吀ematic Semester on New Directions in Lie 吀eory - Université ...
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
C CENTRE
R
M
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 contem- by Alistair Savage will serve to introduce
porary mathematics. 吀e structure and students to the new and exciting field of
representation theory of Lie groups and categorification. Its goal is to prepare stu-
Lie algebras have resulted in impor- dents for the workshop Geometric Repre-
tant applications in physics and other sentation 吀eory and Categorification. In
branches of mathematics and, in turn, the first week, a recent approach to the
Lie theory has benefited from these representation theory of the symmetric
connections. 吀e pioneering work of group will be presented. 吀is will pro-
Kac, Frenkel, Lepowsky and Meurman vide students with a novel approach to
linked the representation theory of the subject that is well suited to the ideas
Kac–Moody algebras to the Monster of categorification. In the second week,
and led to the theory of vertex alge- the basic ideas of categorification will be
bras. Another example is the theory presented, with a special emphasis on ex-
of quantized enveloping algebras initi- plicit examples (drawing, in particular,
ated by Drinfeld and Jimbo, with roots from the material in the first week). 吀e
in the work of Fadeev, Reshetikhin second course of the winter school, In-
and Takhtajan on integrable systems. troduction to Kac–Moody and Related Lie
吀e work of Lusztig and Kashiwara on Algebras by Erhard Neher, will comple-
quantum groups led to major break- ment the typical first course on Lie al-
throughs in the representation theory gebras offered at many universities. It
of finite-dimensional simple Lie alge- will present the basic structure theory of
bras and Kac–Moody algebras. 吀e the- Kac–Moody algebras with a special em-
ory of canonical bases (due to Lusztig) phasis on affine Lie algebras. Some re-
and global crystal bases (due to Kashi- lated non-Kac–Moody algebras, such as
wara) of finite-dimensional simple representations of simple Lie al- toroidal algebras, will also be presented. 吀is will help prepare stu-
gebras resulted in dramatic developments and new areas of research dents for V. Chari’s course (see below).
in geometric and combinatorial representation theory. Recent work A second Winter School will take place February 24–March 7. 吀e
in the area of categorification has had a major impact on both rep- course Representation 吀eory of Semisimple and Affine Kac–Moody
resentation theory and low dimensional topology. Algebras by Vyjayanthi Chari will focus in the first week on the cat-
Lie theory has a long and successful tradition in Canada, as do com- egory O for finite-dimensional simple Lie algebras. 吀e second week
binatorics and the representation theory of associative algebras. 吀e will deal with the case of affine Lie algebras. 吀e focus will be on in-
topics of the program workshops have been chosen to reflect the new tegrable weight representations. 吀e connections between positive
connections between these subjects. 吀e overall goals of the thematic level representations of affine Lie algebras and the finite-dimensional
semester are to highlight current research in Lie theory and its ap- representations of the maximal parabolic subalgebra of the affine Lie
plications to other fields, to foster interaction between Canadian and algebra will be made. Finally, the course will discuss how the meth-
foreign researchers working in this area, and to provide a forum for ods and problems used and studied in category O can be formulated
young mathematicians to learn about the current trends in the sub- and studied to understand the category of finite-dimensional repre-
ject and to interact with the leading experts in this exciting field. sentations of the affine Lie algebra. 吀e second course of the winter
吀e program will begin with a Winter School taking place Jan- school, Vertex Algebras for Mathematicians by Michael Lau (Laval),
uary 6–17. It will feature two courses aimed at graduate students will introduce vertex algebras and operator product expansion from
and postdoctoral fellows. 吀e course Introduction to Categorification first principles, as well as touch on their applications to Lie algebras,
crm.math.ca
representation theory and modular forms. A number of examples gether Lie theorists and mathematical physicists, in the expectation
will be featured, including vertex Lie algebras, la琀ice vertex algebras
that further dialogue will stimulate additional interesting and im-
and W-algebras. If time permits, there will be a discussion of some portant breakthroughs in both domains. 吀e workshop will focus
of the geometry around vertex algebras in the context of conformal on groups and algebras in quantum theory (in particular, infinite-
blocks. dimensional 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 “categorifica-
tween algebraic groups, quantum groups and Kac–Moody Lie groups tion” was coined by Louis Crane and Igor Frenkel. Broadly speak-
is 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 applica-
ory 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 quan-
Two 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 cer-
the structure of these basic objects and their representation theory.
tain class of commutative rings, equipped with distinguished gen-
Many 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 realiza-
as part of an a琀empt to provide an algebraic framework for under-
tion of some algebraic structure) is o昀en a precursor to categorifica-
standing 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 cen-
procedure 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 ex-
examine 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 cate-
and 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 Insti-
theory 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 quan-
field theory in critical phenomena in statistical models is studied by tum groups. Most relevant for the thematic program is his develop-
recent 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.
