Annales de l’institut Fourier (AIF)

Aratyn, Henrik

van de LEUR, Johan

The symplectic Kadomtsev-Petviashvili hierarchy and rational solutions of Painlevé VI

Pirio, Luc

Trépreau, Jean-Marie

Tissus plans exceptionnels et fonctions thêta

Gilligan, Bruce

An obstruction to homogeneous manifolds being Kähler

Vaccarino, Francesco

The ring of multisymmetric functions

Stokolos, Alexander

Zygmund's program: some partial solutions

Vanhaecke, Pol

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L’INSTITUT FOURIER
Pol VANHAECKE
Pierre van Moerbeke: the first 60 years
Tome 55, no 6 (2005), p. VII-XI.
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PIERRE VAN MOERBEKE: THE FIRST 60 YEARS
The university of Poitiers organized a conference on integrable systems
in June 2004 to honor Pierre at the occasion of his sixtiest birthday. Among
the participants, there was an overall consensus that the conference was a
great success, due to the high scientific content of the lectures, delivered
by the very best people in the field.
Secondly, the year-long help and support of my colleagues and co-
organizers Hervé Sabourin and Rupert Yu, of our directors (chairmen)
Abderrazak Bouaziz and Patrice Tauvel, and -last, but not least- of our
secretaries Brigitte Brault and Valérie Chauchet, also contributed to the
success of the conference and led to a climax during that week. Finally,
this event would not have taken place without the financial help from the
following sources:
• Conseil Régional Poitou-Charentes
• C.N.R.S.
• Université de Poitiers
• UFR-SFA, Université de Poitiers.
• Laboratoire de Mathématiques C.N.R.S. UMR 6086
• CAP Communauté d’Agglomération de Poitiers
• National Science Foundation
I hereby express my sincere thanks to all of you, members of the scientific
committee, speakers, participants, co-organizers and sponsors. I also ac-
knowledge the editors of Annales de l’Institut Fourier for having accepted
to edit and print a special volume of their journal for this conference, in
honor of Pierre van Moerbeke. Un grand merci also to Nathalie Catrain,
secrétaire de rédaction des Annales, for an efficient and pleasant collabo-
ration.
1. The early years
Pierre was born on October 1st, 1944 in Leuven (Louvain), an old
university town in the heart of Belgium, near Brussels. At that time the city
was liberated by British soldiers, who were patrolling the city for German
snipers and strongholds. Medieval Leuven had already suffered from the
successive bombings; then a new string of V2-attacks came in the early
spring 1945 and destroyed other parts of the city. Post-war Belgium was
politically agitated, but nevertheless the country’s reconstruction started
and the economy started flourishing and marked a new era for Belgium.
It is in this turbulent period that Pierre grew up, in a bilingual family, as
the youngest of a family of five. The most interesting stories of what Pierre
did during his infancy cannot be told here. What can be said will not come
to a surprise to those who know Pierre: already at a very young age he
had a strong curiosity in a wide range of scientific fields, he also loved the
outdoors (sailing, hiking, skiing,...) and inherited from his parents a taste
for fine arts, including music, theatre and visual arts. But it was his interest
in applied sciences that attracted him to become an engineering student
in his home town university, still a bilingual university in the 60’s. Like
many of us, who started out in a slightly different direction, he got along
the way seduced by the beauty of mathematics, and finally got a degree in
mathematics, in 1966. A research grant from the Belgian government gave
him the possibility to prepare a PhD, first in Leuven, and later, starting in
1969, he became a graduate student at Rockefeller University (New York).
During those years, the late 60’s, most European universities under-
went major transformations; the old “Université Catholique de Louvain”,
founded in 1425 by Pope Martin V, was no exception! Additional political
pressures led to a split of the bilingual university into a Flemish and a
French part; the former stayed in the Flemish city of Leuven, while the lat-
ter settled on rolling hills, mostly wheat and sugar beet fields, some twenty
kilometers from Leuven, and of course at the other side of the newly estab-
lished linguistic frontier. A new city, Louvain-la-Neuve, was born!
