Topic #1 ----------- OP-SF NET 14.4 ----------- July 15, 2007 From: Tom Koornwinder, thk@science.uva.nl Subject: Report on OPSFA 2007, Marseille Report on the 9th Conference on Orthogonal Polynomials, Special Functions and Applications (OPSFA9), Luminy, Marseille, France, 2-6 July 2007 The ninth OPSFA was held during the first week of July 2007 in the nice Internat ional Center for Mathematical Meetings (CIRM) of the French Mathematical Society in Luminy, far out to the southeast from the centre of Marseille, at the boundary of the large park of the Calanques (geologic formations in the form of deep valleys with steep sides, typically of limestone, partly submerged in the Mediterranean). Where and when have the previous OPSFA meetings been held? Curiously enough, there is no permanent OPSFA website giving this information, and neither is there an official OPSFA board, which continues in office after an OPSFA meeting. Still, the attractiveness of the field and the closeness of informal contacts have been strong enough to maintain a tradition of OPSFA meetings for 23 years. Fortunately the OPSFA9 website lists all past OPSFA conferences, but there is immediate confusion, since 11 meetings have preceded Marseille. However, the Granada and Sevilla meetings fit into a special Spanish sequence of SPOA meetings (see OP-SF NET 3.5 #5 for a list), and the Stieltjes meeting in Delft had a much wider scope. So we have the following list. o OPSFA1 Bar-le-Duc (France, 1984, opened by Jean Dieudonné) o OPSFA2 Segovia (Spain, 1986) o OPSFA3 Erice (Italy, 1990) o VII SPOA Granada (Spain, 1991) o OPSFA4 Evian (France, 1992) o Stieltjes Delft (Netherlands, 1994, in honour of Thomas Jan Stieltjes, 1856- 1894) o VIII SPOA Sevilla (Spain, 1997) o OPSFA5 Patras (Greece, 1999, in honour of Theodore Chihara) o OPSFA6 Rome-Ostia (Italy, 2001) o OPSFA7 Copenhagen (Denmark, 2003, in honour of Richard Askey) o OPSFA8 München (Germany, 2005) o OPSFA9 Marseille-Luminy (France, 2007) Some obvious questions arise. How many of the participants in Marseille attended all twelve listed meetings, or all nine OPSFA meetings? Probably nobody. How many of the participants were also in Bar-le-Duc in 1984? I would guess at least ten. Of course, each of these conferences emphasized certain themes, closely related to the choice of invited speakers. It was only while writing this report that I found a web document describing these themes for OPSFA9 much more clearly than in the announcements distributed before the conference; see for instance OP-SF NET 13.6 #1. This web document mentions as the foremost theme "Orthogonal polynomials on the unit circle (OPUC) and spectral theory of Schrödinger operators", writing: "Barry Simon has published in 2005 a full treatise in two volumes on OPUC (AMS Colloquium Publications, 54), in the same series where Szegö published his pioneering book on OP in 1939. This is a kind of a bible on the subject, including the important recent results by Killip and Denissov. These OPUC are the best tool for studying discretizations of Schrödinger equation and have led to very unusual results on the spectrum: Denissov has shown that there exist Schrödinger operators, with square integrable potentials, which exhibit absolutely continuous and singular spectrum on the same spectral interval. It exhibits also the major theorems in the field (Szegö, Rahmanov, Geronimus, Baxter, ...) including modern proofs and tying OPUC theory with spectral theory. This leads to deep results for periodic or exponentially decaying Verblunsky coefficients. This book induces a strong revival in the field and will be well represented at the Conference, since Denissov, Killip and Simon will be lecturing." And indeed, this was really Barry Simon's conference. Not only did he give a plenary lecture on "Fuchsian groups and the spectral theory of finite gap Jacobi matrices or Peherstorfer-Sodin-Yuditskii meet Killip-Simon", but on earlier days he had already given two evening lectures on two "earthquakes": - Lubinsky earthquake: A revolution in universality and OP zeros (papers 199, 206 on http://www.math.gatech.edu/~lubinsky/ SelectedPapers.html ) - Remling earthquake: A revolution in AC spectrum (http://arxiv.org/abs/0706.1101 ) It was a good idea to have also some younger plenary lecturers such as Denissov and Killip, mentioned above. They both gave very good lectures, on continuous analogs of OP's on the unit circle, and on random OP's and random