Fall 2013 - Centre de recherches mathématiques

C
R
M
CENTRE
DE RECHERCHES
MATHÉMATIQUES
Le Bulletin
Automne/Fall 2013 — Volume 19, No 2 — Le Centre de recherches mathématiques
January–June 2014
吀ematic Semester on New Directions in Lie 吀eory
Vyjayanthi Chari (UC Riverside), Erhard Neher and Alistair Savage (University of O琀awa)
Lie theory is a central area of contemporary mathematics. 吀e structure and
representation theory of Lie groups and
Lie algebras have resulted in important applications in physics and other
branches of mathematics and, in turn,
Lie theory has benefited from these
connections. 吀e pioneering work of
Kac, Frenkel, Lepowsky and Meurman
linked the representation theory of
Kac–Moody algebras to the Monster
and led to the theory of vertex algebras. Another example is the theory
of quantized enveloping algebras initiated by Drinfeld and Jimbo, with roots
in the work of Fadeev, Reshetikhin
and Takhtajan on integrable systems.
吀e work of Lusztig and Kashiwara on
quantum groups led to major breakthroughs in the representation theory
of finite-dimensional simple Lie algebras and Kac–Moody algebras. 吀e theory of canonical bases (due to Lusztig)
and global crystal bases (due to Kashiwara) of finite-dimensional simple representations of simple Lie algebras resulted in dramatic developments and new areas of research
in geometric and combinatorial representation theory. Recent work
in the area of categorification has had a major impact on both representation theory and low dimensional topology.
Lie theory has a long and successful tradition in Canada, as do combinatorics and the representation theory of associative algebras. 吀e
topics of the program workshops have been chosen to reflect the new
connections between these subjects. 吀e overall goals of the thematic
semester are to highlight current research in Lie theory and its applications to other fields, to foster interaction between Canadian and
foreign researchers working in this area, and to provide a forum for
young mathematicians to learn about the current trends in the subject and to interact with the leading experts in this exciting field.
吀e program will begin with a Winter School taking place January 6–17. It will feature two courses aimed at graduate students
and postdoctoral fellows. 吀e course Introduction to Categorification
by Alistair Savage will serve to introduce
students to the new and exciting field of
categorification. Its goal is to prepare students for the workshop Geometric Representation 吀eory and Categorification. In
the first week, a recent approach to the
representation theory of the symmetric
group will be presented. 吀is will provide students with a novel approach to
the subject that is well suited to the ideas
of categorification. In the second week,
the basic ideas of categorification will be
presented, with a special emphasis on explicit examples (drawing, in particular,
from the material in the first week). 吀e
second course of the winter school, Introduction to Kac–Moody and Related Lie
Algebras by Erhard Neher, will complement the typical first course on Lie algebras offered at many universities. It
will present the basic structure theory of
Kac–Moody algebras with a special emphasis on affine Lie algebras. Some related non-Kac–Moody algebras, such as
toroidal algebras, will also be presented. 吀is will help prepare students for V. Chari’s course (see below).
A second Winter School will take place February 24–March 7. 吀e
course Representation 吀eory of Semisimple and Affine Kac–Moody
Algebras by Vyjayanthi Chari will focus in the first week on the category O for finite-dimensional simple Lie algebras. 吀e second week
will deal with the case of affine Lie algebras. 吀e focus will be on integrable weight representations. 吀e connections between positive
level representations of affine Lie algebras and the finite-dimensional
representations of the maximal parabolic subalgebra of the affine Lie
algebra will be made. Finally, the course will discuss how the methods and problems used and studied in category O can be formulated
and studied to understand the category of finite-dimensional representations of the affine Lie algebra. 吀e second course of the winter
school, Vertex Algebras for Mathematicians by Michael Lau (Laval),
will introduce vertex algebras and operator product expansion from
first principles, as well as touch on their applications to Lie algebras,
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representation theory and modular forms. A number of examples
will be featured, including vertex Lie algebras, la琀ice vertex algebras
and W-algebras. If time permits, there will be a discussion of some
of the geometry around vertex algebras in the context of conformal
blocks.
gether Lie theorists and mathematical physicists, in the expectation
that further dialogue will stimulate additional interesting and important breakthroughs in both domains. 吀e workshop will focus
on groups and algebras in quantum theory (in particular, infinitedimensional groups and algebras such as Virasoro, Kac–Moody, and
vertex operator (super)algebras), quantum field theory (in particu吀e winter schools will be followed by four week-long workshops.
lar, (super)conformal field theories, string and superstring theories
吀e first of these, on Combinatorial Representation 吀eory, will be
and topological quantum field theories) and statistical mechanics (in
held April 21–25. 吀e representation theory of quantized enveloping
particular, exactly solvable models).
algebras, Kac–Moody Lie algebras, extended affine Lie algebras and
Hecke algebras involves many ideas which have been developed for 吀e semester will conclude with a workshop on Categorification and
algebraic groups and Lie groups. 吀e full impact of the interplay be- Geometric Representation 吀eory, June 9–13. 吀e term “categorificatween algebraic groups, quantum groups and Kac–Moody Lie groups tion” was coined by Louis Crane and Igor Frenkel. Broadly speakis yet to be realized and one of the aims of the workshop is to bring ing, it is the process of realizing mathematical concepts as shadows
together specialists in these areas to explore this in more depth. 吀e of ones with more structure. It has become increasingly clear that
combinatorial aspects of the conference will revolve around the the- categorification is a mathematical phenomenon with broad applicaory of canonical bases and crystal bases introduced and studied by tions. As an example, understanding the categorical representation
Lusztig and Kashiwara. 吀e connections of the subject with the the- theory of affine Lie algebras led to a proof of Broué’s conjecture for
ory of solvable la琀ice models will also be a theme of the conference. symmetric groups, a purely representation theoretic statement. More
generally, the categorification of such mathematical objects as quanTwo weeks later, May 8–12, the thematic semester will feature a
tum groups and Hecke algebras has given us a new understanding of
workshop on Hall and Cluster Algebras. Cluster algebras are a certhe structure of these basic objects and their representation theory.
tain class of commutative rings, equipped with distinguished genMany ideas in categorification are related to geometric methods in
erators called clusters. 吀ey were defined by Fomin and Zelevinsky
representation theory. In fact, geometrization (the geometric realizaas part of an a琀empt to provide an algebraic framework for undertion of some algebraic structure) is o昀en a precursor to categorificastanding Lusztig’s dual canonical bases and total positivity. Neither
tion. For example, constructions of natural bases (such as Lusztig’s
the generators nor the relations among them are given from the start,
canonical bases in quantum groups or the Kazhdan-Lusztig bases in
but they are produced by an iterative process called mutation. 吀is
Hecke algebras) with positivity and integrality properties are a cenprocedure seems to encode a universal phenomenon, which might
tral part of geometric representation theory. 吀e categorifications
explain the explosive development of this topic. 吀e workshop will
that are suggested by such geometric constructions provide rich exexamine such topics as cluster algebras associated to triangulated
planations for the existence of these bases: in the categorification,
surfaces, systems of discrete functional equations called T-systems
basis vectors are reinterpreted as indecomposable objects in a cateand Y-systems (introduced and studied in the Bethe Ansatz method
gory, while structure constants become decomposition numbers, or
for integrable systems) and the representation theory of hereditary
multiplicities. From this point of view, positivity and integrality are
algebras and tilting modules.
manifest. 吀is field is moving forward rapidly and giving exciting
From May 19th to 23rd there will be a workshop on Lie 吀eory and results. 吀e workshop will serve as a venue to discuss this progress.
Mathematical Physics. 吀ere has always been a close and extremely
fruitful interaction between these two topics. 吀is relationship has Aisenstadt Chairholder: Masaki Kashiwara
clearly deepened and blossomed significantly with the arrival of
string theory. Lie theory in particular has been transformed forever Masaki Kashiwara received his Ph.D. from the University of Tokyo
and profoundly by string theory and related areas; conversely, string in 1974. He is currently a Professor Emeritus at the Research Institheory without Lie theory would be a race car without a motor. Ver- tute for Mathematical Sciences in Kyoto. He received the Prize of
tex operator algebras are now coming of age, in that big theorems the Mathematical Society of Japan in 1981 and was elected a foreign
like Verlinde’s formula have been established and big questions like member of the French Academy of Sciences in 2002 and a member
the rationality of orbifolds are beginning to crumble. 吀ere is a deep of the Japan Academy in 2007. In 1978, he was a plenary speaker
relation between topological quantum field theory and conformal at the ICM in Helsinki and, in 1990, was an invited speaker at the
field theory, and the relation of both to twisted equivariant K-theory ICM in Kyoto. He has made fundamental contributions in several
and to Lurie’s cobordism theorem is currently being developed by fields of mathematics, including algebraic and microlocal analysis,
Freed, Hopkins, Teleman, Lurie and others. 吀e role of conformal representation theory, Hodge theory, integrable systems, and quanfield theory in critical phenomena in statistical models is studied by tum groups. Most relevant for the thematic program is his developrecent Fields medalists Smirnov and Werner. In light of these recent ment of the theory of crystal bases which opened up vast areas of
developments, it is an exciting time for a conference to bring to- mathematics, many of which will be part of the program.
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吀ematic Year 2012–2013: Moduli Spaces, Extremality and Global Invariants
Aisenstadt Chairs
吀e Aisenstadt Chair allows us to welcome, in each of the thematic
programs two or three world-famous mathematicians for one-week to
one-semester stay. 吀e recipients of the chair give a series of lectures
on subjects chosen for their relevance and impact within the thematic
program. One of these lectures, in compliance with the donor André
Aisenstadt’s wish, must be accessible to a broad audience.
these developments, and emphasized that the time has come for a
renewed exploration of questions originating in celestial mechanics
and space transportation by using all the sophisticated tools at our
disposal today.
David Gabai
吀e lectures by three of the Aisenstadt chairholders of this thematic year by Steven Boyer (UQAM)
which took place in spring 2013: Helmut Hofer (Institute for Advanced
Study), David Gabai (Princeton University) and Gang Tian (Princeton Professor David Gabai of Princeton University gave a series of four
lectures during the events surrounding the workshop 吀e Topology
University and Peking University) are described below.
of 3-dimensional manifolds. Professor Gabai is a world leader in the
topology and geometry of low-dimensional manifolds, with many
Helmut Hofer
outstanding research contributions throughout a career in which
by Octav Cornea (Université de Montréal)
he has solved major problems and developed powerful techniques
Helmut Hofer obtained his which have had a profound impact in the field. 吀ese include proofs
Ph.D. at the University of of the Seifert fiber space conjecture (1992), the rigidity of homotopy
Zürich in 1981 and is currently hyperbolic 3-manifolds (1994), the Smale Conjecture for hyperbolic
a permanent member of the In- 3-manifolds (2001), and Marden’s Tameness Conjecture (2006). He
stitute for Advanced Study in was awarded the Oswald Veblen Prize in Geometry by the AmeriPrinceton. Among other presti- can Mathematical Society in 2004, and was made a member of the
gious distinctions, he is a mem- American National Academy of Sciences in 2012.
ber of the National Academy
of Sciences and was twice an Professor Gabai’s first two lecHelmut Hofer
invited speaker at an Interna- tures were expository in nature.
tional Congress of Mathematicians, the last time plenary in 1998. 吀e first provided an overview
Hofer is one of the founders of modern symplectic topology. His of Poincaré’s many contribuwork, alone and with collaborators, on Floer theory, on capacities tions to topology, remarkable
and applications to Hamiltonian dynamics and on cases of the We- for their foundational and seminstein conjecture in dimension three was transformational. It led to inal nature as well as their imthe establishment and the further study by countless researchers of pact on subsequent research.
David Gabai
(what is now called) the Hofer geometry of the Hamiltonian diffeo- 吀e second surveyed the still
morphisms groups. More recently, together with Wysocki and Zehn- mysterious field of the volumes of hyperbolic 3-manifolds from
der he is developing polyfold theory, a wide-reaching extension of W. 吀urston’s 1970s proof that the set of such positive real numusual differential geometry particularly adapted to the study of reg- bers is closed and well ordered, to Professor Gabai’s 2009 theorem
ularity properties of moduli spaces of solutions of PDEs. 吀e topic with Robert Meyerhoff and Nathaniel 吀urston that the Weeks manof Hofer’s talks was Hamiltonian Dynamics and Symplectic Rigid- ifold is the unique manifold to realize the minimal possible volume
ity. Hamiltonian systems, which occur frequently in physics, are among closed orientable hyperbolic 3-manifolds. He described the
also of great interest in many branches of mathematics. Symplectic three decades of effort by many mathematicians, using a wide vageometry allows us to formulate certain dynamical questions into riety of techniques, which bridged these results, including the use
geometric and, sometimes, even algebraic problems. 吀is is based of Ricci flow by Ian Agol and Nathan Dunfield to obtain strong
on the properties of the moduli spaces of (perturbed) J-holomorphic volume estimates. 吀e talk concluded with a discussion of open
curves that naturally arise in relation to a given Hamiltonian sys- problems and an approach towards addressing the 吀urston, Weeks,
tem, once the appropriate action functional is defined. Assuming Matveev–Fomenko conjecture that complete low volume hyperbolic
that these moduli spaces are well-behaved, they are amenable to 3-manifolds are of low topological complexity.
combinatorial assembly and the output is presented as certain ho- Professor Gabai’s last two Aisenstadt Chair lectures focused on the
mological type invariants such as Gromov–Wi琀en invariants, Floer topology of the ending lamination space associated to a finite type
homology and others. Understanding of these invariants sheds light hyperbolic surface S. He set the stage by describing its connections
on the dynamical properties of the initial Hamiltonian system. In the with the space of projective measured laminations of S, the Grolast twenty years the symplectic machinery that follows this general mov boundary of the curve complex of S, and the set of doubly
scheme has developed tremendously. Hofer’s talks reviewed part of degenerate hyperbolic structures on S × . Most of his time was
R
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devoted, though, to explaining his recent results in the area. For inInvariant Subspaces of the Shi昀 Operator
stance, if S has genus g and p punctures, then its ending lamination
space is both (n−1)-connected and (n−1)-locally-connected where
26–30 August 2013
2n + 1 = 6g + 2p − 7. Further, when g is zero, its ending lamination space is the Nöbeling space of points in 2m+1 with at most m
Organizers: Emmanuel Fricain (Lille 1), Javad Mashreghi (Laval),
rational coordinates.
William Ross (Richmond)
R
Gang Tian
by Vestislav Apostolov (UQAM)
Professor Gang Tian, who is Eugene Higgins Professor at Princeton and the director of the Beijing International Centre for
Mathematical Research (BICMR), gave a
series of three Aisenstadt lectures during the workshop Extremal Kähler metrics. In global analysis and geometry over
the last 25 years, Professor Gang Tian has
been one of the most influential and versatile figures. From his Ph.D. thesis to his
most recent work, his contributions have
been distinguished by their diversity and
Gang Tian
depth. His most striking work to date concerns the existence of Kähler–Einstein metrics on complex manifolds (which was a main theme of the workshop), but he has made
many other significant contributions. For instance, he established
(with Y. Ruan) the associativity of the quantum cohomology ring
of a symplectic manifold, constructed (with Jun Li) moduli spaces
of curves in algebraic symplectic geometry, and elaborated (with
J. Morgan) a clear and detailed exposition of Perelman’s proof of the
Poincaré conjecture, and developed ideas to link Kähler–Ricci flow
with the Mori program in algebraic geometry. Most relevant to this
CRM workshop, however, is Tian’s work on Kähler–Einstein metrics.