BULLETIN CRM–2crm.math.ca
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 these developments, and emphasized that the time has come for a
programs two or three world-famous mathematicians for one-week to renewed exploration of questions originating in celestial mechanics
one-semester stay. 吀e recipients of the chair give a series of lectures and space transportation by using all the sophisticated tools at our
on subjects chosen for their relevance and impact within the thematic disposal today.
program. One of these lectures, in compliance with the donor André
Aisenstadt’s wish, must be accessible to a broad audience.
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
University and Peking University) are described below. lectures during the events surrounding the workshop 吀e Topology
of 3-dimensional manifolds. Professor Gabai is a world leader in the
Helmut Hofer topology and geometry of low-dimensional manifolds, with many
by Octav Cornea (Université de Montréal) outstanding research contributions throughout a career in which
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 Ameri-
Princeton. 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 lec-
Helmut 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 contribu-
work, 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 sem-
instein conjecture in dimension three was transformational. It led to inal nature as well as their im-
the 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 num-
usual 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 man-
of 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 va-
geometry 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 Gro-
last twenty years the symplectic machinery that follows this general mov boundary of the curve complex of S, and the set of doubly
R
scheme has developed tremendously. Hofer’s talks reviewed part of degenerate hyperbolic structures on S × . Most of his time was
BULLETIN CRM–3crm.math.ca
devoted, though, to explaining his recent results in the area. For in-
stance, if S has genus g and p punctures, then its ending lamination
Invariant Subspaces of the Shi昀 Operator
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 lamina-
R
tion 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)
Gang Tian 吀e main theme of this workshop was the invariant subspaces
by Vestislav Apostolov (UQAM) 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,
Professor Gang Tian, who is Eugene Hig-
de Branges–Rovnyak spaces H(b), where b is an element of the
gins Professor at Princeton and the direc-
closed unit ball of H ∞ , and model spaces KΘ , where Θ is an inner
tor of the Beijing International Centre for
function on the open unit disk, were at the center of a琀ention. 吀e
Mathematical Research (BICMR), gave a
first two days of the conference were devoted to four mini-courses,
series of three Aisenstadt lectures dur-
each three hours long, on the above mentioned spaces. Besides
ing the workshop Extremal Kähler met-
the majority of invited speakers, more than 25 graduate students
rics. In global analysis and geometry over
and postdoctoral fellows participated in these lectures by world-
the last 25 years, Professor Gang Tian has
renowned experts (K. Dyakonov, S. Garcia, D. Timotin, T. Ransford).
been one of the most influential and ver-
satile figures. From his Ph.D. thesis to his 吀e last three days were filled with 21 special talks by invited speak-
most recent work, his contributions have ers. With more than 55 participants, this part was a great success.
been distinguished by their diversity and Many new features of function spaces and operators were discussed
Gang Tian depth. His most striking work to date con- during and also a昀er the sessions. One of the touching parts of the
cerns the existence of Kähler–Einstein metrics on complex mani- conference was the beautiful presentation by Carl C. Cowen on the
folds (which was a main theme of the workshop), but he has made Invariant Subspaces Problem (ISP). Analysts believe that the ISP is
many other significant contributions. For instance, he established one of the great unsolved problems, at least in operator theory. In
(with Y. Ruan) the associativity of the quantum cohomology ring January 2013, Carl C. Cowen and his collaborator Eva Gallarda an-
of a symplectic manifold, constructed (with Jun Li) moduli spaces nounced that they had solved the ISP. Unfortunately, a short while
of curves in algebraic symplectic geometry, and elaborated (with later, a gap was found in their proof. However, their efforts were not
J. Morgan) a clear and detailed exposition of Perelman’s proof of the totally in vain and they have succeeded in opening many new fron-
Poincaré conjecture, and developed ideas to link Kähler–Ricci flow tiers and shedding light on fruitful topics in operator theory. We had
with the Mori program in algebraic geometry. Most relevant to this the privilege of having Carl Cowen among us to generously share
CRM workshop, however, is Tian’s work on Kähler–Einstein metrics. his ideas with the audience.