2. From Brownian into soliton motion
In New York, Mark Kac and Henry McKean had a very profound
influence on Pierre’s mathematical career; Henry McKean became his
advisor. Pierre’s thesis dealt with the rational pricing of stock warrants,
a problem initiated by Nobel laureate Paul Samuelson and put on a solid
mathematical basis by Henry McKean in a widely noticed appendix to
Samuelson’s paper. A warrant is a contract conferring the right to purchase
a common stock at a fixed unit price at any time prior to some expiration
date. Taking the logarithmic Brownian motion as model for the stock prices,
the optimal strategy is to buy the stock as soon as the current price exceeds
a certain time dependent curve; the question is ultimately transformed into
a free boundary problem for the heat equation and led to Pierre’s first
paper, which appeared in the Swedish Acta Mathematica.
A 1971 lecture by Ludwig Faddeev at Rockefeller University on the
transformation of the Schrödinger equation to scattering data arose Pierre’s
interest in the Korteweg-de Vries equation. The Feynman-Kac formula for
Brownian motion, which he learned from Mark Kac, in the context of the
stochastic optimization problems, ought to be a generating function for the
KdV invariants: that was Pierre’s first paper in that new field, and some
jointly with Mark Kac. Then together with Henry McKean, he linearized
the periodic KdV equation in terms of Hill’s equation, ∂2/∂x2+q(x) on the
circle, and hyperelliptic theta functions; geometrically speaking, KdV is a
linear flow on Jacobians of hyperelliptic curves, whose branch points are
the Floquet spectrum of Hill’s equation. This is now known as the finite gap
integration for the Korteweg-de Vries equation and was carried out almost
at the same time by two other groups, in Leningrad (Its and Matveev) and
Moscow (Dubrovin and Novikov).
Thus new objects as algebraic curves, Abelian varieties and theta
functions appeared in the context of the Schrödinger operator and the
Korteweg-de Vries equation. A very fruitful era of integrability emerged!
3. Classical mechanics and algebraic integrability
After having completed his thesis on optimization and in the middle
of solving problems connected to the Korteweg-de Vries and Toda lattice
equations, Pierre returned to Belgium, where he accepted in 1972 a
professorship at the Center for Operations Research and Econometrics
(CORE) of the University of Louvain. But restless as he was, his return
was short, but just long enough to steal the heart of Bernadette Van der
Schueren, whom he married in 1977, and to take her to Berkeley. Indeed,
in the years after his thesis he occupied several positions, ranging from
short term to permanent positions in centers, such as the Institute for
Advanced Study (Princeton), Stanford University, Paris XI (Orsay) and
the University of California (Berkeley).
During this period another interesting shift took place in Pierre’s re-
search. Since the Korteweg-de Vries equation has, on the one hand, all
formal properties of a finite-dimensional integrable system, in the tradition
of Euler, Jacobi, Hamilton, Liouville, Lagrange and Kowalevski, and, on
the other hand, allows algebraic geometric solutions, which are obtained
by the finite gap integration method, one naturally wonders whether this
method could be applied to a finite-dimensional context. In a series of
papers by himself and jointly with Mark Kac, he did this for the Toda lat-
tice and related lattices; the latter is a finite-dimensional discretization of
the Korteweg-de Vries equation. The deep connection between integrabil-
ity and algebraic geometry, appearing at the horizon, gets a further boost
when David Mumford and Pierre study, in a systematic way, the isospec-
tral deformations of band matrices, generalizing the standard Toda lattice
for tridiagonal matrices. They show how to linearize Lax equations with a
spectral parameter on the Jacobian of the spectral curve.
Andrew Lenart and Franco Magri’s discovery that the KdV equation
has two compatible symplectic structures and Pierre’s formulation of the
symplectic structure for Toda led Mark Adler to show a deep and striking
connection between Lie algebra splittings and integrability, a result which
is now cristallized under the name of the Adler-Kostant-Symes Theorem;
it applies to finite-dimensional and infinite-dimensional systems.
In the summer of 1978, Mark Adler and Pierre meet at Berkeley and
