## Extract from OP-SF NET

Topic #16 --------------- OP-SF NET ---------------- November 9, 1995

From: Tom H. Koornwinder thk@fwi.uva.nl
Subject: Impression from the Mittag-Leffler Institute, Sweden .

During almost two months I am staying here at the Mittag-Leffler Institute in Djursholm, Sweden. Djursholm is a suburb north of Stockholm, with luxurious villas on big lots, and beautifully located at the Stockholm archipelago. The institute is housed in the villa which was built in 1890 for the Swedish mathematician Gösta Mittag-Leffler (1846-1927) and which was drastically reconstructed and extended several times during the next fifteen years. Mittag-Leffler and his Finnish wife Signe, née af Lindfors, lived here for the rest of their life. The villa is towering on a hill and makes the impression of a small castle. [See line drawing - Web Editor.] From the beginning the library took a central place in the villa, and the whole architecture is a function of the necessity to house a library which was the largest private mathematics library in the world.

Mittag-Leffler is well-known by his work in function theory, but his greatest merit for mathematics is probably in his extensive international contacts with the top mathematicians of that time and in his founding (in 1882) and editing of the journal Acta Mathematica. From the beginning, first-rate contributions from leading mathematicians in France, Germany and other countries were obtained, and the journal is still considered as one of the highest ranking mathematics journals in the world.

In 1916 Mittag-Leffler and his wife set up a foundation to promote research in pure mathematics in the Scandinavian countries (see G. & S. Mittag-Leffler, Testament 16/3 1916, Acta Math. 40 (1916), III-X. The foundation was to maintain the large library in the villa and to support a research institute there with several professors, and with fellowships for younger mathematicians. In 1916 the plans for the Institute were realistic at least in that Mittag-Leffler's financial resources were adequate for the task. However, in 1922 there was a large financial crash related to the economic crisis in Europe at the time. The crash brought Mittag-Leffler near to bankruptcy and at his death in 1927 the resources did not allow the realisation of his original intentions.

The Royal Swedish Academy of Sciences, which had incorporated the Institute in 1919, appointed Torsten Carleman as Director of the Institute. Until 1969, the activities of the Institute were mainly restricted to maintaining the library and editing Acta Mathematica. From 1969 on outward funding was obtained by which the new director Lennart Carleson could finally realise Mittag-Leffler's intentions. From then on, a topic is chosen for every year. Within this field experts are invited to work at the Institute for periods of one or more months, and fellowships are made available for post-docs and graduate students. When one looks at the list of programs of the past twenty-five years, one sees topics from analysis dominating, while algebraic geometry has also been repeatedly a topic of concentration. The program for the present academic year is Analysis on Lie Groups.

The library is really marvellous, both as a piece of architecture and because of its wealth of older books and journals. It also tries to keep up with the present-day flood of books and journals, but it succeeds of course only partially. Applied mathematics is not represented very strongly. The library does not subscribe to any of the SIAM journals, not even to the SIAM Journal on Mathematical Analysis.

As for Acta Mathematica, it is very interesting to look at the "Table genérale des tomes 1-35" which was published in 1913. For this task the young Marcel Riesz, coming from Hungary, was hired by Mittag-Leffler. For all authors in alphabetical order one finds there a vitae and a list of publications which appeared in Acta. There is a picture gallery of the authors as well. Various well-known authors in orthogonal polynomials and special functions can be found there, for instance Thomas Stieltjes. In Volume 2 (1883) one already finds a paper by E. Goursat "Sur une classe de fonctions representées par des intégrales définies", and in Volume 3 (1883) a short note by H. Mellin "Eine Verallgemeinerung der Gleichung $\Gamma(1+x) \Gamma(1-x) = \pi x / \sin(\pi x)$". Also in the recently published issue of Acta Mathematica special functions do occur in E. Opdam's paper "Harmonic analysis for certain representations of graded Hecke algebra", Acta Math. 175 (1995), 75-121. Although the outsider would not guess this from the title, what really happens here is generalizing the Plancherel formula and the Paley-Wiener theorem for the Mehler-Fock transform to an integral kernel involving Jacobi functions associated with an arbitrary root system (the so-called Heckman-Opdam hypergeometric functions). More information about the Mittag-Leffler Institute can be found on WWW: Tom Koornwinder