Topic #7 ------------ OP-SF NET 7.4 ------------- July 15, 2000 ~~~~~~~~~~~~~ From: OP-SF NET EditorSubject: Reports on Special Functions 2000 Special Functions 2000: Current Perspective and Future Directions, Arizona State University, May 29 to June 9, 2000. 1. From Erik Koelink The 2-week conference actually consisted of three parts. A NATO Advanced Study Institute, a NSF Research Conference and a series of lectures on Computer Algebra. The NATO ASI talks were plenary 1-hour talks on various subjects. These talks ranged from introductory talks to more advanced talks on recent results. To mention a few, I very much liked the talks by Christian Krattenthaler (a great performer with transparencies and figures), Mizan Rahman (associated orthogonal polynomials and Askey-Wilson operators), Simon Ruijsenaars (solutions of the Askey-Wilson difference operators with q on the unit circle), Percy Deift (Riemann-Hilbert problems, and their application to all kinds of problems), Ken Ono (recent exiting results in number theory), Slava Spiridonov (a very impressive account of factorisations and their use), Sergei Suslov (q-Fourier series and a q-Riemann zeta function), Alexei Zhedanov (biorthogonal rational functions), Hjalmar Rosengren (dynamical Yang-Baxter equation and n-j symbols, n=3,6,9). The above list consists more of the talks on more recent results, and there were also some very good introductions by Dennis Stanton and Mourad Ismail. Maybe I should mention all speakers in this programme, since the talks were in general very good and very interesting. The half-hour talks in the NSF-programme were organised in parallel sessions, so that it's impossible to attend them all. Some of my personal favourites were Jan Felipe van Diejen on a multivariable summation formula for elliptic hypergeometric series conjectured by Ole Warnaar that he could almost prove, Katsuhisa Mimachi on representations of the Hecke algebra on twisted homology, Andre Unterberger on relativistic quantisation applied to special functions, Michitomo Nishizawa on all kinds of generalisations of the gamma function and Joaquin Bustoz on q-Bessel functions and q-Lommel polynomials. The talks in the Computer Algebra part were usually scheduled in the evening, which is one of the reasons that I missed a number of them. Some of these talks were presentations by people from Mathematica who discussed their huge posters on special functions. The chief of the (local) organisation was Sergei Suslov and he has made a tremendous effort in making the conference such a success. His daughter Liliya has been a great help in organising. All in all the organising committee has done a very nice job. The Tempe surroundings were very pleasant, but also very hot. The Grand Canyon was one of the touristic events and, being an inhabitant of a flat country, I was really impressed with it. 2. From Kathy Driver <036KAD@cosmos.wits.ac.za> Over 100 mathematicians gathered in Tempe to discuss Current Perspectives and Future Directions in the area of Special Functions. The meeting was remarkable from several different perspectives, perhaps the most striking feature being the diversity of areas in which talks were presented. The old maxim that "special functions are everywhere" gained considerable credibility as a variety of topics unfolded both in the main presentations and also during the parallel sessions. Orthogonal polynomials , special functions of one and several variables, asymptotics, continued fractions, applications to number theory, combinatorics and mathematical physics, integrable systems, harmonic analysis and quantum groups, Painleve classification were listed as some of the topics to be covered and that was no exaggeration--these and many others featured in a lively and well- organised programme. Many of the well-established masters in the area presented talks, mostly for two separate hours which facilitated more than just a glimpse of their ideas and expertise and attendance by graduate students was noteworthy. Richard Askey commented in his speech at the banquet that he was grateful to those present for carrying the banner of special functions forward over the past ten years and it was easy to see why he is pleased with developments in the field. The venue was comfortable and suitable, the organisational details were taken care of in exemplary fashion and the power failure was thankfully short in duration, given the formidable heat in the desert at that time of year. attended. This was an extremely successful meeting and bodes very well for continuing vigorous interest in this area. 3. From William Connett It was hot. The sun was a terrifying presence. Your reporter would look out the window of the conference center and he could see for two miles down Apache Boulevard, and often not a single person could be seen in the open in this very modern city. Although the temperature of the air was over 100 F, nobody went into the swimming pool during the day because the temperature of the pool was above 90 F and the sun was so intense that it would give you skin cancer in five minutes. On the other hand it was not the hottest mathematical meeting that your reporter ever attended. I remember one epic meeting in Morocco in July when there was no air conditioning in the hotel, no water