In his early career, he solved the existence question for Kähler–Einstein metrics on compact complex surfaces with positive first Chern
class, and showed that Fano varieties with a Kähler–Einstein metric
must be stable in the sense of geometric invariant theory, thus confirming part of a suggested correspondence by Yau. Professor Tian
gave a precise formulation of this correspondence, known today as
the Yau–Tian–Donaldson conjecture, identifying the obstruction to
the existence of Kähler–Einstein metrics with what is now called the
K-stability of the Fano variety. His ideas have inspired a tremendous
amount of work in recent years, and spectacular progress has been
made, culminating in the complete resolution of the conjecture by
Tian himself, and by Xiuxiong Chen, Simon Donaldson and Song
Sun.
Professor Tian’s first lecture was expository in nature, and reviewed
the theory and latest progress related to the existence of Kähler–Einstein metrics. 吀e second lecture was concerned with a more detailed
exposition of the key ideas of Professor’s Tian most recent work regarding the existence of Kähler–Einstein metrics on a K-stable Fano
variety. 吀e third lecture featured recent results on convergence of
the Kähler–Ricci flow on a Fano variety.
吀e main theme of this workshop was the invariant subspaces
of the shi昀 operator S, or its adjoint, on certain function spaces,
e.g., Hardy spaces H p or the Dirichlet space D. In particular,
de Branges–Rovnyak spaces H(b), where b is an element of the
closed unit ball of H ∞ , and model spaces KΘ , where Θ is an inner
function on the open unit disk, were at the center of a琀ention. 吀e
first two days of the conference were devoted to four mini-courses,
each three hours long, on the above mentioned spaces. Besides
the majority of invited speakers, more than 25 graduate students
and postdoctoral fellows participated in these lectures by worldrenowned experts (K. Dyakonov, S. Garcia, D. Timotin, T. Ransford).
吀e last three days were filled with 21 special talks by invited speakers. With more than 55 participants, this part was a great success.
Many new features of function spaces and operators were discussed
during and also a昀er the sessions. One of the touching parts of the
conference was the beautiful presentation by Carl C. Cowen on the
Invariant Subspaces Problem (ISP). Analysts believe that the ISP is
one of the great unsolved problems, at least in operator theory. In
January 2013, Carl C. Cowen and his collaborator Eva Gallarda announced that they had solved the ISP. Unfortunately, a short while
later, a gap was found in their proof. However, their efforts were not
totally in vain and they have succeeded in opening many new frontiers and shedding light on fruitful topics in operator theory. We had
the privilege of having Carl Cowen among us to generously share
his ideas with the audience.
Apart from the ISP, many other questions and topics on the frontiers
of analysis and operator theory were highlighted in the last three
days. In particular, we mention the following topics: the multivariate case of Hilbert modules by R. Douglas; review and continuation
of the results of Hunt–Fefferman–Cwikel–Alexandrov on weak L1
and Weak H 1 spaces by J. Cima and A. Nicolau; truncated Toeplitz
operators and complex symmetric operators by S. Garcia and D. Timotin; treatment of results of Paley–Wiener, Kadet–Ingham, Seip and
Borichev–Lyubarskii on sampling and interpolation, and presenting
the new approaches in Fock spaces by A. Hartmann; numerous open
questions on classical function spaces by D. Khavinson, which have
been the focus of research by him and his students for more than 30
years; the new achievements on compositions operators on function
spaces by Yu. Lyubarskii, H. 儀effélec and L. Khoi; the great presentation of Crofoot–Sarason polynomials by N. Makarov (which immediately constituted the basis of a thesis at Université Laval); the study
of invariant subspaces on the Bergman space by S. Richter; compact
operators on the Bergman space by B. Wick; and cyclicity and the
Brown¬Shields conjecture in Dirichlet spaces by C. Beneteau and
T. Ransford.
吀e proceedings of the workshop are expected to appear in the Contemporary Mathematics series, published by the AMS.
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2013 André-Aisenstadt Prize Recipient
Spyros Alexakis (University of Toronto)
Dr. Alexakis obtained a B.A. degree from the University of Athens in
1999 and a Ph.D. from Princeton University, under the supervision of
Charles Fefferman, in 2005. He held a Clay Research Fellowship as
well as a Sloan Fellowship, and has been at the University of Toronto
since 2008. Working in the areas of analysis and mathematical physics,
alone and with collaborators he has obtained striking results in at least
three different directions. His main contribution, published as a research monograph in the prestigious Annals of Mathematics Studies of
Princeton University, is a solution to a conjecture of Deser and Schwimmer regarding the structure of global conformal invariants. Secondly,
together with Klainerman and Ionescu, he made important progress
in the understanding of the Kerr solutions to Einstein’s equations. Finally, jointly with Mazzeo, he obtained deep results concerning minimal surfaces with bounded Wilmore energy. 吀is impressive research is
described below, in more detail, by Dr. Alexakis himself.
• Pfaff(Rijkl ) is the Pfaffian of the curvature
R tensor Rijkl of g,
since by the Chern–Gauss–Bonnet theorem M Pfaff(Rijkl ) dVg =
Cn χ(M ), for some dimensional constant Cn . Here χ(M ) is the
Euler number of M .
In a long series of works [1–5] I confirmed the stipulation of Deser
and Schwimmer [13]:
吀eorem 1. Any Riemannian invariant P (g) of weight −n whose integral over closed manifolds satisfies (1) can be expressed as a sum of
the form (2).
吀e ideas in the proof relied on the deep progress made in the last
three decades on local conformal invariants (in particular the ambient metric construction of Fefferman an Graham, [14]), together
with a new, explicit, and useful formula for the first variation of P (g)
under conformal changes, derived in [2]. 吀e argument then proceeded by an (essentially algorithmic) construction of the decomposition. One inductively constructs local conformal invariants and diIntegral Conformal Invariants
vergences of vector fields which, upon subtraction from P (g), yield
A classical theorem (essentially due to H. Weyl, [23]) in Riemannian a new, simpler integral conformal invariant. At the very last step of
geometry asserts that local intrinsic scalars (scalar-valued polyno- this procedure one shows that the remaining term is a multiple of
mial expressions in the metric, its inverse and its derivatives, which the Pfaffian.
are invariant under changes of the underlying coordinate system,
While the method of proof is tailored to the case of integral conand have a given weight) can be expressed as linear combinations of
formal invariants, it is possible that some of the ideas could be aptensor products and complete contractions of the curvature tensor
plicable to the understanding of other integral invariants which apand its covariant derivatives. A further natural question is to underpear naturally in other geometries. One such possibility is the understand the space of invariant scalars for other geometries. 吀is quesstanding of the local structure of the terms in the Tian–Yau–Zelditch
tion arose naturally in understanding the expansions of heat kernels
expansion [24]. Another is the recent discovery by Hirachi of invariin Riemannian geometry and of the Bergman and Szegő kernels in
ance properties of a term in the expansion of the Szegő kernel [18].
CR geometry. In particular the space of local conformal invariants
has received much interest [14].
A generalization of these local questions which first appeared in the
physics literature [13] is to understand all global conformal invariants. 吀e challenge here is to understand
the space of Riemannian
R
scalars P (g) for which the integral M P (g) dVg over any closed
Riemannian manifold remains invariant under conformal changes
g → e2φ g, φ ∈ C ∞ (M ). In other words, we require:
Z
Z
2φ
P (e g) dVe2φ g =
P (g) dVg .
(1)
M
M
吀e obvious candidates P (g) which have this property are sums of
the form
P (g) = W (g) + divi X i (g) + C · Pfaff(Rijkl ),
where
(2)
The black hole uniqueness question
A natural question that has been studied for nearly four decades
concerns the possible vacuum, stationary black hole solutions to
Einstein’s equations in general relativity. 吀e interest in this question was spurred by two factors: firstly, the discovery of the Kerr
2-parameter family of such solutions [21](parametrized by total mass
and angular momentum), which generalized the Schwarzschild solutions (corresponding to zero angular momentum). Secondly, the expectation (formulated in [17]) that precisely such solutions would
be the possible final states of smoothly evolving black holes. 吀e
heuristic argument in [17] asserts that generic dynamical black holes
should radiate energy towards infinity and into the black hole region
before approachng a final state; therefore (we are told) the final state
would have to be nonradiating and stationary. Hence the interest in
the possible stationary black holes is due to their asserted relevance
as potential final states of dynamical space-times.
• W (g) is a local conformal invariant of weight −n,Ri.e., W (e2φ g) =
e−nφ W (g). In that case the global invariance of M W (g) dVg is
obvious in view of the transformation of the volume form dVe2φ g =
enφ dVg .
儀estion 1. Are the (subextremal) Kerr solutions the only possible
• divi X i (g) is a divergence
of
a
(Riemannian)
vector
field,
since
stationary, vacuum single-black hole solutions to Einstein’s equaR
Stokes’ theorem implies M divi X i (g) dVg = 0.
tions, under suitable regularity assumptions?
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吀e above has been answered in the affirmative in a series of works
over the past decades, always under certain additional assumptions
on the space-time. 吀e case of static, rather than just stationary
space-time was se琀led by Israel in [20]. 儀estion 1 was also answered in the affirmative for space-times which in addition to being
stationary are also axi-symmetric, in that they admit an additional
Killing field whose orbits are closed with a fixed period; this is the
Carter–Robinson theorem, [11, 12, 22] and references therein.
Y . A starting point of my work with Mazzeo [8] was that this renormalized area turned out to be essentially equivalent to the total curvature or Willmore energy of the surface:
Z
1
Ren.Area[Y ] = −2πχ(Y ) −
|Â|2 dVY .
(3)
2 Y
吀is allowed us to study the first and second variations and the critical points of this functional.
吀e main question addressed in [9] is to find the correct analogue
of bubbling in the space of such minimal (and more generally Willmore) surfaces with unprescribed boundaries in ∂∞ 3 , and with an
upper bound on the total energy. In particular, in many variational
geometric PDE (i.e. solutions of the Euler–Lagrange equations corresponding to a geometric energy functional), one is interested in understanding the behaviour of sequences of solutions whose energy
In joint work with A. Ionescu and S. Klainerman [6, 7], we succeeded is bounded above. A typical result (for harmonic maps from a surin relaxing this assumption substantially by merely assuming close- face (Σ2 , h) → (M n , g)) is that the sequences converges (away from
a set of “bad points”) to a new solution. 吀e failure of smooth conness to the Kerr family of solutions in a C 2 -sense:
vergence at the bad points is due to bubbling of energy phenomena,
吀eorem 2. Assume that (M, g) is a stationary, vacuum, single-black
where a nonzero amount of energy concentrates at the bad points.
hole space-time with suitable regularity assumptions. Assume further
that (M, g) is close to one of the Kerr solutions, as measured by the 吀e question that [9] sought to address was whether similar bubbling
smallness of the Mars-Simon tensor. 吀en (M, g) is isometric to a Kerr phenomena can be expected towards the boundary at infinity, in the
se琀ing of minimal (and Willmore) surfaces in 3 . We showed that
black hole exterior.
this phenomenon does persist, under the assumption of a slightly
More recently yet, we have relaxed the additional assumption to the
weighted version of the energy. A key difference is that arbitrarily
stationary Killing field being suitably small on the event horizon.
small amounts of energy can now bubble off towards infinity.
吀is can be thought of as implying that the angular momentum on
吀e natural question that arises is whether this result is optimal, and
the horizons is small.
the extent to which it is a general feature of variational problems on
吀eorem 3. Let (M, g) be a stationary, vacuum, single black-hole
manifolds with boundary, with no apriori assumptions on the reguspace-time with suitable regularity assumptions. Assume further that
larity of the solution at the boundary.
the Killing field T is suitably small on the (future and past) event horizons, N , N . 吀en (M, g) is isometric to one of the Kerr black hole ex- [1] S. Alexakis, On the decomposition of global conformal invariants. I, Ann. of
Math. (2) 170 (2009), no. 3, 1241–1306.
teriors.
吀is extra condition can be relaxed, as observed in [15, 17], who
showed that stationarity along with a bifurcate horizon implies that
the space-times must admit a jet of a second, rotational Killing field,
tangent to the (past and future) event horizons. 吀us, as explained in
detail in [12], if one restricts a琀ention to real-analytic space-times,
儀estion 1 can be answered in the affirmative.
H
H
[2]
吀e ideas of the above rely on unique continuation techniques for
wave equations, where a notion of pseudo-convexity, namely convexity with respect merely to null geodesics, is central. 吀e technique is to construct a foliation of the black-hole exterior by level
sets of a regular function, whose leaves are T-conditionally pseudoconvex, i.e. convex with respect to T-normal null geodesics. (吀at
this weaker notion of pseudo-convexity suffices for the problem at
hand was already noticed in [19]). 吀e key obstacle is the presence of
an ergoregion, where the Killing field T is space-like. 吀e challenge
of answering 儀estion 1 in suitable generality remains.
Minimal surfaces in
H
3
and boundary regularity
[3]
[4]
[5]
[6]
[7]
[8]
吀e work on this topic concerns minimal surfaces in hyperbolic
3-space 3 , with a boundary at infinity. 吀e study of these was ini- [9]
tiated by M. Anderson in the 1980s [10], who solved the analogue of
the Plateau problem for such surfaces, with a boundary at infinity. [10]
H
An interesting notion regarding these surfaces was that of the renor- [11]
malized area, introduced by Graham and Wi琀en in [16]: although
the area of any such minimal surface is necessarily infinite, one can [12]
nonetheless perform a Hadamard regularization and obtain a welldefined notion of renormalized area Ren.Area[Y ] of such a surface
BULLETIN CRM–6
, On the decomposition of global conformal invariants. II, Adv. Math.
206 (2006), no. 2, 466–502.
, 吀e decomposition of global conformal invariants, Ann. of Math. Stud.,
vol. 182, Princeton Univ. Press, Princeton, NJ, 2012.
, 吀e decomposition of global conformal invariants: some technical
proofs. I, SIGMA Symmetry Integrability Geom. Methods Appl. 7 (2011), Paper
019.
, 吀e decomposition of global conformal invariants: some technical
proofs. II, Pacific J. Math. 260 (2012), no. 1, 1–88.
S. Alexakis, A. D. Ionescu, and S. Klainerman, Hawking’s local rigidity theorem
without analyticity, Geom. Funct. Anal. 20 (2010), no. 4, 845–869.
S. Alexakis, A. D. Ionescu, and S. Klainerman, Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces, Comm. Math.
Phys. 299 (2010), no. 1, 89–127.
S. Alexakis and R. Mazzeo, Renormalized area and properly embedded minimal surfaces in hyperbolic 3-manifolds, Comm. Math. Phys. 297 (2010), no. 3,
621–651.
, Complete Willmore surfaces in H 3 with bounded energy: boundary
regularity and bubbling, available at arXiv:1204.4955.
M. T. Anderson, Complete minimal varieties in hyperbolic space, Invent. Math.
69 (1982), no. 3, 477–494.
B. Carter, An axy-symmetric black hole has only two degrees of freedom, Phys.
Rev. Le琀. 26 (1971), 331–333.
P. T. Chruściel and J. L. Costa, On uniqueness of stationary vacuum black holes,
Astérisque 321 (2008), 195–265.
