SIAM AG on Orthogonal Polynomials and Special Functions


Extract from OP-SF NET

Topic #7  ------------    OP-SF NET 7.4   -------------   July 15, 2000
From: OP-SF NET Editor 
Subject: 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 <>

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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