in the rest rooms, the temperature one day got up to 130F, and all the lectures were in French. By that standard, this meeting was a cake walk. The Holiday Inn was a very pleasant venue. There was so much air conditioning in the conference hall that most participants wore their jackets, the food was serviceable and easy to obtain, and the lay out of the conference with all lectures, food, and rooms in one location made it a very pleasant meeting. The weather kept all the participants in the motel, so that mathematical conversations were spontaneous and quite easy. This was one of the most complex meetings that I have ever attended. It was concurrently: first, a NATO funded Advanced Study Institute, second, a NSF funded Research Conference, and third, a mini conference on Computer Algebra and Special Functions on the Web, supported by Wolfram Research and other sources. This may become the new paradigm for organizing a conference. The field of special functions has grown so enormously that it is difficult to remember the time when the few enthusiasts could easily fit into a small seminar room to discuss the problems of common interest. Now there is a cast of hundreds, working in dozens of areas. The specialty meeting now take on more of the character of the large national meetings. And the total experience was quite enjoyable. The NATO funded Advanced Study Institute featured a number of hour long talks which were intended to introduce a topic, and bring a sophisticated audience up to a certain level of competence on a particular problem. For example, Luc Vinet gave two lectures entitled "Advances in multivariable special functions and mathematical physics", but actually he had the courage to ignore the physics, and work through several concrete examples of the new families of symmetric polynomials called atoms, related to the t-Kostka polynomials. The examples were carefully done, and the audience was very appreciative of the care with which they were explained. Two other speakers in this section that I really enjoyed were Christian Krattenthaler who gave a lovely series of lectures on plane partitions, orthogonal polynomials, and hypergeometric series. Christian certainly wins the prize for the most innovative use of the overhead projector in his presentations. Even if I did not enjoy the topic, I would be fascinated by his implementation of ur-animation in his talks. His screen reminds me of some of the early Loony Toons cartoons with the jerky but eye-catching animation. The other speaker was Alexander Kitaev, who introduced the audience to the six versions of the Painleve equation and their solutions. I was very appreciative of his effort to explain to the outsider what was going on in this important area. The NSF research Conference included many more traditional research type talks, from this feast of topics, I will mention two that I found particularly memorable: Yuan Xu talked about problems in Fourier expansions in several variables, and Khalifa Trimeche worked out the harmonic analysis associated with a singular differential-difference operator (a generalization of the Dunkl operator on the real line). There were many other excellent talks. The final part of the conference were the sessions on computer algebra. real indication of the interest in these topics (or perhaps just the weather) that even though the meetings started at 8:00am and went all day with only an hour for lunch and dinner, there would frequently be over one hundred people in the lecture hall at 9:00pm to hear Oleg Marichev or Michael Trott from Mathematica talk about their product, or Lance Littlejohn or Axel Riese talk about some new software that they had produced to simplify certain calculations. The wealth of computational tools now available is truly impressive. Many talks were given in many areas, and this brief note can only mention a few of them. On the other hand, I think it is important to try and see what the new tools or new areas where great progress is being made. I will mention three. First, it is quite clear from the talks of Dunkl, Xu, Littlejohn, Kill, Haine and others that finally a theory of multivariable polynomials is beginning to emerge. We may not agree on which of these polynomials to call classical, but we are beginning to see the clear lines of the theory. I look forward to the new book from Dunkl and Xu. Second, it was clear from the talks of Percy Deift and Walter Van Assche that the techniques developed to solve the Riemann-Hilbert problem are providing powerful new tools for the study of orthogonal polynomials. Finally, I have gone to many meeting and never heard mentioned the solutions of the Painleve equation. Such solutions were not on everyone's lips at this meeting, but they were mentioned in at least five different talks, and they were the subject of two hours of plenary talks. We will hear much more about "the Painleve Transcendents". Finally we must thank the organizing committee: Bustoz, Ismail, Koornwinder, Spiridonov, Suslov, and Vinet for a splendid program, and the gracious hosts from Arizona State University, Sergi Suslov and Joaquin Bustoz for a wonderful scientific adventure in a very hot corner of the world. Hot mathematics in a hot place!

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