(continued on page 12)
crm.math.ca
Organizers: F. Lutscher (O琀awa), J. Bélair (Montréal), M. Lewis (Alberta), J. Wu (York) and J. Watmough (New Brunswick)
Aisenstadt Chair: Simon A. Levin
In his first lecture, Prof. Levin explored some specific examples of
collective phenomena, from universality in bacterial pa琀ern formation to collective motion and collective decision-making in animal
by Frithjof Lutscher (O琀awa) and Frédéric Guichard (McGill)
groups. 吀ese examples showcased the contribution of mathematics
Simon A Levin is the George to biology, and the importance of collective phenomena for resolving
M. Moffe琀 Professor of Biol- fundamental and applied biological problems.
ogy at Princeton University in Many ecosystems that provide important services to humans are also
the Department of Ecology and at risk, and one challenge is to predict abrupt shi昀s in ecosystem
Evolutionary Biology. He held state in response to gradual environmental change. In savannahs,
an Aisenstadt Chair in July 2013 arid vegetation retains soil and water and displays patchy pa琀erns
and was an invited speaker at that can be used to understand dramatic changes in vegetation cover
the workshop Biodiversity in in response to precipitation. Staver and Levin (2012) showed how
a Changing World. During his feedbacks between tree growth and fire regimes can lead to bistable
tenure of the chair, Prof. Levin equilibria and even heteroclinic cycles where large shi昀s in vegeSimon A. Levin
gave three closely related lectures, during which he took his audi- tation cover (between equilibria) can be triggered by rain falls. 吀e
ence on a fascinating whirlwind tour of some of the major challenges potential for large ecosystems to spontaneously undergo cycles of
that humanity faces, their ecological and evolutionary perspectives shi昀s between multiple states is of particular importance for their
and the potential that mathematics has to offer to their solution. 吀e management.
titles of the individual lectures were
1. Collective phenomena, collective motion, and collective action in Collective behaviour in groups of individuals is another well documented emerging property in ecological and social systems. Such
ecological systems
2. Evolutionary perspectives on discounting, public goods and col- collective behaviour emerging from individual decisions can explain
cooperation and adaptive response to predation in social organisms.
lective behaviour
Nabet et al. (2009) represented individual decision within groups as
3. 吀e challenge of sustainability and the promise of mathematics
a set of coupled oscillators similar to the classic Kuramoto equation.
Here are a brief summary of the first two lectures and a more elabo- 吀ey showed how diverging opinions can lead to the emergence of
rate overview of the third lecture. Reference to some of the relevant stable groups of individuals that ‘align’ their behaviour. 吀is result
publications by Prof. Levin allow the readers to delve into the subject sheds light on how groups of organisms resolve conflict by forming
areas as deeply as they desire.
complex leadership structures.
Collective phenomena
吀ese examples suggest that studies of emergence, scaling and critical transitions in physical systems can inform the analysis of similar
phenomena in ecological systems, while raising new challenges for
theory. 吀is first lecture made clear that the growing recognition that
ecosystems’ properties emerge from the collective behaviour of individuals is associated with a major shi昀 in management strategies:
local interactions and feedbacks give rise to macroscopic properties
that undergo strong fluctuations and sudden shi昀s. We are part of,
and have to manage, very dynamic ecosystems.
Predicting the dynamics and pa琀erns found in ecological systems
o昀en fails when we consider populations, communities and ecosystems as fundamental units to understanding persistence, species diversity, and ecosystem productivity. Instead, important questions in
basic and applied ecology alike involve complex adaptive systems,
in which localized interactions among individual agents give rise
to emergent pa琀erns that feed back to affect individual behaviour.
In such systems where ‘more is different,’ a central challenge is to
scale from the ‘microscopic’ to the ‘macroscopic’ in order to under- Public goods
stand the emergence of collective phenomena, the potential for critical transitions, and the ecological and evolutionary conflicts between Ecological and economic systems are alike in that individual agents
compete for limited resources, evolve their behaviours in response
levels of organization.
BULLETIN CRM–7
crm.math.ca
to interactions with others, and form exploitative as well as cooperative interactions as a result. In these complex adaptive systems,
macroscopic properties like the flow pa琀erns of resources (such as
nutrients and capital) emerge from large numbers of microscopic interactions, and feed back to affect individual behaviours. Contagion
can lead to critical transitions from one basin of a琀raction to another,
as for example with eutrophication, desertification, pest outbreaks,
and market collapses. In both sorts of systems, evolution of one type
or another leads to the differentiation of roles and the emergence of
system organization, but with no guarantee of robustness. It is crucial to understand how evolutionary forces have shaped individual
behaviours in the face of uncertainty.
serve every species of tree, but we need to preserve trees as a source
of building material, wildlife habitat, oxygen generation and carbon
storage, among others.
tainable future as one of intergenerational equity: can we enjoy
economic growth without negative consequences for future generations? Sustainability, of course, applies to many different areas, for
example, the financial sector, energy and natural resources, biological and cultural diversity, or ecosystems services. 吀e la琀er is o昀en
particularly difficult to value, so mathematics may provide particularly useful insights.
吀e potential that mathematics has to offer to those challenges is in
developing a statistical mechanics framework of ecological communities and socio-economic systems; describing and detecting emergent pa琀erns; finding indicators of critical transitions; and supporting governance of multiscale systems.
One of the great challenges when considering ecosystem function is
the question of scales. How do we relate phenomena across different spatial and temporal scales? And how robust are small-scale
results on larger scales? One approach is to begin with a detailed
model (such as a forest simulator model) and then scale up and
compare to global vegetation models, where world-wide pa琀erns
are tracked rather than individual species. Another example is the
DARWIN model (http://darwinproject.mit.edu/) that tracks
phytoplankton, zooplankton and nutrient densities in oceans on a
Prof. Levin began his second lecture by defining public goods in global scale (Follows et al., 2007).
a very broad sense, from fisheries, aquifers and air quality to the 吀e potential that mathematics has to offer to those challenges is
effectiveness of vaccines, antibiotics and information. 吀e first two rigorous frameworks and methods for scale transitions: the theory
case studies considered how humans cooperate to make insurance of coarse-graining or aggregating; the transition from Lagrangian
arrangements against environmental uncertainty. Evolving water- to Eulerian models; theories of moment closure; and equation-free
use strategies, for example, lead to prudent behaviour only when in- methods.
teractions are sufficiently local; global mixing typically leads to less
conservation (Zea-Cabrera et al., 2006). Sharing of grazing grounds Many pa琀erns in ecological communities arise exogenously, but
of ca琀le farmers in Africa can work as such an insurance against some are driven by endogenous mechanisms. In the la琀er case, two
drought, but whether it evolves depends on the discounting rate locally stable steady states can occur, so that prediction and control
(Dixit et al., 2012). Mathematically, these questions are related to of the desired state become more difficult. For example, forests and
game theory and constraint optimization. 吀e question emerges as savanna in Africa represent two locally stable states. Human activito how one can design mechanisms for self-reinforcing rules for co- ties and natural causes (wildfire) can trigger the transition between
them (Staver et al., 2011). A similar bistable situation occurs in shaloperation.
low lakes where algal levels can be either low or extremely high.
Prof. Levin spoke about the tragedy of the commons and why we fail Climate change has the potential to trigger the transition of lakes
to preserve public goods in the context of biodiversity, and specifi- from low to high algae concentration and thereby suffocate many of
cally the emergence of resistance against antibiotics. A game theo- the lake’s other living organisms. Such state transitions are typical
retic approach to resistance in hospitals showed how prevention of for complex adaptive systems and how they react to external forcing.
resistance can only emerge when patients are highly likely to be ad- Together with two colleagues, Simon Levin pioneers the dialogue bemi琀ed to the same hospital when sick a second time (Laxminarayan tween science and economics to enhance economists’ understanding
et al., 2005). In many more examples, Prof. Levin explored the com- of local interactions and global feedback within the complex systems
mon features of these systems, especially as they involve the evolu- that support life on earth (May et al., 2008).
tion of intragenerational and intergenerational resource allocation
and the evolution of cooperation in dealing with public goods, com- Mathematicians and ecologists have studied in detail how such phase
transitions occur and whether they can be detected by early warning
mon pool resources and collective movement.
signals. M. Scheffer et al. (2012) identified a critical slowing down, an
increasing variance, and a flickering, as such signals. Recent theory
Sustainability
by Boe琀inger and coauthors expands these indicators to the global
In the third lecture, Prof. Levin presented the challenge of a sus- scale and the entire biosphere.
When evaluating ecosystems services, one places the focus not on
preserving every single species and aspect but rather certain general functions, particularly those that are of great importance to humans. For example, we might not be able to preserve every species
of fish, but we should aim to preserve the overall contribution of fish
as a protein source to our diet. Similarly, we may not be able to preBULLETIN CRM–8
吀e mathematics of governance includes optimal control, voting theory, collective motion, games and negotiation. All too o昀en, there
is clear scientific consensus, but action is lacking. In part, this lack
of action stems from missing commitment to the public good. Many
aspects of how to set and enforce commitment to the public good
were discussed in the second lecture in detail. 吀e overall question
(continued on page 14)
crm.math.ca
La Terre gronde… Les mathématiciens écoutent
Christiane Rousseau (Université de Montréal)
C’est sous ce titre qu’Ingrid Daubechies a donné la quatrième conférence Mathématiques de la planète Terre de la série Simons le 10 avril
à Montréal dans la salle du Cœur des sciences. Pas moins de 400 personnes s’étaient déplacées pour l’événement. Elle a livré sa splendide conférence en français et a enregistré elle-même le doublage
en anglais devant une salle vide. Des vidéos français et anglais de la
conférence peuvent être écoutés à http://www.videocrm.ca/.
Trop souvent, on limite Mathématiques de la planète Terre 2013 aux
changements climatiques et au développement durable. La conférence d’Ingrid Daubechies cadrait au contraire parfaitement sous le
premier thème « Une planète à découvrir ». Elle relatait la coopération de la conférencière avec des géophysiciens, et leurs résultats
très récents sur la compréhension du processus de formation des îles
volcaniques isolées. Sur le fond des océans, les roches les plus récentes se retrouvent le long des dorsales où les plaques tectoniques
divergent. L’activité volcanique le long de ces dorsales fait remonter du magma depuis le manteau, lequel forme de nouvelles roches.
Mais, il existe des îles volcaniques isolées comme Hawaï, Tahiti, les
Açores, le Cap Vert, etc. Si l’on regarde l’archipel d’Hawaï, les îles
sont alignées par ordre d’âge décroissant, avec la plus grande île,
et la plus récente à l’extrémité est de l’archipel. Ceci a suggéré aux
géophysiciens la conjecture que ces îles sont formées par un panache
volcanique, c’est-à-dire une sorte de cheminée volcanique au travers
du manteau. Rappelons que la profondeur du manteau est environ la
moitié du rayon de la Terre. Puisque la plaque tectonique de surface
se déplace, ceci expliquerait la formation successive d’îles alignées,
dont la différence d’âge pourrait être calculée par la distance entre les
îles et la vitesse de la plaque tectonique. Mais il faut rajouter d’autres
éléments de preuve pour que la conjecture soit définitivement acceptée par la communauté scientifique, l’un de ces éléments pouvant
être de « voir » le panache. Un outil pour explorer l’intérieur de la
Terre est la télédétection : on envoie des signaux et on analyse les signaux réfléchis ou réfractés sur les différentes couches dans le sol. La
technique est utilisée pour chercher du pétrole. Mais, les panaches
sont si profonds sous la croûte terrestre que les signaux artificiels
ne sont pas assez puissants pour une telle analyse. Les seuls signaux
suffisamment puissants pour analyser les détails à une telle profondeur sont les ondes sismiques générées par les grands tremblements
de terre. L’équipe autour d’Ingrid Daubechies a eu accès à de grandes
bases de données contenant les enregistrements des ondes sismiques
captées par les stations sismiques autour de la planète.
que l’analyse des ondes sismiques perme琀ait de détecter
des zones de perturbation des
ondes de pression (P-ondes) des
tremblements de terre. De telles
régions avaient été identifiées,
qui chevauchaient exactement
les régions avec des îles volcaniques isolées, et la température du fond océanique était
plus grande dans ces régions.
Mais, comme les panaches sont
si petits et la perturbation de la
vitesse des P-ondes si faible, le
risque d’erreurs est grand dans
Ingrid Daubechies
la reconstruction numérique de
la structure interne de la Terre, sauf si on utilise des outils suffisamment performants. C’est là que les ondele琀es sont si utiles. Elles sont
l’outil parfait pour analyser de petits détails locaux. De plus, l’analyse
en ondele琀es permet de concentrer toute l’énergie sur ces petites régions et de négliger les autres régions.
Dans sa conférence, Ingrid Daubechies a donné un mini-cours sur
l’analyse en ondele琀es adaptée aux images digitales composées de
pixels. Une image en tons de gris est simplement une matrice de
nombres donnant le ton de gris de chaque pixel. À partir de ce琀e matrice, on construit quatre matrices plus petites. La première contient
les moyennes horizontales et verticales des pixels voisins pris deux
à deux, la seconde les moyennes horizontales et les différences verticales des pixels voisins pris deux à deux, la troisième les moyennes
verticales et les différences horizontales des pixels voisins pris deux
à deux, et la dernière, les différences horizontales et verticales des
pixels voisins pris deux à deux. On itère le processus sur la première
matrice (celle des moyennes horizontales et verticales). Jusque-là, pas
de perte d’information. Ingrid Daubechies a expliqué comment les
ondele琀es perme琀ent de compresser l’information et comment on
peut extraire les détails fins dans une petite région, tout en ayant
compressé beaucoup l’information. L’utilisation des ondele琀es pour
la reconstruction d’images permet d’éliminer les erreurs de reconstruction numérique et d’être certain que les zones singulières identifiées dans l’image sont effectivement spéciales. Ingrid Daubechies a
montré des images « propres » obtenues grâce aux ondele琀es, dans
lesquelles les régions artificielles ont été enlevées, et elle a pu annonDonc, les données existent. Il ne manque qu’un bon outil pour les
cer, « hot off the press », qu’elle et ses collaborateurs avaient obtenu
analyser. Le problème est loin d’être trivial. Les panaches sont très
les premiers résultats sur l’ensemble de la Terre avec des vraies donfins et, de plus, la différence de vitesse d’une onde sismique au travers
nées !
d’un panache n’est que de l’ordre de 1%. Les sismologues Tony Dahlen et Guust Noleta ont approché Ingrid Daubechies en 2005 pour Le public a posé de nombreuses questions dans la salle de confévoir si les ondele琀es ne pourraient pas les aider dans leur étude. En rences avant de poursuivre les discussions autour d’une réception.
effet, les résultats prome琀eurs de Raffaella Montelli avaient montré
BULLETIN CRM–9
crm.math.ca
Workshop on Planetary Motions,
Satellite Dynamics, and Spaceship Orbits
Organizers: Alessandra Celle琀i (Tor Vergata), Walter Craig (Mc- Edward Belbruno explained the mathematics of low energy transMaster), Florin Diacu (Victoria), Christiane Rousseau (Montréal)
fer trajectories between planets using ballistic capture: the idea is
to target the weak stability boundary, with no breaking necessary to
It is not the tradition in the community of people working in celestial
be captured by a celestial body. He described the recent achievement
mechanics to organize meetings bringing together people from many
of showing the existence of low energy routes allowing transfer of
areas of the domain, and the workshop was unique in that regard. In
material between planetary systems, from which we cannot exclude
fact, while the workshop brought together the major players of the
that the origin of life on Earth could have come from a remote planfield, many of these people had never met before and the workshop
etary system.
really helped in structuring the community of scientists a琀ached to
the theme of the workshop. 吀e lectures were all of exceptional qual- Too o昀en, we have the image of celestial mechanics as non dissipaity, and the workshop played the role of a school for those not very tive. But it is in fact slightly dissipative (for instance because of the
familiar with the general theme, or with a special sub-theme. 吀is atmosphere around the Earth which slows down its rotation around
was the case of the organizer Christiane Rousseau, a specialist in its axis) and KAM theory has been adapted to treat these cases. 吀is
dynamical systems but an amateur in celestial mechanics, and for dissipation plays a major role in ge琀ing stable motions and allows
whom the workshop was an exceptionally rewarding experience.
one to provide rigorous proofs of the stability of these motions with
integer-arithmetic numerical techniques.
