the years 1977-1987. The organizers did a great job. Still a few critical remarks may be in order. - The opening session was scheduled to last for one hour, but it was finished after 15 minutes. I would have enjoyed to hear more from the mouths of local rectors, deans and chairmen about the history of the three Roman universities, about the number of mathematics students, about the reason why most mathematics students in Italy are female, but most mathematics professors are male, and whether the Museo della Matematica housed in the Dipartimento di Matematica of Universita di Roma "la Sapienza", and mentioned in the very comprehensive booklet Tesori di Roma, is meant as something serious or as a kind of joke. - The topics of lectures (more so for the contributed than for the invited lectures) remained somewhat classical and traditional, with emphasis on one-variable theory and analytic methods. Some more follow-up of fascinating developments about which one could hear last year in Tempe, Arizona and some more spin-off of things going on during last half year at the Newton Institute in Cambridge, UK about symmetric functions and Macdonald polynomials might have been appropriate. - A generic criticism of common practice in math meetings is that transparencies are displayed too briefly, so that it is impossible to take notes and to digest their full contents. During the plenary lectures this effect might have been softened by bringing in a second projector. At some meetings xerox copies of transparencies of plenary lectures are distributed. A cheaper alternative might be to scan the transparencies and put them on the web or hotel TV system. Next porno flashes on the TV's in the hotel rooms might be replaced by flashes from the transparencies of the plenary lectures. After paying 20000 Lire one might then get the full view of the lecture contents on one's TV screen. - One more thing about the web. It would have been nice if the full schedule would have been on the web some days before the beginning of the meeting. From: William Connett (connett@arch.cs.umsl.edu) Like a visitor to ancient Herculaneum in 79 A.D., who had come to town for the very good theater, your reporter arrived in Rome on the eve of the climactic game of the Italian Soccer cup completely unaware of the real drama that was about to unfold, thinking only of polynomials, and the opportunity to visit a few historical monuments (where is the Forum anyway?) and to pay the proper obeisance to a number of Christian monuments, St. John Lateran, St. Lawrence outside the walls, the Basilica of Sts. Cosmos and Damian, etc., and perhaps to recite "Ode to Melancholy" on the grave of "a Young English Poet", when like the eruption of Vesuvius, the triumph (by the score 3-1) of AS Roma over Parma late that Sunday afternoon (June 17), changed my world, and the pandemonium let loose on the streets of Rome by the hundreds of thousands of hysterical fans was the most memorable single event of the trip. I was staying in Trastevere, and I will never forget the torrent of modern charioteers cascading down the Lungotevere di Anguillara, singing, shouting, blowing trumpets, waving enormous yellow and red flags from what were actually very small motorbikes, crossing the Ponte Palatino, and pooling in the center of Rome. Little did I realize, that the pool of people meant that the trolley lines could not run, and soon the bus lines could not run either, and my euphoria changed to the grim realization that the only way to get to the Porta San Paola, and the train to Lido di Ostia, was to tramp, carrying my suitcase, some three miles from Monteverde to Piramide. I caught the last train, and collapsed amid a throng of very tired and somewhat inebriated supporters for the thirty minute ride out to the Hotel Satellite in Lido di Ostia. This was quite a dramatic beginning to the Sixth International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA-VI). In spite of a long evening of rumpus and ruction in the streets, the some one hundred and fifty mathematicians appeared Monday morning, for the opening ceremony of a very interesting meeting. There were seven plenary lectures, ninety two more technical research seminars, and an open problem session on the last day of the meeting. The following is a very impressionistic overview of some of the highlights of the scientific meeting. One of the topics in Dick Askey's talk which opened the meeting, was the problem of finding bounds for the maximum values of the polynomials in an orthogonal family. He reminded us of the argument due to Sonine, I believe, for the Jacobi polynomials, that gives a bound for the maximum value of each polynomial in the interval of support, and suggested several ways that this argument might be generalized to other orthogonal families. I was intrigued. The scientific committee (de Bruin, Laforgia, Marcellan, Muldoon, Ricci, and Siafarikas) are to be congratulated for their efforts to bring speakers to these meetings who have found new and interesting uses for the classical mathematics. The excellent talk of David Sattinger was a good example of this. I am still not sure what a "multipeakon" might be, but found his application of the classical moment problem to this problem in fluid dynamics a delight. In another direction, I was also intrigued by the improbable idea presented by Walter Van Assche, of considering a family of polynomials to be orthogonal with respect to two different measures. These ideas were later elaborated on by Els Coussement, and Jonathen Coussement in research seminars. I must give the prize for innovation to Franz Peherstorfer for his very exciting talk on the distribution of the zeroes of polynomials that are orthogonal with respect to a weight supported on disjoint intervals of the real line. I have used a simple version of this problem for years as a summer project for college students, and although they (and I) have learned much from the experience, nothing I knew prepared me for the complexity of the machinery from complex variables that he employed to give a definitive resolution to this problem. Well done! Nonlinear special functions are alive and well. The talk of Peter Clarkson did an excellent job pulling together a number of facts about the solutions to the six Painleve equations, including the nonlinear recurrence relations (Baecklund transformations) for the solutions of several of them. I think that there is much more to come here. I also enjoyed the seminar of Mohammed Sifi who did some very nice harmonic analysis showing that the action of a particular Dunkl operator could be considered as a multiplier that satisfied the Hormander condition, and therefore was bounded in L^p for a range of p. Multiplier operators first got me interested in special functions, and it is nice to see new approaches to these old questions. The physical arrangements for the meeting were excellent. The hotel was comfortable, the food quite good, and the location away from the bustle of the center of Rome, but on the Tyrrhenian Sea, was perfect for a scientific meeting. In the gathering twilight, groups of mathematicians could be seen strolling along the beach, sampling the excellent gelato and, as is there wont, filling napkins with illegible calculations. Arrivederci, Roma! From: Peter Clarkson (P.A.Clarkson@ukc.ac.uk) The sixth International Symposium "Orthogonal Polynomials, Special Functions and Applications" (OPSFA) was held in Rome, Italy, in June 201. This was attended by about 150 scientists from around the world. The plenary lectures were given by R. Askey, C.F. Dunkl, D. Sattinger, D. Stanton, S.K. Suslov, N.M. Temme and W. van Assche. Further there was a commemoration of A. Elbert by A. Laforgia, M. Muldoon and P.D. Siafarikas. The contributed talks were given in four parallel sessions. The structure of the meeting was similar to many others that I have attended. It was extremely well organized. Locating the OPSFA meeting near to a city such as Rome is certainly an additional attraction. On a personal note that was the first OPSFA meeting which I have attended and I enjoyed it very much (despite three hours delays for both my London-Rome and Rome-London flights and missing luggage!). I had met a number of the other participants previously at other meetings, in particular the "Symmetries and Integrability of Difference Equations" series of meetings (Esterel, Canada, 1994; Canterbury, UK, 1996; Sabaudia, Italy, 1998; Tokyo, Japan 2000). My own research field is the study of nonlinear differential equations and nonlinear difference equations, in particular exact solutions and asymptotics. Frequently we use results from "Special Functions" and "Orthogonal Polynomials", despite being linear equations in the solution of the nonlinear problems. I was pleased and encouraged that there were some talks on nonlinear problems at this meeting, including a plenary lecture by David Sattinger. I believe that there are many mathematicians and physicists who have research interest in "Special Functions" and "Orthogonal Polynomials" though not as their main field of research. I feel that the involvement of such scientists in the OP-SF activity group and participation in future OPSFA meetings should be strongly encouraged.

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