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

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