Planetary motions are usually modelled through the N -body problem, which is the study of trajectories of N mass particles submi琀ed Can we explain why the Solar System is exactly as we observe it?
to Newton’s gravitational law. 吀e underlying dynamical system is Several lectures addressed this issue. While energy is dissipated,
non integrable as soon as N > 2. 吀e lectures of the workshop cov- angular momentum is preserved. Hence, what is the minimal enered the whole spectrum from N = 3, and even the restricted 3-body ergy configuration for a N -body system with fixed angular momenproblem which is the limit case when one mass is put equal to zero, tum? Dan Scheeres showed that this ill-posed question becomes well
to N very large.
posed if instead the question is formulated accounting for finite density distributions, thus leading to a natural “granular mechanics” exIn the case of N = 3, it is known that there are five families of
tension of celestial mechanics. 吀e lecture of Vladislav Sidorenko
periodic synchronous motions for the three bodies: a昀er a change
addressed the problem of understanding the quasi-satellite regime
of coordinates to a moving frame, these special trajectories become
of small celestial bodies such as asteroids, and the route from the
equilibrium positions in the new frame, called Lagrange equilibrium
formation of the Solar system to its present state.
points (also libration points). 吀eir associated invariant manifolds
play an essential role in organizing the dynamics and the different 吀e case with N large was covered by a spectrum of applications.
types of motions. In applications they provide low-energy pathways Stanley Dermo琀 presented the erosion of the asteroid belt under
for interplanetary missions and are associated to the weak bound- Martian resonances; Martin Duncan presented a model of core acary capture of celestial objects. Invariant manifolds were studied cretion for giant planet formation from billions of planetesimals and
both analytically (lecture of E昀hymiopoulos) and numerically (lec- its numerical simulations; Anne Lemaitre explained the challenges
ture of Doedel). Marian Gidea showed how their existence can ex- of understanding the dynamics of the tens of thousands of space deplain Arnold diffusion.
bris with diameter between 1 cm and 10 cm, which are too numerous
to be followed individually, but sufficiently large to represent a real
Several lectures described the normal forms and their applications.
danger: the motion of the debris is simulated with an accurate symIn particular, Gabriella Pinzari described her recent results with
plectic integration scheme and a model which takes into account
L. Chierchia, showing the existence and nondegeneracy of a Birkhoff
the effects for solar radiation pressure and Earth shadow crossings.
normal form for the planetary problem and its consequence on the
吀e goal is to understand where these debris are more likely to acexistence of a large measure set of stable motions and lower dimencumulate. Jacques Laskar discussed the paleoclimate reconstruction
sional elliptic tori in phase space, thus solving a problem open for
through the past planetary motions of the Solar System: a strong
more than 50 years.
resonance between the asteroids Ceres and Vesta prevents any preTwo lectures described near collision orbits: at the limit, the system cise reconstruction beyond 60 Myr, but a more regular oscillation of
becomes singular and a desingularization process is necessary to un- the eccentricity of the Earth with period 405 kyr can nevertheless be
derstand the phenomenon. A geometric desingularization was pre- used for calibrating climates over the whole Mezozoic era.
sented by Richard Moeckel, while the lecture of Sergey Bolotin explained how a variational approach allows to transform the problem
to a billiard type problem with elastic collisions.
BULLETIN CRM–10
crm.math.ca
Les ponts de Königsberg, les digues de Hollande et
la chute de Wall Street
Christian Genest et Johanna G. Nešlehová (Université McGill)
Les Grandes Conférences du CRM a琀irent toujours beaucoup de
monde, même un vendredi soir. À plus forte raison le 10 mai, alors
que l’événement était jumelé à la 8e édition des « 24 heures de
science ». Tandis que la foule s’assemblait pour entendre l’exposé
de Paul Embrechts, professeur de mathématiques à l’École polytechnique fédérale de Zürich, tout le monde se demandait quel lien il
pouvait bien y avoir entre les éléments évoqués dans le titre accrocheur de sa présentation : « Les ponts de Königsberg, les digues de
Hollande et la chute de Wall Street ».
Paul Embrechts
Paul Embrechts est un spécialiste de la théorie des valeurs extrêmes
et de la gestion quantitative du risque. Il est l’auteur de plusieurs
ouvrages influents et de plus de 150 articles sur ces thèmes de recherche. Les institutions financières et les organismes réglementaires internationaux le consultent régulièrement. Son exposé visait
à illustrer en termes simples et concrets comment les mathématiques
peuvent contribuer à l’élaboration de mesures de protection contre
les épidémies, les inondations, les crises financières et autres catastrophes du genre par l’examen méticuleux des facteurs de risque,
l’estimation de la probabilité de dépassement de niveaux critiques
et l’étude des mécanismes de contagion et de propagation du risque
dans les réseaux bancaires et sociaux.
L’exposé s’est ouvert sur le célèbre problème des sept ponts de Königsberg : est-il possible de parcourir la ville en passant une et une
seule fois sur chacun de ces ponts ? En répondant à ce琀e question par
la négative, Euler a donné une technique à la fois simple et générale
de résoudre le problème, quel que soit le nombre de ponts ou leur
configuration. Ainsi est née la théorie des graphes. Dans son exposé,
Paul Embrechts a montré l’utilité des graphes pour la visualisation
du réseau complexe d’interdépendance entre les institutions financières et l’évaluation du risque qu’en raison d’engagements croisés,
une faillite provoque une réaction en chaîne menant à l’effondre-
ment du système financier mondial. En s’appuyant sur des travaux
parus dans Nature en août 2012, il a aussi expliqué comment des
outils statistiques et graphiques ont permis d’évaluer la hausse du
risque systémique lors de la crise des prêts hypothécaires, pendant
laquelle le risque qu’une seule défaillance déclenche un effet domino
est devenu considérable.
Au plan conceptuel, la protection des banques contre les risques
de faillite s’apparente à la défense du plat pays de la mer du Nord
contre les marées de tempête et les inondations. Un peu partout
dans le monde, les agences de réglementation du secteur bancaire
exigent maintenant des institutions financières qu’elles affectent des
réserves de capitaux suffisantes pour couvrir le risque de lourdes
pertes monétaires. Ces fonds de réserve jouent essentiellement le
même rôle qu’un système de digues le long d’un li琀oral. Par suite
du raz-de-marée dévastateur de 1953, le gouvernement des Pays-Bas
a entrepris la construction de grands ouvrages de protection côtière.
En tenant compte des coûts et de diverses contraintes techniques,
la Commission des travaux du Delta a fixé un niveau acceptable de
risque d’inondation pour chacune des régions concernées. À titre
d’exemple, il a été convenu qu’en Hollande méridionale, les digues
devraient perme琀re de se prémunir contre une inondation ne se
produisant qu’au plus une fois tous les 10 000 ans. Mais comment
peut-on estimer un tel risque quand les données nous manquent
pour juger de l’ampleur d’une inondation déca-millénaire ? Comme
Paul Embrechts l’a expliqué, c’est ici que la théorie des valeurs extrêmes intervient car elle permet d’extrapoler au-delà de la limite
de l’observable. Fait insolite : les travaux réalisés dans le domaine
sont en bonne partie l’œuvre de probabilistes et de statisticiens qui
ont grandi près de la mer du Nord. Paul Embrechts lui-même est né
quelques jours après l’inondation de 1953, tout près de la zone sinistrée.
En finance, les livres à succès de Nassim Nicholas Taleb ont popularisé l’expression « cygnes noirs » pour faire référence à des événements rares et imprévisibles à haute incidence. Il n’y a donc rien
d’étonnant à ce qu’en finance, la théorie des valeurs extrêmes puisse
être employée pour prédire, évaluer et se prémunir contre de tels
événements.
On dit qu’un malheur ne vient jamais seul. À l’instar des grandes
inondations qui affectent plusieurs régions, les mouvements boursiers entraînent souvent la chute simultanée de nombreux actifs et
plusieurs débiteurs peuvent tout à coup se trouver incapables de respecter leurs engagements. Lorsqu’elles négligent ce fait, les institutions financières tendent à sous-estimer leur risque. En 2009, la
presse a a琀ribué la chute de Wall Street à la « copule gaussienne »
que les analystes financiers utilisaient pour calculer la probabilité de
multiples défauts de paiement ; voir entre autres « 吀e Formula that
(suite à la page 14)
BULLETIN CRM–11
crm.math.ca
Le 15e congrès annuel des jeunes chercheurs de l’IMS
Johanna G. Nešlehová (Université McGill) et Aarti Singh (Carnegie Mellon University)
Fondé en 1935, l’IMS est une société savante à caractère international
vouée au soutien, à la promotion et à la dissémination de la recherche en
statistique et en probabilités. Elle compte plus de 4 000 membres dans
le monde et publie certaines des revues les plus prestigieuses dans le
domaine, dont 吀e Annals of Statistics et 吀e Annals of Probability.
(Steve Fienberg, Pi琀sburgh). De plus, les jeunes chercheurs ont tous
eu l’occasion de présenter brièvement leurs travaux de recherche
couvrant un large spectre, de la statistique fondamentale à la génétique, en passant par la biostatistique, l’analyse multidimensionnelle,
la modélisation spatio-temporelle et la théorie de l’échantillonnage.
Le dernier jour, les participants ont pu entendre l’allocution de Jingchen Liu, lauréat 2012 du prix Tweedie, récompense annuelle remise par l’IMS à un chercheur s’étant particulièrement illustré au
cours des cinq années suivant l’obtention de son doctorat. Hansruedi
Künsch a ensuite prononcé un vibrant plaidoyer en faveur des sociétés savantes et de l’importance d’y adhérer. Enfin, quatre ateliers
avaient été organisés pour conseiller les jeunes chercheurs en matière d’enseignement, de recherche, de supervision et de rédaction de
demandes de subventions. Ces ateliers ont été animés par les quatre
conférenciers vede琀es, ainsi que par Larry Wasserman (Pi琀sburgh)
et Dave Stephens (Montréal). Des porte-parole du CRSNG, de MITACS et de diverses agences américaines (NIH, NSA, NSF) étaient
aussi sur place pour répondre aux questions.
À l’issue du congrès, les participants ne cachaient pas leur enthousiasme. Ils garderont un excellent souvenir de l’accueil qui leur a été
réservé par le personnel du CRM et la grande qualité des installations. Pour de plus amples renseignements concernant le congrès,
Le 15e congrès annuel des jeunes chercheurs de l’IMS a eu lieu au y compris un programme détaillé et des photos, voir http://www.
CRM du 1er au 3 août. Ce琀e rencontre, parrainée par l’Institute of math.mcgill.ca/nrc2013/.
Mathematical Statistics, a été organisée avec le soutien logistique et
financier du CRM et de son laboratoire de statistique. La National
Spyros Alexakis
Science Foundation (NSF), les National Institutes of Health (NIH),
(continued from page 6)
l’Institut des sciences mathématiques (ISM) et la Société statistique
du Canada ont aussi contribué à financer l’événement, qui a réuni 48 [13] S. Deser and A. Schwimmer, Geometric classification of conformal anomalies in
arbitrary dimensions, Phys. Le琀. B 309 (1993), no. 3-4, 279–284.
jeunes chercheurs en probabilités et en statistique, dont 20 femmes.
[14] C. Fefferman and C. R. Graham, 吀e ambient metric, Ann. of Math. Stud.,
Le congrès visait à stimuler la communication entre eux et à les renvol. 178, Princeton Univ. Press, Princeton, NJ, 2012.
seigner sur divers aspects de la carrière universitaire. Les partici- [15] H. Friedrich, I. Rácz, and R. M. Wald, On the rigidity theorem for spacetimes
pants, triés sur le volet, provenaient pour la plupart du Canada et
with a stationary event horizon or a compact Cauchy horizon, Comm. Math.
Phys. 204 (1999), no. 3, 691–707.
des États-Unis, mais aussi de France, de Suisse et du Japon.
Le programme du congrès, réparti sur trois jours, comportait quatre
grandes conférences et autant de séances de communications invitées, toutes en plénière. Après avoir été accueillis par le président
de l’IMS, Hansruedi Künsch, et au nom du CRM par le directeur de
l’ISM, Christian Genest, les participants ont été un peu déstabilisés
au départ par le discours délibérément provocateur de Terry Speed,
directeur du Département de bioinformatique à l’Institut Walterand-Eliza-Hall de recherche médicale, en Australie. « Devriez-vous
continuer à vous intéresser à la statistique mathématique et si oui,
pendant combien de temps ? » a-t-il demandé aux participants. Les
autres conférenciers vede琀es ont exprimé, chacun à leur manière,
leur passion pour la discipline en décrivant de récentes découvertes
ou de fascinantes applications de l’analyse de données fonctionnelles (Aurore Delaigle, Melbourne), du calcul intensif par méthodes
de Monte-Carlo (Jeffrey Rosenthal, Toronto) et de l’analyse causale
BULLETIN CRM–12
[16] C. R. Graham and E. Wi琀en, Conformal anomaly of submanifold observables in
AdS/CFT correspondence, Nuclear Phys. B 546 (1999), no. 1-2, 52–64.
[17] S. W. Hawking and G. F. R. Ellis, 吀e large scale structure of space-time, Cambridge Monogr. Math. Phys., vol. 1, Cambridge Univ. Press, London, 1973.
[18] K. Hirachi, Logarithmic singularity of the Szegő kernel and a global invariant of
strictly pseudoconvex domains, Ann. of Math. (2) 163 (2006), no. 2, 499–515.
[19] A. D. Ionescu and S. Klainerman, On the uniqueness of smooth, stationary black
holes in vacuum, Invent. Math. 175 (2009), no. 1, 35–102.
[20] W. Israel, Event horizons in static vacuum space-times, Phys. Rev. Le琀. 164
(1967), 1776–1779.
[21] R. P. Kerr, Gravitational field of a spinning mass as an example of algebraically
special metrics, Phys. Rev. Le琀. 11 (1963), 237–238.
[22] D. C. Robinson, Uniqueness of the Kerr black hole, Phys. Rev. Le琀. 34 (1975),
905–906.
[23] H. Weyl, 吀e classical groups. 吀eir invariants and representations, Princeton
Univ. Press, Princeton, NJ, 1939.
[24] S. Zelditch, Szegő kernels and a theorem of Tian, Internat. Math. Res. Notices 6
(1998), 317–331.
crm.math.ca
Séminaire de Mathématiques Supérieures 2013
Physics and Mathematics of Link Homology
Organizers: Sergei Gukov (Caltech), Mikhail Khovanov (Colum- gories of coherent sheaves on quiver varieties and on convolution
bia), Johannes Walcher (McGill)
varieties of affine Grassmannians, or the enumerative geometry of
(relative) Donaldson–吀omas invariants of Calabi–Yau threefolds.
Symplectic geometers might study Fukaya–Floer categories associated with Heegard spli琀ings or with Lagrangians on quiver varieties.