In his early career, he solved the existence question for Kähler–Ein- Apart from the ISP, many other questions and topics on the frontiers
stein metrics on compact complex surfaces with positive first Chern of analysis and operator theory were highlighted in the last three
class, and showed that Fano varieties with a Kähler–Einstein metric days. In particular, we mention the following topics: the multivari-
must be stable in the sense of geometric invariant theory, thus con- ate case of Hilbert modules by R. Douglas; review and continuation
firming part of a suggested correspondence by Yau. Professor Tian of the results of Hunt–Fefferman–Cwikel–Alexandrov on weak L1
gave a precise formulation of this correspondence, known today as and Weak H 1 spaces by J. Cima and A. Nicolau; truncated Toeplitz
the Yau–Tian–Donaldson conjecture, identifying the obstruction to operators and complex symmetric operators by S. Garcia and D. Tim-
the existence of Kähler–Einstein metrics with what is now called the otin; treatment of results of Paley–Wiener, Kadet–Ingham, Seip and
K-stability of the Fano variety. His ideas have inspired a tremendous Borichev–Lyubarskii on sampling and interpolation, and presenting
amount of work in recent years, and spectacular progress has been the new approaches in Fock spaces by A. Hartmann; numerous open
made, culminating in the complete resolution of the conjecture by questions on classical function spaces by D. Khavinson, which have
Tian himself, and by Xiuxiong Chen, Simon Donaldson and Song been the focus of research by him and his students for more than 30
Sun. years; the new achievements on compositions operators on function
Professor Tian’s first lecture was expository in nature, and reviewed spaces by Yu. Lyubarskii, H. 儀effélec and L. Khoi; the great presen-
the theory and latest progress related to the existence of Kähler–Ein- tation of Crofoot–Sarason polynomials by N. Makarov (which imme-
stein metrics. 吀e second lecture was concerned with a more detailed diately constituted the basis of a thesis at Université Laval); the study
exposition of the key ideas of Professor’s Tian most recent work re- of invariant subspaces on the Bergman space by S. Richter; compact
garding the existence of Kähler–Einstein metrics on a K-stable Fano operators on the Bergman space by B. Wick; and cyclicity and the
variety. 吀e third lecture featured recent results on convergence of Brown¬Shields conjecture in Dirichlet spaces by C. Beneteau and
the Kähler–Ricci flow on a Fano variety. T. Ransford.
吀e proceedings of the workshop are expected to appear in the Con-
temporary Mathematics series, published by the AMS.
BULLETIN CRM–4crm.math.ca
2013 André-Aisenstadt Prize Recipient
Spyros Alexakis (University of Toronto)
Dr. Alexakis obtained a B.A. degree from the University of Athens in • Pfaff(Rijkl ) is the Pfaffian of the curvature
R tensor Rijkl of g,
1999 and a Ph.D. from Princeton University, under the supervision of since by the Chern–Gauss–Bonnet theorem M Pfaff(Rijkl ) dVg =
Charles Fefferman, in 2005. He held a Clay Research Fellowship as Cn χ(M ), for some dimensional constant Cn . Here χ(M ) is the
well as a Sloan Fellowship, and has been at the University of Toronto Euler number of M .
since 2008. Working in the areas of analysis and mathematical physics, In a long series of works [1–5] I confirmed the stipulation of Deser
alone and with collaborators he has obtained striking results in at least and Schwimmer [13]:
three different directions. His main contribution, published as a re-
search monograph in the prestigious Annals of Mathematics Studies of 吀eorem 1. Any Riemannian invariant P (g) of weight −n whose in-
Princeton University, is a solution to a conjecture of Deser and Schwim- tegral over closed manifolds satisfies (1) can be expressed as a sum of
mer regarding the structure of global conformal invariants. Secondly, the form (2).
together with Klainerman and Ionescu, he made important progress 吀e ideas in the proof relied on the deep progress made in the last
in the understanding of the Kerr solutions to Einstein’s equations. Fi-three decades on local conformal invariants (in particular the am-
nally, jointly with Mazzeo, he obtained deep results concerning mini- bient metric construction of Fefferman an Graham, [14]), together
mal surfaces with bounded Wilmore energy. 吀is impressive research is with a new, explicit, and useful formula for the first variation of P (g)
described below, in more detail, by Dr. Alexakis himself. under conformal changes, derived in [2]. 吀e argument then pro-
ceeded by an (essentially algorithmic) construction of the decompo-
Integral Conformal Invariants sition. One inductively constructs local conformal invariants and di-
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 con-
and have a given weight) can be expressed as linear combinations of
formal invariants, it is possible that some of the ideas could be ap-
tensor products and complete contractions of the curvature tensor
plicable to the understanding of other integral invariants which ap-
and its covariant derivatives. A further natural question is to under-
pear naturally in other geometries. One such possibility is the under-
stand the space of invariant scalars for other geometries. 吀is ques-
standing 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 invari-
in 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].