exchange ideas while hiking in the Sierra; this was the beginning of a
long-lasting scientific collaboration. Soon after their first encounter, they
give birth to a twin paper, one of the most cited papers in the subject,
where they introduce Kac-Moody Lie algebras into mechanics and explain
their integrability, in terms of algebraic geometry, through Lax equations
(with a spectral parameter). Almost all integrable examples from classical
mechanics fit into this mold. Their results get amplified soon afterwards,
when they reveal the geometric contents of Kowalevski’s method, and
introduce the notion of algebraic integrability; this sets the stage for a
rich interaction between algebraic integrability, geometry and Lie theory.
At the 1982 Warszaw International Congress of Mathematicians, postponed
at the last minute to 1983, Pierre gives a lecture on integrability in the Lie
algebra section. In 1988, he gets the Francqui prize, the highest scientific
distinction in Belgium, from King Beaudouin. When Brandeis University
came up with the idea of offering a part time position, besides his duties at
Louvain-la-Neuve, Pierre chooses for this linear interpolation between his
family life, teaching and research, with one foot in each continent.
4. Matrix integrals, the string equation,
combinatorics and integrable lattices
An MIT lecture by Edward Witten in the early 90’s on the connection
between the topology of moduli space of curves and the KdV equation
again stimulated Pierre’s interest. The precise conjecture was established
by Maxim Kontsevich and led to the famous Kontsevich integral, a matrix
integral with a matrix Airy integrand. This conjecture states that the com-
mon solution to the Korteweg-de Vries equation and the string equation
contains topological information on the moduli spaceMg,d of genus g Rie-
mann surfaces, with d marked points. The work of Kontsevich and Witten
and much earlier work by Bessis, Itzykson and Zuber was the starting point
for a new research direction on matrix models, orthogonal polynomials and
integrable equations. Soon later, around 1994, an Oberwolfach lecture on
random matrices by Craig Tracy again led to new horizons; Mark Adler
and Pierre wrote a series of papers -some with Takahiro Shiota and some
with Emil Horozov- on random matrices and the new underlying integrable
lattices. Their approach, based on integrable equations and Virasoro con-
straints yielded PDE’s for the distribution of the spectrum of the random
matrices for various ensembles; it also led to PDE’s for the transition prob-
abilities of stochastic processes arising in the context of Dyson’s Brownian
motion, a dynamic version of random matrix theory, when the matrices get
very large.
5. A personal note
I was introduced to Pierre in 1987 when I was a first year graduate
student in the Flemish part of the “Université Catholique de Louvain”. He
walked in my office, gave me a three hour lecture on algebraic integrability
and presented me with a list of interesting problems, some of which are still
unsolved. Starting from that very first day he became a friend and felt like a
collaborator, more than a supervisor. He engaged me in the most beautiful
mathematics, gently slowing down to my pace, which made him stroll,
while I still had to run. But mathematics is not the only thing I learned
from Pierre, quite on the contrary. Between two spoons of mathematics
he taught me human values, he shared with me his love and passion for
the outdoors and fine arts, he was there when I needed any kind of help,
throwing a flashlight to get me out of the dark. Happy 60th birthday Pierre
and thanks, in the name of all your students, collaborators and friends for
what you have given us.
October 1st, 2004
Pol Vanhaecke


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The symplectic Kadomtsev-Petviashvili hierarchy and rational solutions of Painlevé VI

Tissus plans exceptionnels et fonctions thêta

An obstruction to homogeneous manifolds being Kähler

The ring of multisymmetric functions

Zygmund's program: some partial solutions

Pierre van Moerbeke: the first 60 years

The Chern character for Lie-Rinehart algebras

Periodic billiard orbits in right triangles

Singularité et intégrabilité des fonctions plurisousharmoniques

Stabilité de l'inégalité de Faber-Krahn en courbure de Ricci positive

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