Researchers with a background in representation theory will focus
on the li昀ing of quantum groups and their representation theory to
higher categories. As refreshing as this diversity may be, it represented a barrier for newcomers to enter the field. So one of the goals
of the SMS was to address the sorely felt need for a pedagogical introduction aimed at intermediate to advanced graduate students interested in the mathematics of knot homology.
吀roughout recent history, the theory of knot invariants has been
a fascinating melting pot of ideas and cultures, blending mathematics and physics, geometry, topology and algebra, gauge theory and
quantum gravity. Participants of this year’s Séminaire de Mathématiques Supérieures can confirm that the explosion of activity we are
currently witnessing most likely is just the beginning of a larger and
much more uniform story.
One of the pervasive themes of the recent developments is “aiming
for higher dimensions” — 吀e original construction of the Jones polynomial invariant of knots and links (one-dimensional objects nontrivially embedded in three-dimensional space) in 1984 was firmly
rooted in two-dimensional mathematical physics, but did not make
all symmetries manifest. An intrinsically three-dimensional interpretation was given by Wi琀en in 1988, representing the Jones polynomial as a physicist’s path-integral over the space of connections
with Chern-Simons action. Following the categorification paradigm,
Khovanov in 1998 li昀ed the Jones polynomial to a homological invariant of link cobordisms living in four dimensions. Around the
same time, developments initiated by Gopakumar and Vafa began to
expose relations to higher-dimensional geometric invariants of interest in superstring theory. Spli琀ing M-theory’s eleven dimensions
in various ways makes room for a large number of vantage points to
help explain the origin of lower-dimensional phenomena.
And while the physics perspective promises to ultimately help clarify
the relations between all these approaches, initially it only seems to
complicate ma琀ers. In fact, one of the main a琀ractions of homological knot invariants for mathematical physics is precisely that it mixes
so many different parts of topological quantum theory. Not only do
knot homologies provide a bridge between topological field theories of cohomological and Chern-Simons type, the relation with the
enumerative geometry of Calabi–Yau threefolds is one of the incarnations of the celebrated large-N (or gauge/gravity) duality that has
come to dominate formal theoretical physics in the last 15 years. Providing an access point into the relevant parts of theoretical physics
was a declared aim of the school as well. 吀e organizers made a particular effort to encourage interactions and fruitful exchanges between the communities in order to facilitate the development of the
unified picture.
Clearly, these are a lot of different topics to cover in a two-week summer school, aiming at a coherent and unified overview of the subject!
With necessary gaps, the enthusiastic response and feedback from
junior as well as senior participants alike is testimony to the fact that
the programme amply succeeded in balancing the various interests.
吀e 15 main lecturers were given the leisure of 3 (first week) or 2
(second week) times 75 minutes to explain the background that they
felt most necessary or useful for their particular subject, sketch their
main results and end with explaining those challenges that they view
as most promising. 吀ree topical seminar style lectures rounded out
the programme. All the while, the consistently streamlined schedule
of the school allowed for the organization of a number of sponta吀is two week long programme on “Physics and mathematics of link neous discussion sessions between students from mathematics and
homology” brought together leading researchers in mathematics and physics backgrounds, as well as several social outings around Montmathematical physics, in order to provide an opportunity to educate réal.
a new generation of scientists in this growing field. 吀e challenges 吀ere were countless cross-references between the various lectures.
were remarkable.
Indeed, the group of speakers is very well in tune regarding each
吀eories of link homology are studied from a large variety of mathe- other’s lecture topic. 吀is is due to a number of recent events with
matical approaches and backgrounds. Topologists will find the ax- a similar topic, albeit at a rather higher level. In particular, a Siiomatic framework of topological quantum field theory most ac- mons Symposium on Knot Homologies and BPS States was held in
(continued on page 17)
cessible. Algebraic geometers could be interested in derived cateBULLETIN CRM–13
crm.math.ca
Simon A. Levin
(continued from page 8)
Follows, M.J., S. Dutkiewicz, S. Grant and S.W. Chisholm. 2007. Emergent biogeography of microbial communities in a model ocean, Science,
315:1843-1846
Laxminarayan, R., D.L. Smith, L. A. Real, and S.A. Levin. 2005. On the im-
is one of equity: we always discount the future or the interest of portance of incentives in hospital infection control spending. Discovery
others; how should we counteract the negative consequences of this Medicine 5(27):303-308.
May, R.M., S.A. Levin, and G. Sugihara. 2008. Ecology for bankers. Nature
discounting? (Arrow et al., 2004)
吀is question can be broken down into several steps. At first, one
wants to determine how much a given generation is allowed to consume before they jeopardize their own survival or that of future generations (Arrow and Levin, 2009). 吀en, with the goals set, one needs
to achieve inter-generational cooperation. Such problems o昀en lead
to game-theoretic approaches and can be related to the Prisoner’s
dilemma. One difficulty o昀en encountered is that public goods are
involved and social costs are o昀en neglected. Analytical and simulation approaches indicate that the principle of “Mutual coercion,
mutually agreed upon” is a highly efficient way to manage such systems. 吀is insight leads to the third question of how social norms
get established. 吀is question falls under the more general topic of
aggregation, collective motion and grouping that was discussed in
detail in the first lecture.
Empirical and theoretical studies show that animal groups can be led
by very few leaders. Roughly 5% of individuals are sufficient to lead
the remaining 95%. 吀e same result holds for humans. In both cases,
so-called un-opinionated individuals are crucial to form consensus
and to enhance success. If a small but stubborn group tries to obtain a
certain outcome against a larger but not so heavily commi琀ed group,
then the number of un-opinionated individuals decides where the
overall decision will fall. 吀e larger the number of un-opinionated
individuals, the more likely that the opinion of the larger group will
prevail. 吀e empirical study was done with fish; the theoretical study
is based on an adaptive network model (Couzin et al., 2011).
451:893-895
Nabet, B., N.E. Leonard, I.D. Couzin, and S.A. Levin. 2009. Dynamics of
decision making in animal group motion. Journal of Nonlinear Science
19:399-435.
Scheffer, M. et al. 2012. Anticipating critical transitions. Science
338:334-348.
Staver, A.C. and S.A. Levin 2012. Integrating theoretical climate and fire
effects on savanna and forest systems. American Naturalist 180:211-224.
Staver, A.C., S. Archibald, and S.A. Levin. 2011. 吀e global extent and
determinants of savanna and forest as alternative biome states. Science
334:230-232.
Zea-Cabrera, E., Y. Iwasa, S. Levin, and I. Rodriguez-Iturbe. 2006. Tragedy
of the commons in plant water use. Water Resources Research. 42, W06D02,
DOI: 10.1029/2005WR004514.
Paul Embrechts
(suite de la page 11)
Killed Wall Street » dans le Financial Times du 24 avril 2009. Pour citer Paul Embrechts, « C’est comme si on blâmait la formule E = mc2
de Einstein pour les dommages causés par la bombe atomique. » Il
est aussi ironique, a-t-il dit, que les chercheurs universitaires aient
été accusés d’avoir indirectement provoqué la crise alors qu’ils n’ont
cessé de rappeler les carences des méthodes employées dans le secteur financier. De fait, Paul Embrechts et son équipe avaient euxmêmes signalé dès 1998 que « la formule qui a tué Wall Street » sousProfessor Levin closed his fascinating presentation with the re- estimait considérablement le risque de simultanéité d’événements
minder that ecological and socio-economic systems are complex extrêmes.
adaptive systems. To manage them, we need to integrate bo琀om-up À l’issue de cet exposé passionnant, beaucoup d’auditeurs ont sans
and top-down mechanisms to achieve adaptive, polycentric gover- doute médité le commentaire de clôture du professeur Embrechts,
nance and agreements. He pointed out how similar these approaches qui portait sur l’importance de la communication. L’excellence scienare to our human immune system. Adam Smith’s invisible hand does tifique et les meilleures pratiques vont de pair, a-t-il rappelé ; elles
not protect society nor ecological systems. New institutions need to
nécessitent un dialogue constructif. Les modèles stochastiques et les
be adaptive, based on trust and cooperation. Historically, cooperatechniques d’inférence statistique très puissants dont nous disposons
tion within groups o昀en emerged to fight against other groups. We
aujourd’hui ne sauraient être employés à leur plein potentiel que si
need to establish cooperation in the absence of a fight against other
l’on est à l’écoute des chercheurs plutôt que de les voir comme des
groups to achieve a sustainable future.
savants fous réfugiés dans des tours d’ivoire. En même temps, il est
vrai que le monde ne peut plus se perme琀re les chercheurs ingénus
Arrow, K and S.A. Levin. 2009. Intergenerational resource transfers with
random offspring. Proceedings of the National Academy of Sciences et irresponsables caricaturés par l’humoriste américain Tom Lehrer
lorsqu’il chante :
106(33):13702–13706
Arrow, K. et al. 2004. Are we consuming too much? J. Economic Perspectives 18(3):147-172
“Once the rockets are up,
who cares where they come down ?
Couzin, I.D. et al. 2011. Uninformed individuals promote democratic con- 吀at’s not my department,”
sensus in animal groups. Science 334:1578-1580
says Wernher von Braun.
Dixit, A.K., S. Levin, and D.I. Rubenstein. 2012. Reciprocal insurance among
Kenyan pastoralists. 吀eoretical Ecology: DOI: 10.1007/s12080-012-0169-x.
BULLETIN CRM–14
crm.math.ca
MAGE : Modèles et algorithmes pour la génomique évolutive
Conférence en l’honneur des 50 ans de contribution scientifique de David Sankoff
Nadia El-Mabrouk (Université de Montréal)
Samedi matin à l’hôtel château Bromont, dans la belle région des
Cantons de l’Est, juste en face des pistes de ski, ou plutôt des glissades d’eau en ce琀e fin de mois d’août. Le soleil est au rendez-vous,
ainsi que tous les participants à cet événement réunissant un panel
de chercheurs, parmi les plus renommés de la biologie computationnelle, sans oublier la relève constituée de chercheurs plus jeunes et
d’étudiants. Le ton est donné par la présentation d’Anne Bergeron
qui met en perspective la carrière de David Sankoff, incluant un petit
tournage maison pour tester un algorithme de reconnaissance musicale développé par David. La présentation suivante est donnée par
Gene Myers, bien connu dans le cadre du projet de séquençage du
génome humain, et un acteur principal dans le développement de
BLAST, probablement l’outil le plus utilisé par les biologistes pour
analyser les séquences moléculaires. Sa présentation met en perspective les avancées algorithmiques à l’origine du logiciel et de son succès. Après une pause-café, suivie de quatre présentations de grande
qualité, ce琀e première matinée s’achève par un repas servi au restaurant de l’hôtel. Alors que dans le jardin un vol de colombes clôture
une cérémonie de mariage, dans la salle de conférence les présentations se poursuivent, ponctuées par des retours sur la carrière de
David. À noter, en particulier, ce vidéo mémorable, envoyé par Joe
Felsenstein, rappelant les débuts de la bio-informatique au CRM avec
David Sankoff et Robert Cedergren, qui en a ému plus d’un. Tout est
en place pour la session posters « vin et fromage » de la fin d’aprèsmidi. La journée s’achève par un repas du soir servi dans la salle
« chasse à courre ». Dimanche est une autre journée bien chargée.
Nous commençons par une présentation de David sur ses travaux
les plus récents liés à l’évolution des plantes par polyploïdie, et nous
poursuivons avec un panel de sujets couvrant divers aspects de la
biologie évolutive, de la modélisation (inférence de génomes ancestraux, inférence de distances évolutives), à l’application (évolution
du génome de la tomate, du génome Utricularia gibba, et d’autres Eudicots). Les présentations sont à saveur biologique (Joseph Nadeau,
Aoife McLysaght), bio-informatique pratique (Eric Lyons, Victor Albert), méthodologique et algorithmique (Pavel Pevzner, Dannie Durand, Binhai Zhu et autres). Il ne faut pas oublier la photo de groupe !
Pas de mariage aujourd’hui, le jardin est à nous pour la pause-café
de l’après-midi. Et la salle de mariage est à nous pour le banquet
de la conférence. Un micro et un pupitre, c’est le moment du petit
mot de Pavel Pevzner, suivi de David, et de moi-même pour remercier chaleureusement tous les partenaires de ce琀e conférence. Lundi
matin, nous clôturons ces trois journées de conférence par une session riche en sujets ouvrant sur des perspectives de recherche prome琀euses, en commençant par la présentation de Ron Shamir sur
l’étude des réarrangements dans les cellules cancéreuses. Après un
dernier repas au restaurant de l’hôtel, l’autobus est à l’heure, en ce
début d’après-midi pluvieux, pour le voyage de retour vers l’aéroport et l’Université de Montréal. De l’avis de tous, cet événement a
été une occasion unique de se rassembler, de retracer les débuts de
ce琀e jeune discipline qu’est la bio-informatique et de discuter des défis qui nous a琀endent, encore plus grands que les réalisations. À la
suite de ce琀e conférence, j’ai reçu d’innombrables messages qui témoignent de l’impact de ce琀e conférence sur les participants : « probablement la meilleure conférence depuis longtemps », « réellement
mémorable ». Cet événement a également été l’occasion d’éditer un
livre, dont la publication par Springer dans la série Computational
Biology est prévue pour le mois de novembre 2013, regroupant 14
chapitres écrits par des participants à MAGE. Ce livre est très a琀endu
par la communauté. Pour ma part, certains chapitres me servent déjà
de support pour mes cours gradués de bio-informatique. Signalons
également qu’un article a été consacré à la conférence MAGE dans
le numéro du 26 août de FORUM, la revue hebdomadaire de l’Université de Montréal.
David Sankoff is known as one of the founding fathers of bioinformatics and computational biology. He has been at the origin of numerous fields in bioinformatics, starting with the fundamental problem
of sequence alignment. In particular, he contributed to the early introduction of dynamic programming to computational biology. In addition, he has laid the foundations of numerous fields in comparative
genomics, such as: sequence alignment, RNA folding, phylogeny reconstruction, gene order rearrangement, and long-term genomic evolution.
David Sankoff currently holds the Canada Research Chair in Mathematical Genomics at the University of O琀awa. He joined the new
Centre de recherches mathématiques (CRM) at Université de Montréal
in 1969 and was also Professor in the Mathematics and Statistics Department from 1984 to 2002. He is a Fellow of the Royal Society of
Canada and of the International Society for Computational Biology
and is active in the Canadian Institute for Advanced Research. He is
recipient of the Award for Excellence in Research from the University
of O琀awa, medalist of the “Association francophone pour le savoir,”
recipient of the first Senior Scientist Accomplishment Award by the International Society for Computational Biology and the Weldon Memorial Prize from Oxford University. He was founding editor of Language
Variation and Change (Cambridge) and serves on the editorial boards
of a number of bioinformatics, computational biology and linguistics
journals. David published his first paper in 1963, during his B.Sc. at
McGill University. 吀is conference celebrated his 50th year of research
contributions.
BULLETIN CRM–15
crm.math.ca
Les Annales sont mortes ! Vive les Annales !
Claude Levesque (Université Laval)
D’entrée de jeu, rappelons que la revue
Annales des sciences mathématiques du 儀ébec (ASMQ)
est devenue en janvier 2013 une revue Springer avec le nouveau nom
Annales mathématiques du 儀ébec (AMQ).
La revue est chapeautée par le Centre de recherches mathématiques
(CRM) et par l’Institut des sciences mathématiques (ISM) du 儀ébec.
C’est feu le Pr Serge Vasilach (Université Laval) qui a lancé et défendu l’idée de créer un journal mathématique au 儀ébec, et c’est le
CRM qui a créé et démarré officiellement ce琀e revue biannuelle en
janvier 1978.