matrices, respectively. A further broad area of mutually interacting themes concerned the keywords Hermite-Padé approximants, rational approximation and interpolation, Riemann-Hilbert problems, varying weights, multiple orthogonal polynomials, connections with random matrix theory, and numerical aspects (plenary speakers Aptekarev, Beckermann, Kuijlaars, Magnus, Stahl and Van Assche). The French school on Fuchsian differential equations and differential Galois theory, and their q-analogues, was suitably represented by Ramis. Functional analytic aspects were covered by Lasser and Golinskii. Approximation theory in n-dimensional setting was presented by Plesniak. Grunbaum talked on matrix orthogonal polynomials, Martínez-Finkelshtein lectured on information measures of OP's, while more classical aspects of orthogonal polynomials could be heard in the lectures by Berg, Ismail and Marcellán. Anny Cuyt reported in her lecture on the "Handbook of continued fractions for special functions", which will come out soon, both in book form and online; see already http://www.cfhblive.ua.ac.be/ Mourad Ismail made a short announcement of the Askey-Bateman project, a multi-volume series of books which will be a successor to "Higher Transcendental Functions" (the Bateman Project). In an evening session he gave interested persons a further briefing on this project. This was followed by a meeting of the SIAM Activity Group on OPSF, where present chair Peter Clarkson heard many good suggestions, about which he will probably report in OP-SF NET. On Friday afternoon Mourad headed a problem session, and Dick Askey lectured on problems from special functions suitable for high school teachers, in particular easy forms of addition formulas. These two events were additional to the official conference program. Then there were the contributed lectures, always five in parallel, so that you were sure to miss some that you would like to have heard. But from those that I could attend, and from what I read in the abstracts, I can say that there was a lot of good stuff. Social activities included an aperitif before lunch on Tuesday, and an excursion to the nearby charming seaside village of Cassis, from which we made a boat trip to the calanques, and which was followed by a conference dinner in a very pleasantly located restaurant. The organizers Galliano Valent, Jacek Gilewicz and Roland Triay, supported by an international scientific committee, really did a great job in making this conference into an important scientific event. The excellent facilities of CIRM (meals, lodging, library, computers, wireless network, lecture rooms) were very helpful in making this week a success. Of course, nothing is perfect, so let me list a few things which may have annoyed some people and from which the organizers of the next OPSFA (maybe in Leuven, Belgium) can learn. There were about 150 participants, which is more than can be accommodated by CIRM. The less privileged participants had to stay in student dormitories on the nearby campus, and take their meals in student cafeterias. Another consequence of this large number was that the plenary lectures on the first day were held in a large lecture room on campus with inferior projectors. Fortunately, on the succeeding days we fitted without problems into the smaller, very nice CIRM auditorium. One draw-back of the CIRM auditorium was the difficult control of the beamer. Technical assistance was not always available. A somewhat confusing feature was the absence of a central registration area at the beginning of the conference. Instead, during the first days of the conference, one had to visit two different places in the building (a window and a desk) to pay or receive money. With the emphasis on some scientific themes, of course some others will receive less coverage. I missed several variables, and connections with root systems, groups and quantum groups. Some of the plenary computer presentations were really excellent, for instance those by Kuijlaars and by Martínez-Finkelshtein. But taking notes is more difficult with this medium. I have attended other conferences where the sheets of the lectures were put on the conference website. I want to recommend this also to OPSFA. The conference book is on the web, see http://www.cirm.univ-mrs.fr/liste_rencontre/programmes/ProgValent0727juin.pdf Finally I suggest that the reader visit http://www.morewords.com/word/opsfa/

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