C’était la belle époque où les mathématiciens du 儀ébec se réunissaient deux fois l’an, pendant la journée du samedi, au Colloque des
sciences mathématiques du 儀ébec (CSMQ) pour présenter leurs
travaux, échanger entre eux et surtout resserrer les liens d’amitié. Autres temps, autres mœurs, ces pèlerinages annuels ont cessé
d’exister vers 2005.
Suite aux recommandations des membres du comité L3 (Réginald Lavoie, Raymond Leblanc et feu Pierre Leroux), le « Projet de fondation
d’une revue de mathématiques » fut officiellement adopté le 15 mars
1975 lors de l’assemblée générale du 3e CSMQ. En 1978, l’Association
mathématique du 儀ébec crée le Groupe des chercheurs en sciences
mathématiques (GCSM), un « groupe d’intérêt » avec un statut officiel légal pour gérer les Annales.
Les buts de la revue y furent définis… pour le meilleur et pour le
pire…, c’est le cas de le dire : publier principalement les travaux effectués par des mathématiciens du 儀ébec, et ce, en français. Les temps
ont bien changé, vous en conviendrez.
Les méchantes langues (dont et surtout la mienne) ont trouvé que ledit groupe légal, sans président élu, sans assemblées dûment convoquées, sans liste officielle de ses membres, formait un groupe fantôme, mais rendons à César ce qui appartient à César, la formule a
bien fonctionné.
subventions fédérales du CRSNG (le CNR à l’époque), jusqu’au jour
où ces programmes de subventions furent abolis.
Contre vents et marées, la revue a résisté à plusieurs tempêtes. Si la
revue n’est pas passée de vie à trépas, nous en sommes tout particulièrement redevables au travail de missionnaire de René Ferland.
À chaque année, à l’occasion du CSMQ, René cherchait quelqu’un
pour occuper différents postes au sein des Annales, et il ne trouvait
personne. Lorsque les problèmes financiers ont commencé à poindre,
notre cher René nous annonçait à chaque colloque que nous convergions vers la mort de la revue. En fin psychologue, Ibrahim réussissait chaque fois à convaincre René de ne pas abandonner le navire.
En fait, il y a environ cinq ans, la revue en est venue à « un quart
d’heure de la mort ». Notre vénérable René (pour ne pas amicalement dire notre damné René) a concocté une machination machiavélique pour trouver des candidats pour les postes à pourvoir au sein
des Annales. Il a donné une date d’échéance : si à 23h59 au jour J, il
n’y avait pas de candidats pour le poste de rédacteur en chef et pour
celui de rédacteur gérant, René annonçerait officiellement la mort
des Annales, sa le琀re était déjà rédigée, toute prête à être envoyée.
Il courait un grand risque, calculé bien sûr, celui de n’avoir aucun
candidat, mais il était confiant que les membres de la communauté
mathématique, acculés au mur aussi brutalement, réagiraient positivement. Ce ratoureux de René a vu juste et c’est tout à son honneur.
C’est à minuit moins quart au jour J que (pas un, mais) deux bons
larrons se sont pointés : Claude Levesque pour le poste de rédacteur
en chef et François Huard au comité de direction (et plus tard David
Smith comme rédacteur gérant).
Une fois en poste, mon but avoué était de faire des ASMQ un journal
viable de 4 numéros par an, et d’en faire même une revue Springer, le
passage chez Springer perme琀ant de régler tous les tracas financiers
et de bénéficier d’une reconnaissance internationale instantanée.
Il a d’abord fallu trouver de l’argent pour survivre. Dans un élan
de générosité, l’École Polytechnique de Montréal et l’Université BiDepuis 1978, le journal a publié deux numéros par année, et les ré- shop’s ont contribué respectivement de 2 000 $ et 1 000 $, ce qui a
dacteurs en chef en ont été successivement :
remis le train sur les rails. Nous sommes aussi infiniment redevables
Serge Dubuc (Montréal) : 1978-1980 (Numéros 1-1 à 3-1) ;
à Octav Cornea, alors directeur de l’ISM, pour avoir cru au futur de
Bernard Aupetit (Laval) : 1980-1983 (Numéros 3-2 à 6-1) ;
la revue au point d’impliquer financièrement l’ISM. Les directeurs de
Robert Cléroux (Montréal) : 1983-1986 (Numéros 6-2 à 9-2) ;
l’ISM qui ont suivi, Olivier Collin et Christian Genest, ont maintenu
Gilbert Labelle (UQAM) : 1987-2000 (Numéros 10-1 à 23-1) ;
cet appui et François Lalonde, alors directeur du CRM, a emboîté le
Ibrahim Assem (Sherbrooke) : 2000-2008 (Numéros 23-2 à 31-1) ; pas. Sans ces généreux montants octroyés au fil des ans, les Annales
Claude Levesque (Laval) : 2008-2012 (Numéros 31-2 à 36-2).
feraient aujourd’hui partie du folklore.
Nous sommes redevables à TOUS les rédacteurs en chef pour leurs Il y a 4 ans, François Lalonde et moi-même avons échangé plusieurs
efforts fort louables. En particulier, Gilbert Labelle a fait un travail longs courriels pour définir ce que devrait être un excellent journal
de moine, et Ibrahim Assem y a mis tout son cœur de 儀ébécois.
mathématique québécois. L’adjectif « québécois » est un peu superPendant plusieurs années, la revue a pu compter sur des subventions flu, mais nous avions en tête d’avoir un journal représentatif des ind’organismes québécois (FAC, FAR, FCAC,…, les noms changeaient térêts des mathématiciens québécois. L’histoire pourra dire que ces
souvent se rappelleront les « vieux de la vieille »), et même sur des deux lascars (FL et CL) concoctaient alors ce qui est devenu maintenant une réalisation : les Annales mathématiques du 儀ébec.
BULLETIN CRM–16
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Le passage à Springer ne fut pas évident. Plusieurs mathématiciens
boyco琀ent les grandes maisons d’édition. Il semble par contre que
certains détracteurs parmi les plus féroces aient accepté l’idée que
les Annales ont plus à gagner qu’à perdre en passant à Springer… du
moins pour le moment. Bien sûr, comme l’affirme avec conviction
mon bon ami Henri Darmon, une revue comme Publications mathématiques de Besançon fait ses frais sans passer par une maison d’édition professionnelle. En passant, ce琀e revue a été créée en 1974, mais
a quand même mis plusieurs années avant d’être reconnue officiellement… (en dehors du cercle des algébristes et des théoriciens des
nombres). Aurions-nous ce琀e même patience ?
et G. Perbet ; C. Junkins et M. Kolster. Pour 37-2 les auteurs sont :
K. I. Ikeda, R. Langlands, E. Ullmo et A. Yafaev.
Il y a plus de deux ans, j’ai donc amorcé des négociations avec Springer et nous en sommes venus à une proposition de contrat. Il me
fallait toutefois préciser à qui les ASMQ appartenaient légalement,
et moi de répondre : « Elle appartient au Groupe des chercheurs en
sciences mathématiques, mais c’est un organisme fantôme ». 儀elle
ingratitude de ma part ! C’est alors que Christian Genest, au début
de son mandat de directeur de l’ISM, a découvert, au terme de nombreuses démarches, que les ASMQ ont un propriétaire légal : l’Association mathématique du 儀ébec. De justesse, je fus sauvé des eaux…
tumultueuses de poursuites légales potentielles grâce aux efforts de
Christian qui a alors finalisé et signé le contrat avec Springer. De
plus, Christian a entrepris des négociations avec l’Association mathématique du 儀ébec et a obtenu que la Fondation Carl-Herz devienne propriétaire des anciennes et des nouvelles Annales.
Ici se termine une autre page de la belle histoire des mathématiques
au 儀ébec.
Il ne faut pas cacher la plus belle partie de ce琀e histoire. Christian
(quel coquin !) a fait accepter une clause à l’effet que Springer doit
verser annuellement des royautés à l’ISM. Le montant n’est pas très
élevé, mais il est loin d’être négligeable.
Lorsque vint le moment de monter un comité éditorial pour
les AMQ, l’expertise de François Lalonde, ses grandes connaissances et son expérience comme éditeur au sein du Mathematische Zeitschri昀 ont été fort utiles. Il fut alors établi que le comité éditorial serait formé des mathématiciens dont on trouve la
liste sur la toile : http://www.springer.com/mathematics/
numbers/journal/40316?detailsPage=editorialBoard. Il
a été convenu que du sang nouveau devait être injecté dans ce comité
éditorial et dans ce琀e perspective, mon mandat d’éditeur-en-chef ne
sera que de deux ans.
Le but avoué du comité de direction (CRM et ISM) de ces nouvelles
Annales est d’en faire un journal (plutôt généraliste) de haut niveau
dont la vocation principale est de publier les meilleurs articles dans
tous les domaines des mathématiques pures, avec parfois des articles
à la frontière des mathématiques et de domaines comme les mathématiques appliquées, la physique mathématique ou l’informatique, à
condition que ces articles satisfassent aux critères de rigueur et d’excellence en mathématiques. Il pourra arriver que certains numéros
du journal soient thématiques, donc portent sur un thème donné. Ce
sera le cas des deux premiers numéros officiels de 2013 des AMQ
qui contiennent neuf articles portant sur la théorie des nombres (au
sens large) et dédiés au Pr Paulo Ribenboim. En voici les auteurs
pour 37-1 : A. Balog, A. Granville et K. Soundararajan ; Y. Bugeaud,
M. Cipu et M. Migno琀e ; H. Cohen et D. Zagier ; J.-F. Jaulent, C. Maire
Nous profitons de l’occasion pour remercier les membres du comité
éditorial des Annales des sciences mathématiques du 儀ébec pour leur
contribution des quatre dernières années : http://www.labmath.
uqam.ca/~annales/redaction.html
Les Annales mathématiques du 儀ébec viennent de prendre un tournant majeur. Il y a eu quelques décisions difficiles à prendre, et
avouons-le, c’est parfois difficile de plaire à Dieu, au diable, à sa
femme et à celle du voisin.
Séminaire de mathématiques supérieures 2013
(continued from page 13)
April 2012, and an MSRI program on Knot homology was hosted in
2010. 吀is arrangement of content contributed to a very coherent
school, and conversely, the school was a very welcome opportunity
for transmi琀ing the results discussed at these events to the larger
group containing the next generation of professional researchers.
To enhance the pedagogical value of the school, the organizers strove
to capitalize on two effects: first of all, the interactions between
mathematics and physics, which has been a spectacularly powerful
force driving progress in the theory of knot invariants. Not only did
students discuss with lecturers a昀er the talks, during coffee breaks,
and beyond, but they also organized several “math-physics clinics,”
in which students with different backgrounds explained some basic
concepts from their respective field to each other. One can be very
confident that these seeds and contact will grow during the coming years. 吀e second notable feature was the many open questions
that each lecturer highlighted towards the end of their lectures. And
these are as many opportunities for the formation of the next generation! As participants prepared to leave Montréal laden with a wealth
of impressions, new friendships, and as much additional knowledge
as they could carry, the organizers realized once again that the ultimate success of the school will really only be measured, several years
down the road, in its contribution to the launch of the participants’
research careers!
As its immediate predecessors, this year’s edition of the SMS
was made possible by financial support from a consortium of
North American Mathematics Institutes, including the Centre de
recherches mathématiques, the Fields Institute, the Mathematical
Sciences Research Institute, the Pacific Institute for the Mathematical Sciences, the Institut des sciences mathématiques, the Canadian
Mathematical Society, the Centre Interdisciplinaire de Recherche en
Géométrie et Topologie. In addition, we received a generous contribution from the Simons Foundation for the a琀ribution of Fellowships
to 20 particularly promising students.
吀e proceedings of the school are expected to appear in the CRM Proceedings–Contemporary Mathematics series, published by the AMS.
BULLETIN CRM–17
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Atelier « Les mathématiques pour une biodiversité en évolution »
16 au 20 septembre 2013
Organisateurs : Amaury Lambert (CIRB, Collège de France), Jona- mathématique de la dynamique adaptative décrivant l’évolution des
than Davies (McGill), Nicolas Lartillot (LBBE, Lyon 1)
traits et, plus récemment, sur des processus de branchement visant
à décrire la diversification des espèces ; enfin, Nicolas Lartillot, spéL’atelier avait pour objectif de jeter des ponts, à la fois entre plusieurs
cialiste de l’application de l’inférence probabiliste bayésienne en géthématiques et entre plusieurs communautés de recherche. Du côté
nétique évolutive, faisant ainsi la jonction entre les perspectives emthématique, la question fondamentale abordée durant l’atelier était
piriques et mathématiques apportées par ses deux collaborateurs.
la suivante : comment comprendre la biodiversité à partir des processus écologiques ? Comment l’écologie se proje琀e-t-elle dans les pro- Globalement, l’atelier s’est avéré un succès. D’après les retours que
cessus opérant à l’échelle macroévolutive ? Enfin, comment faire se nous avons eus, il semblerait que début septembre ne soit pas forcérejoindre les perspectives mécanistique et phénoménologique clas- ment un bon choix pour organiser un colloque ou un atelier, à cause
siquement adoptées sur ces questions ? La perspective mécanistique des enseignements et autres responsabilités académiques qui sont
cherche à analyser les processus écologiques à l’œuvre dans l’évolu- particulièrement prenantes à ce琀e époque de l’année, juste après la
tion des espèces et de leur diversité, tandis que la perspective phé- rentrée des classes. Malgré tout, et en grande partie grâce à nos têtes
noménologique vise plutôt à bien caractériser de façon empirique d’affiches, nous avons fait quasiment salle pleine sur les 3 premiers
le type de distributions décrivant adéquatement la biodiversité en jours de l’atelier, avec une assemblée un tout petit peu plus réduite
évolution. Ultimement, les deux perspectives doivent se rejoindre, (une cinquantaine de personnes) lors de la dernière journée.
le mécanistique expliquant in fine les motifs empiriques effectiveLa semaine semble avoir été intense pour tout le monde. De par sa
ment observés. En pratique, toutefois, la recherche actuelle procède
nature trans-disciplinaire et l’étendue des problématiques abordées,
indépendamment depuis ces deux perspectives, via des communaul’atelier a pris un tour parfois quelque peu éclectique. Ce trait de
tés distinctes. Or, aujourd’hui, le terrain semble mûr pour opérer une
caractère a peut-être même été amplifié par le fait que, afin d’incisynthèse, et cet atelier avait pour but d’en susciter les prémisses.
ter le plus activement possible les interactions et éviter une possible
L’objectif général ainsi décliné est par nature très ambitieux. En assiduité sélective aux conférences en fonction des spécialisations
effet, les questions abordées ici font intervenir aussi bien des mathé- de chacun, nous avions volontairement choisi d’alterner entre des
maticiens théoriciens travaillant sur des questions théoriques très conférences très théoriques, d’orientation plutôt matheuse, et des
pointues concernant les processus stochastiques utilisés pour dé- conférences à caractère plutôt empirique. Toutefois, d’après les recrire l’évolution du vivant, que des biologistes d’orientation beau- tours que nous en avons eus, la nature mosaïque de l’ensemble de
coup plus empiriste. Les deux communautés correspondantes ne se l’atelier ne semble pas avoir été perçue comme quelque chose de
parlent finalement pas si souvent, et ont pour l’essentiel développé problématique mais, au contraire, comme une source de stimulation
des langages et des méthodes distincts. De plus, même à l’intérieur de intellectuelle. Du côté des orateurs, le défi consistant à s’adresser à
chacune des deux communautés, plusieurs lignes de partage existent, une assemblée aussi hétérogène a pu sembler parfois difficile à releen particulier entre ceux qui s’intéressent avant tout à l’évolution ver. Mais d’un autre côté, les interactions ont été très nombreuses,
des traits au sein de chacune des espèces (dynamique adaptative), avec de nombreuses questions posées durant le fil des conférences,
ceux qui se focalisent plutôt sur la dynamique de diversification des afin de favoriser au maximum le dialogue et la compréhension muespèces (forme et structure des arbres phylogénétiques), ou ceux tuelle entre les différentes communautés.
qui me琀ent l’accent sur l’analyse des processus écologiques, le tout,
Enfin, nous avions suggéré aux conférenciers invités de consacrer
sans toujours voir les éventuels couplages entre ces différents phé5 à 10 minutes de leur temps imparti pour aborder des questions
nomènes.
de nature plus spéculative, questions pour lesquelles ils n’auraient
L’atelier était co-organisé par trois chercheurs, dont les formations pas encore de réponses, mais qui leur sembleraient être des quesrespectives incarnaient de façon idéale l’objectif trans-disciplinaire tions fondamentales à me琀re sur l’agenda collectif de la recherche
de l’exercice : Jonathan Davies, biologiste de l’Université McGill, pos- des prochaines années. Ce琀e suggestion a été suivie par certains de
sédant une très bonne connaissance des données empiriques, ainsi nos orateurs, et a eu pour conséquence de vivifier l’ensemble de l’ateque de très bons contacts avec l’ensemble de la communauté des lier, en augmentant la dose de pensée spéculative, ainsi qu’en faisant
biologistes d’orientation empiriste dans le domaine de l’écologie et partager avec l’assemblée la recherche en action, plutôt que celle déjà
de l’évolution ; Amaury Lambert, mathématicien chevronné, proba- accomplie.
biliste ayant travaillé à la fois sur des questions de formalisation
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CRM–ISM 2013–2014 Postdoctoral Fellows
CRM–ISM 2013–2014 Postdoctoral Fellows
James Maynard, Ph.D. University of Oxford, 2013
Supervisor: Andrew Granville (Montréal)
I work on problems in analytic number theory. My main research focus is on using sieve
methods to approach classical questions on
the distribution of primes. 吀is has included
bounds on the number of primes in an arithmetic progression, estimates for the size of
gaps between primes and almost-prime approximations to the Hardy–Li琀lewood prime
k-tuples problem. Currently, in addition to extensions of the above, I am working on the issue of existence of infinitely many primes in certain ‘thin’ polynomial sets.
David Belius, Ph.D. ETH Zürich, 2013
Supervisors: Louigi Addario-Berry (McGill), Louis-Pierre Arguin
(Montréal)
My main research interest is probability theory, especially problems with a “discrete
flavour.” During my Ph.D. I have studied random walks on graphs and the random interlacement model. One particular question
I have looked at is how long it takes for a
random walk to visit every vertex of a finite
graph. 吀is random time is called the cover
time. My thesis contains central limit theorem
type results about the fluctuations of the cover time around its typi- CRM Thematic Postdoctoral Fellows
cal value for some particular graphs.
Yu-Ting Chen, Ph.D. University of British Columbia, 2013
吀omas Bothner, Ph.D. IUPUI, 2013
Supervisors: Louigi Addario-Berry (McGill), Sabin Lessard (MontSupervisors: Marco Bertola, John Harnad, Dmitry Korotkin (Con- réal), Lea Popovic (Concordia)
cordia)
My research interests are in probability theMy work is related to the modern theory
ory and stochastic processes, with a foof integrable systems, dealing with problems
cus on problems from mathematical biology.
from the theory of integrable nonlinear ODEs
More specifically, the stochastic processes
(the Painlevé type equations), from exactly
which I study include Lévy processes, intersolvable statistical physics models (unitary
acting particle systems, superprocesses, and
and normal random matrix ensembles as well
stochastic partial differential equations. In the
as exact solutions to the six-vertex model)
last few years, I have made progress in diverse
and from random growth models (directed
problems. 吀e problems concern some conpolymers in random media and the asymmet- nections between voter models on finite spatial structures and evoric simple exclusion process). 吀is work falls lutionary game theory, and the pathwise uniqueness problem for the
into the category of mathematical physics and stochastic partial differential equation (SPDE) of one-dimensional
uses tools from various areas of mathematics, such as probability super-Brownian motion with immigration. My curiosity for intertheory, asymptotical and spectral theory of differential equations acting particle systems in biological models subsequently led me to
and Riemann surface theory. I plan to continue working on various the study of voter models on large spatial structures and evolutionproblems connected to the aforementioned topics but at the same ary dynamics in general structured populations. Besides, I am now
time wish to widen my scope through collaborations with the mem- interested in SPDEs arising from stochastic models closely related to
bers of the mathematical physics laboratory at CRM.
branching particle systems.
Michael Brandenbursky, Ph.D. Technion, 2010
Supervisors: Octav Cornea, François Lalonde (Montréal), Steven
Boyer, Olivier Collin (UQAM)
My research is in the areas of geometry and
topology. I am interested in a variety of topics which include: finite type knot invariants,
braid and mapping class groups of surfaces,
groups of Hamiltonian and symplectic diffeomorphisms of symplectic manifolds and interesting metrics on these groups. 吀is past
year, I was trying to understand properties of
the autonomous metric on groups of Hamiltonian diffeomorphisms. 吀e understanding of
this metric requires sophisticated tools such
as quasi-morphisms and configuration space integrals.
Daniele Rosso, Ph.D. University of Chicago, 2013
Supervisors: Erhard Neher, Alistair Savage (O琀awa), Vyjayanthi
Chari (UC Riverside)
My research is in the representation theory of Lie groups, using also methods from
algebraic geometry and combinatorics. In
particular, I am interested in constructions
related to flag varieties, especially in the
‘mirabolic’ se琀ing. Analogues of the Robinson–Schensted–Knuth correspondence appear when studying the geometry of these
varieties. I am also interested in convolution
algebras over flag varieties, which include Iwahori–Hecke algebras
and other related algebras, like affine Hecke algebras.
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Rajendran Venkatesh, Ph.D. IMSc., Chennai, 2013
Supervisors: Erhard Neher, Alistair Savage (O琀awa), Vyjayanthi
Chari (UC Riverside)
My research interests are in representation
theory of infinite-dimensional Lie algebras.
In my thesis I studied the problem of unique
factorization of tensor products for representations of Kac-Moody algebras. Currently
I am interested in studying the category of
finite dimensional representations of current
algebras. One of the original motivations for
the study of this category is that it is closely
related to the representation theory of the
corresponding quantum affine algebra. I have recently obtained a
generators-and-relations description for a class of modules given by
fusion products.
Yuxiang Zhang, Ph.D. Memorial University, 2012
Supervisors: Frithjof Lutscher (O琀awa), Frédéric Guichard
(McGill)
My research interests mainly focus on mathematical ecology and epidemiology. More
specifically, I use the theory of differential
equations and dynamical systems to study
dynamical behaviours of various biological
models. 吀e research involves mathematical modelling, mathematical analysis and numerical simulations. In my recent work, the
existence of spreading speeds and traveling
waves has been investigated for the ecological and epidemiological models with spatial dispersal. Currently,
I am working on a joint project with my postdoctoral supervisors.
In this project, we are interested in the stability and synchronization
of a predator-prey metacommunity model with density-dependent
dispersal and travel time delay. Some interesting theoretical results
as well as numerical results have been obtained.
In Memoriam
Muriel Pasquale琀i
de Béatrice Kowaliczko-Young
Le 26 mai 2013, est décédée à Montréal aux soins palliatifs de l’hôpital
Maisonneuve-Rosemont, mon ancienne collègue et amie Muriel Pasquale琀i-Kozikowski. Nous avions
été collègues pendant quatre ans
au Centre de recherches mathématiques. (…)
Pendant ses premières semaines au
CRM, en 1998, Muriel disparaissait
à l’heure du lunch soit pour rentrer
chez elle soit pour aller à deux ou
Muriel
trois stations de métro de l’université faire des mots croisés ou Sudokus en mangeant un sandwich.
Une sorte de fuite. Pour affronter le froid, elle avait acheté deux
manteaux Kanuk d’un coup : un beige pour aller jusqu’à −10 et un
noir pour aller jusqu’à −30 et plus. Puis, s’acclimatant, elle resta autour du bureau à midi. On l’entendait baisser le rideau métallique du
secrétariat pour assurer sa tranquillité, mais on l’entendait rire de
temps à autre avec sa collègue bien-aimée, sa chum, Michèle Gilbert.
Pour faire couleur mathématicienne les deux comparses s’étaient rebaptisées Mme Wavele琀e et Mme Bootstrap, termes qu’on entendait ou
lisait souvent sans, évidemment, savoir trop ce que ça voulait dire…
Ce fut le début du phénomène Muriel dont je ne peux témoigner
que jusqu’en 2002, date à laquelle j’ai qui琀é l’U de M. Travailler
au CRM était un plaisir. Dirigé par l’équipe Vinet–Hurtubise–SaintAubin soutenue par les piliers Louis Pelletier, Louise Letendre, Diane
BULLETIN CRM–20
Brulé, Josée Laferrière, André Montpetit, Daniel Ouimet, Diane Poulin, Vincent Masciotra, l’activité y était intense, membres, visiteurs,
postdocs souvent du monde entier, avaient tous affaire au secrétariat
où Muriel régnait, accueillant tout le monde gracieusement, patiemment avec l’humour fin et parfois corrosif qui la caractérisait. Elle
introduisit la tradition des petites gâteries et fleurs déposées en permanence sur le comptoir, celle des cartes postales affichées sur le
tableau (gare à ceux qui n’envoyaient rien pendant leur voyage !).
Elle avait inventé avec Louis Pelletier des concours de bêtises pour
leur chien respectif La Chloé et Molière : ils leur faisaient faire des
tests qu’ils avaient concoctés le soir et le lendemain comparaient les
scores. Ils concluaient la plupart du temps « ils sont vraiment cons
ces chiens ! »
Elles avaient des petites a琀entions pour tous, pour les anniversaires
notamment. À cet égard, elle m’en a beaucoup voulu quand j’ai malencontreusement divulgué qu’elle allait avoir 50 ans. Née un 31 décembre c’était donc toujours la double fête : la Saint-Sylvestre et son
anniversaire. Il faut dire que la fête, elle aimait ça Muriel. Elle adorait
boire, danser, fumer, rire… On est d’origine polonaise ou on ne l’est
pas. Je la vois encore se précipiter vers le centre de la piste de danse
dès que ça jouait I need your love de Marc Anthony ou Mambo #5…
儀and sa maladie a été diagnostiquée, elle a a琀eint une grandeur
exemplaire par sa résignation, générosité, humilité, n’arrêtant pas
de souligner la chance qu’elle avait d’être si bien soignée par du personnel si gentil, faisant face à l’inéluctable avec clairvoyance, lucidité, organisation. Elle s’excusait presque d’être malade. Elle disait
« Je n’ai pas peur. »
crm.math.ca
Fi昀h Montréal Problem Solving Workshop — A CRM–Mprime Event
August 19–23, 2013
Odile Marco琀e (CRM)
Since 2007 the CRM has organized five problem solving workshops.
吀e word “industrial” appeared in the title of each of the first four
workshops, but the fi昀h workshop, which took place from August 19
to August 23 at the CRM, broke new ground (at least for Montréal) by
including a section on problems arising in connective tissue physiology. Indeed mathematics was used for modelling natural phenomena
long before it was applied to industrial problem solving. 吀e section
on physiology was organized by Svetlana Komarova (from the Faculty of Dentistry at McGill University) and Nilima Nigam (who holds
a Canada Research Chair in Applied Mathematics at Simon Fraser
University). Together they found six problems pertaining to connective tissue physiology (including the physiology of bone formation
and resorption).
吀ree of the physiology problems were presented by members of the
Matrix Dynamics Group (University of Toronto). Yongqiang Wang
was looking for a model of the cell-cell fusion stage within the differentiation process of osteoclasts. (Osteoclasts are cells that are responsible for the destruction of mineralized tissues such as bone and
teeth.) Hamid Mohammadi was looking for a model that will allow
researchers to understand the mechanical behaviour of collagens under adherent contractile gels. Christopher McCulloch wanted to understand how the presence or absence of certain focal adhesion proteins affects focal adhesion size and interleukin-related downstream
signals.
吀e three other physiology problems dealt with issues that are very
important for the understanding of bone-related pathology. Stephen
Sims wanted to develop, with mathematicians, a model that takes
into account the function of different regulators of calcium levels
in the differentiation of osteoclast precursors. Frank Rauch wished
to model the action of a protein called PEDF (Pigment-Epithelium
Derived Factor) on bone mineralization. Such a model would shed
light on the disease Osteogenesis imperfecta type VI, which leads
to a large number of fractures in children suffering from it. Laura
Stone has developed a mouse model of progressive low back pain,
which is related to degenerative disc disease and neuronal plasticity. In describing her problem at the workshop, she was hoping to
get some help in the design of a statistical model for determining
the relationships (including causal relationships) between pain, disc
degeneration, and neuronal plasticity in mice as they age.
tin Frasch, a researcher at the Sainte-Justine hospital for children in
Montréal, explained that the measurement of a foetus’s heart rate
is not sufficient for detecting acidemia, a condition associated with
an increase in the risk of brain injury at birth. His research team
has shown that a fetal electroencephalogram (EEG) can be obtained
during labour and could improve the chances of detecting acidemia.
For Martin Frasch, the goal of the workshop was to develop a neuronal model for mimicking the EEG signals obtained in animal experiments.
吀e problem proposed by Jean-Claude Rizzi and Stéphane Alarie,
the representatives from IREQ, concerned the reduction of energy
losses in the Hydro-儀ébec transmission network. Such losses depend upon the electrical resistance of power lines and transformers,
as well as the currents going through the equipment. Energy losses
must be reviewed every hour because the current varies during the
day. Tackling this problem requires the formulation and solution of
a difficult nonlinear programming problem. Patrick St-Louis, from
GIRO, proposed a problem that arises in the planning of transportation for persons with reduced mobility. 吀e sequence of stops along
the route has already been determined, but not the time at which
the transportation provider (bus, taxi, etc.) will visit a specific stop.
Given the utility functions of the customers and the various constraints involved (time windows, maximum span of a customer trip,
etc.), the problem consists of finding the stop times maximizing the
total utility.
Finally, Éric Presco琀-Gagnon, from JDA so昀ware, presented a packing problem faced by major retail chains. Such a chain must pack
“boxes” containing different mixes of clothing items (say). On the
one hand, the number of box types must not be too large (in order
to keep the operating costs within a reasonable range). On the other
hand, the number of box types must be large enough to enable the
retail chain to send to each store the mix of clothes that has been
requested (more or less). 吀e work of the team studying this problem, led by Louis-Martin Rousseau, was a wonderful illustration of
the synergy present in problem solving workshops. 吀e team formulated the problem as an integer programming problem and tried to
solve it by integer programming techniques. 吀e first a琀empts were
not successful! 吀e team members were helped by two participants
whose primary expertise is not in integer programming: Chris Breward (from Oxford) and Winston Sweatman (from New Zealand).
吀e workshop also included a general section, featuring a problem
吀ey quickly found a solution by looking at the data! All the refrom obstetrics and three problems from industry (supplied by IREQ,
searchers involved are now trying to generalize the method used for
GIRO, and JDA So昀ware, respectively). Both IREQ and GIRO had
this particular data set…
already proposed problems in the 2008 and 2009 workshops. Mar-
BULLETIN CRM–21
crm.math.ca
Le Bulletin du CRM
Volume 19, No 2
Automne 2013
Le Bulletin du CRM est une lettre d’information à contenu scientifique, faisant le point sur les actualités du Centre
de recherches mathématiques.
ISSN 1492-7659
Le Centre de recherches mathématiques
(CRM) a vu le jour en 1969. Actuellement dirigé par Luc Vinet, il a pour objectif de servir de centre national pour
la recherche fondamentale en mathématiques et leurs applications. Le personnel scientifique du CRM regroupe
plus d’une centaine de membres réguliers et de boursiers postdoctoraux. De
plus, le CRM accueille chaque année
entre mille et mille cinq cents chercheurs du monde entier.
Le CRM coordonne des cours de cycles
supérieurs et joue un rôle prépondérant (en collaboration avec l’ISM) dans
la formation de jeunes chercheurs. On
retrouve partout dans le monde de nombreux chercheurs ayant eu l’occasion de
parfaire leur formation en recherche au
CRM. Le Centre est un lieu privilégié de
rencontres où tous les membres bénéficient de nombreux échanges et collaborations scientifiques.
Le CRM tient à remercier ses divers partenaires pour leur appui financier à sa
mission : le Conseil de recherches en
sciences naturelles et en génie du Canada, le Fonds de recherche du 儀ébec– Nature et technologies, la National Science Foundation, l’Université
de Montréal, l’Université du 儀ébec à
Montréal, l’Université McGill, l’Université Concordia, l’Université Laval, l’Université d’Ottawa, l’Université de Sherbrooke, le réseau Mitacs, ainsi que les
fonds de dotation André-Aisenstadt et
Serge-Bissonnette.
Directeur : Luc Vinet
Directrice d’édition : Galia Dafni
Conception : André Montpetit
Centre de recherches mathématiques
Université de Montréal
C.P. 6128, succ. Centre-ville
Montréal, QC H3C 3J7
Téléphone : 514.343.7501
Courriel : [email protected]
Le Bulletin est disponible à :
crm.math.ca/docs/docBul_fr.shtml.
BULLETIN CRM–22
Publications
New CRM–AMS Agreement
吀is year a new version of the agreement between the CRM and the American Mathematical Society, dating from 1992, came into
effect, placing the CRM Proceedings and Lecture Notes (CRMP) within the AMS’s prestigious Contemporary Mathematics series. 吀e
last volume in the old CRMP series (volume 56)
appeared earlier this year. 吀e two proceedings
volumes described on this page will be the first
to appear in the new CRMP-CONM series.
Tropical and Non-Archimedean
Geometry
Omid Amini, Ma琀hew Baker, and
Xander Faber, editors
Over the past decade, it has become apparent that tropical geometry and nonArchimedean geometry should be studied in
tandem; each subject has a great deal to say
about the other.
Women in Numbers 2
Research Directions in Number
Theory
Chantal David, Matilde Lalín, and
Michelle Manes, editors
吀e second Women in Numbers workshop
(WIN2) was held November 6–11, 2011, at the
Banff International Research Station (BIRS)
in Banff, Alberta, Canada. During the workshop, group leaders presented open problems
in various areas of number theory, and working groups tackled those problems in collaborations begun at the workshop and continuing long a昀er.
吀is volume collects articles wri琀en by participants of WIN2. Survey papers wri琀en
by project leaders are designed to introduce areas of active research in number theory to advanced graduate students and recent PhDs. Original research articles by the
project groups detail their work on the open
problems tackled during and a昀er WIN2.
Other articles in this volume contain new research on related topics by women number
theorists.
吀is volume is a collection of articles dedicated to one or both of these disciplines.
Some of the articles are based, at least in part,
on the authors’ lectures at the 2011 Bellairs
Workshop in Number 吀eory, held from May
6–13, 2011, at the Bellairs Research Institute,
吀e articles collected here encompass a wide
Holetown, Barbados.
range of topics in number theory including
Lecture topics covered in this volume include Galois representations, the Tamagawa numpolyhedral structures on tropical varieties, ber conjecture, arithmetic intersection forthe structure theory of non-Archimedean mulas, Mahler measures, Newton polygons,
curves (algebraic, analytic, tropical, and the Dwork family, elliptic curves, cryptograformal), uniformization theory for non- phy, and supercongruences.
Archimedean curves and abelian varieties,
and applications to Diophantine geometry. WIN2 and this Proceedings volume are part
Additional articles selected for inclusion in of the Women in Numbers network, aimed
this volume represent other facets of cur- at increasing the visibility of women rerent research and illuminate connections be- searchers’ contributions to number theory
tween tropical geometry, non-Archimedean and at increasing the participation of women
geometry, toric geometry, algebraic graph mathematicians in number theory and retheory, and algorithmic aspects of systems of lated fields.
polynomial equations.
Call for Proposals
吀e Centre de recherches mathématiques is soliciting applications for scientific activities
to take place at the CRM. Proposals should be sent to the CRM by email at: proposal@
crm.umontreal.ca. More details and instructions for the various types of proposals
can be found at http://crm.math.ca/en/act/form/propositions_an.shtml.
crm.math.ca
Luc Vinet, New Director of the CRM
Dr. Geneviève Tanguay, Vice-Rector (Research, Creation and Innovation) at the Université de Montréal, and Chair of the Board of Directors of the Centre de recherches mathématiques (CRM), announced
the appointment of Professor Luc Vinet as Director of the CRM. 吀is
nomination took effect on July 20, 2013. Dr. Vinet succeeds Professor
François Lalonde who headed the CRM from 2004 to 2008 and from
2011 to 2013.
A distinguished mathematical physicist, Luc Vinet has dedicated a
significant part of his career to the organization, and development
of research.
Luc Vinet obtained a first doctorate in theoretical physics at the Université Pierre et Marie Curie in 1979 and a second one at the Université de Montréal in 1980. He was subsequently a postdoctoral fellow
at the Massachusse琀s Institute of Technology (MIT) and he returned
to Montréal in 1982 as University Research Fellow of the Natural
Sciences and Engineering Research Council of Canada (NSERC). In
1992, he was appointed full professor at the Université de Montréal.
His scientific work focuses generally on the exact solution of physical models through the study of symmetries and their description
in terms of algebraic structures. He has achieved major advances in
gauge theory, algebraic combinatorics, supersymmetry, superintegrability, special function theory and quantum transport.
funding doubled. He established the Réseau québécois de calcul de
haute performance (RQCHP), as well as the Network for Computing and Mathematical Modeling (ncm2 ) and the Laboratoire universitaire Bell (LUB). He was one of the founding directors of the national network of centres of excellence MITACS.
In 1999, he became Vice-Principal (Academic) and Provost at McGill
University. He returned to the Université de Montréal in 2005 as its
Rector until 2010. Luc Vinet has received numerous distinctions. In
particular, he was awarded the prestigious Prix Armand-Frappier by
the 儀ébec government in 2009. He was also the recipient in 2012
of the CAP-CRM Prize for his research in theoretical and mathematical physics. He is presently Aisenstadt Professor of Physics at the
Université de Montréal.
As Director of the CRM, Luc Vinet will be responsible for all the operations of the institute and in particular, for its scientific programming.
We are pleased to welcome Professor Vinet as Director of the CRM.
We wish him the greatest success in his functions.
We also wish to take this opportunity to warmly thank outgoing
Director François Lalonde, who has succeeded remarkably in furthering the CRM’s development.
Luc Vinet has already been Director of the Centre de recherches
mathématiques from 1993 to 1999. Under his leadership, the CRM
To Appear
CRM Monograph Series, volume 32
Random Matrices and the Six-Vertex Model
Pavel Bleher and Karl Liechty
CRM Monograph Series, volume 33
Classification and Identification of Lie Algebras
Libor Šnobl and Pavel Winternitz
吀is book provides a detailed description of the Riemann–Hilbert
approach (RH approach) to the asymptotic analysis of both continuous and discrete orthogonal polynomials, and applications to random matrix models as well as to the six-vertex model. 吀e RH approach was an important ingredient in the proofs of universality in
unitary matrix models. 吀is book gives an introduction to the unitary matrix models and discusses bulk and edge universality. 吀e
six-vertex model is an exactly solvable two-dimensional model in
statistical physics, and thanks to the Izergin–Korepin formula for the
model with domain wall boundary conditions, its partition function
matches that of a unitary matrix model with nonpolynomial interaction. 吀e authors introduce in this book the six-vertex model and
include a proof of the Izergin–Korepin formula. Using the RH approach, they explicitly calculate the leading and subleading terms in
the thermodynamic asymptotic behavior of the partition function of
the six-vertex model with domain wall boundary conditions in all
the three phases: disordered, ferroelectric, and antiferroelectric.
吀e purpose of this book is to serve as a tool for researchers and practitioners who apply Lie algebras and Lie groups to solve problems
arising in science and engineering. 吀e authors address the problem
of expressing a Lie algebra obtained in some arbitrary basis in a more
suitable basis in which all essential features of the Lie algebra are directly visible. 吀is includes algorithms accomplishing decomposition
into a direct sum, identification of the radical and the Levi decomposition, and the computation of the nilradical and of the Casimir
invariants. Examples are given for each algorithm.
For low-dimensional Lie algebras this makes it possible to identify
the given Lie algebra completely. 吀e authors provide a representative list of all Lie algebras of dimension less or equal to 6 together
with their important properties, including their Casimir invariants.
吀e list is ordered in a way to make identification easy, using only
basis independent properties of the Lie algebras. 吀ey also describe
certain classes of nilpotent and solvable Lie algebras of arbitrary finite dimensions for which complete or partial classification exists
and discuss in detail their construction and properties.
BULLETIN CRM–23
crm.math.ca
Mot du directeur
« Vingt ans après » est le titre du livre d’Alexandre Dumas qui donne
suite à son ouvrage intitulé « Les Trois Mousquetaires ». Vingt ans
après ma première nomination en 1993, on me fait à nouveau l’honneur de me nommer directeur. Avec ce clin d’œil, souhaitons au CRM
pour les années à venir une suite d’aventures scientifiques fort heureuses dans la foulée de celles qu’il n’a cessé de connaître.
Si une chose me frappe en revenant, c’est l’épanouissement et la vitalité dont fait preuve le CRM. Ce développement qu’il a connu est
largement imputable à François Lalonde qui a tenu la barre pendant
de longues périodes au cours des dix dernières années. J’en profite
pour le remercier chaleureusement du talent et de l’énergie avec lesquels il a mené les destinées du CRM. Je remercie également ceux et
celles qui ont effectué des intérims durant ce琀e période.
rapport qui sera présenté au comité d’évaluation des ressources qui
transme琀ra ses recommandations au CRSNG. À tout évènement, la
préparation de ce琀e demande ARTCMS a permis de mesurer la très
haute tenue du CRM et de vérifier qu’il fait partie de l’élite mondiale
des instituts en sciences mathématiques.
Le CRM, l’IF et PIMS préparent de plus conjointement une autre
demande auprès du CRSNG. Ce琀e demande est intitulée Initiative
d’Innovation des Instituts et l’on s’y réfère en l’appelant initiative
Triple « I ». Elle vise à obtenir du financement afin de développer une
écologie propice à la collaboration entre chercheurs en sciences mathématiques des milieux académiques et vis-à-vis de l’industrie et du
gouvernement. La demande sera présentée dans le cadre d’un nouveau programme nommé Plateforme d’innovation d’instituts. Il s’agit
a priori d’un programme pilote d’une durée de trois ans et doté d’un
financement total de 1,5M $. Une le琀re d’intention a été transmise
et a déjà donné lieu à des discussions entre les trois directeurs et
les officiers responsables du CRSNG. Une version remaniée de ce琀e
le琀re d’intention sera transmise au CRSNG d’ici la fin novembre.
Il est prévu qu’une partie du comité de visite d’évaluation des ressources discute aussi de ce琀e demande avec les trois directeurs lors
de la rencontre à O琀awa en janvier avant que l’application ne soit
développée dans sa forme complète.
Le CRM tire une très grande fierté du leadership extraordinaire qu’a
démontré Christiane Rousseau en faisant en sorte que le projet Mathématiques de la planète Terre (MPT) 2013 qu’elle a conçu, soit
adopté par nombre de pays et d’instituts et ait eu, au cours de l’année
qui s’achève, une portée internationale tout à fait remarquable. Plusieurs activités du CRM se sont déroulées sous ce琀e bannière ce琀e
année : trois ateliers dans le cadre du programme pan-canadien sur
les Modèles et méthodes en écologie, épidémiologie et santé publique,
un autre en Mécanique céleste, diverses conférences pour le grand public et un semestre thématique intitulé Biodiversité et évolution qui se Voilà donc un peu d’information sur ce qui nous occupera aux cours
conclut. Il faudra donner des suites à l’initiative MPT 2013 pour op- des prochaines semaines. Je termine ce mot en exprimant ma recontimiser les retombées de ce琀e importante mobilisation.
naissance envers vous tous (personnel, chercheurs, administrateurs
L’année 2014 s’ouvrira au CRM avec un semestre dédié aux Nou- et autres) qui contribuez à faire du CRM un endroit particulièrement
velles avenues en théories de Lie. Les activités débuteront dès janvier propice à la recherche et à la formation et ce, avec la conviction réet comprendront 4 écoles préparatoires qui seront suivies de quatre confortante que l’on pourra continuer de compter sur votre précieuse
collaboration.
ateliers.
Cet automne, le CRM a été mobilisé par la rédaction de la demande de
subvention au Programme d’appui aux ressources thématiques et collaboratives en mathématiques et en statistique (ARTCMS) du CRSNG.
Le CRM reçoit présentement 1,2M $ par année du CRSNG et dépend
de façon cruciale de ce financement. Le programme ARTCMS a été
créé dans la foulée de l’adoption par le CRSNG du plan à long terme
pour la recherche en mathématiques et en statistique au Canada. La
ronde de ce琀e année implique le CRM, l’Institut Fields (IF), PIMS
ainsi que AARMS (Atlantic Association for Research in the Mathematical Sciences) et l’INCASS (Institut canadien des sciences statistiques). L’enveloppe totale est de 3,5M $ sur une base annuelle, ce qui
correspond au financement actuel du CRM, IF et PIMS pris ensemble.
Il faut se réjouir de ce琀e reconnaissance des sciences mathématiques
et statistiques par le CRSNG et de la stabilité du financement qui y
est associé. Il faut observer cependant que nos budgets deviennent
de plus en plus contraints. Un défi qui nous incombe pour les années à venir est de convaincre de l’importance de davantage laisser
libre cours à cet engin d’innovation qu’est le CRM et de persuader
conséquemment du bien fondé d’accroitre son financement. Revenant à notre demande, nous recevrons les rapports des évaluateurs
externes vers la fin décembre et nous rencontrerons à O琀awa au début de janvier le comité de visite d’évaluation. Ce dernier rédigera un
BULLETIN CRM–24
Luc Vinet, directeur
Le vendredi 25 octobre 2013, François Lalonde a donné une conférence dans le cadre
du Colloque CRM-ISM. Le CRM a profité de l’occasion pour remercier François suite
au travail remarquable qu’il a accompli au cours de ses années à la direction du
CRM, avec une réception en son honneur. Luc Vinet lui a remis un cadeau en présence de Serge Brochu (vice-recteur adjoint à la recherche, Université de Montréal)
et de Jacques Hurtubise (McGill, aussi ancien directeur du CRM).