The black hole uniqueness question
A generalization of these local questions which first appeared in the
physics literature [13] is to understand all global conformal invari- A natural question that has been studied for nearly four decades
ants. 吀e challenge here is to understand the space of Riemannian concerns the possible vacuum, stationary black hole solutions to
scalars P (g) for which the integral M P (g) dVg over any closed
R
Einstein’s equations in general relativity. 吀e interest in this ques-
Riemannian manifold remains invariant under conformal changes tion was spurred by two factors: firstly, the discovery of the Kerr
g → e2φ g, φ ∈ C ∞ (M ). In other words, we require: 2-parameter family of such solutions [21](parametrized by total mass
and angular momentum), which generalized the Schwarzschild solu-
Z Z
P (e g) dVe2φ g =
2φ
P (g) dVg . (1)
M M
tions (corresponding to zero angular momentum). Secondly, the ex-
吀e obvious candidates P (g) which have this property are sums of pectation (formulated in [17]) that precisely such solutions would
the form be the possible final states of smoothly evolving black holes. 吀e
heuristic argument in [17] asserts that generic dynamical black holes
P (g) = W (g) + divi X i (g) + C · Pfaff(Rijkl ), (2) should radiate energy towards infinity and into the black hole region
where before approachng a final state; therefore (we are told) the final state
• W (g) is a local conformal invariant of weight −n,Ri.e., W (e2φ g) = would have to be nonradiating and stationary. Hence the interest in
e−nφ W (g). In that case the global invariance of M W (g) dVg is the possible stationary black holes is due to their asserted relevance
obvious in view of the transformation of the volume form dVe2φ g = as potential final states of dynamical space-times.
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 equa-
Stokes’ theorem implies M divi X i (g) dVg = 0. tions, under suitable regularity assumptions?
R
BULLETIN CRM–5crm.math.ca
吀e above has been answered in the affirmative in a series of works Y . A starting point of my work with Mazzeo [8] was that this renor-
over the past decades, always under certain additional assumptions malized area turned out to be essentially equivalent to the total cur-
on the space-time. 吀e case of static, rather than just stationary vature or Willmore energy of the surface:
space-time was se琀led by Israel in [20]. 儀estion 1 was also an-
Z
1
swered in the affirmative for space-times which in addition to being Ren.Area[Y ] = −2πχ(Y ) − |Â|2 dVY . (3)
2 Y
stationary are also axi-symmetric, in that they admit an additional 吀is allowed us to study the first and second variations and the crit-
Killing field whose orbits are closed with a fixed period; this is the ical points of this functional.
Carter–Robinson theorem, [11, 12, 22] and references therein.
吀e main question addressed in [9] is to find the correct analogue
吀is extra condition can be relaxed, as observed in [15, 17], who of bubbling in the space of such minimal (and more generally Will-
showed that stationarity along with a bifurcate horizon implies that
the space-times must admit a jet of a second, rotational Killing field,
more) surfaces with unprescribed boundaries in ∂∞ 3 , and with an H
upper bound on the total energy. In particular, in many variational
tangent to the (past and future) event horizons. 吀us, as explained in geometric PDE (i.e. solutions of the Euler–Lagrange equations corre-
detail in [12], if one restricts a琀ention to real-analytic space-times, sponding to a geometric energy functional), one is interested in un-
儀estion 1 can be answered in the affirmative. derstanding 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 sur-
in relaxing this assumption substantially by merely assuming close- face (Σ2 , h) → (M n , g)) is that the sequences converges (away from
ness to the Kerr family of solutions in a C 2 -sense: a set of “bad points”) to a new solution. 吀e failure of smooth con-
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
black hole exterior. se琀ing of minimal (and Willmore) surfaces in 3 . We showed thatH
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
the horizons is small. 吀e natural question that arises is whether this result is optimal, and
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 regu-
space-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 hori-
zons, 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.
[2] , On the decomposition of global conformal invariants. II, Adv. Math.
吀e ideas of the above rely on unique continuation techniques for 206 (2006), no. 2, 466–502.
wave equations, where a notion of pseudo-convexity, namely con- [3] , 吀e decomposition of global conformal invariants, Ann. of Math. Stud.,
vol. 182, Princeton Univ. Press, Princeton, NJ, 2012.
vexity with respect merely to null geodesics, is central. 吀e tech-
[4] , 吀e decomposition of global conformal invariants: some technical
nique is to construct a foliation of the black-hole exterior by level proofs. I, SIGMA Symmetry Integrability Geom. Methods Appl. 7 (2011), Paper
sets of a regular function, whose leaves are T-conditionally pseudo- 019.
convex, i.e. convex with respect to T-normal null geodesics. (吀at [5] , 吀e decomposition of global conformal invariants: some technical
this weaker notion of pseudo-convexity suffices for the problem at proofs. II, Pacific J. Math. 260 (2012), no. 1, 1–88.
hand was already noticed in [19]). 吀e key obstacle is the presence of [6] S. Alexakis, A. D. Ionescu, and S. Klainerman, Hawking’s local rigidity theorem
without analyticity, Geom. Funct. Anal. 20 (2010), no. 4, 845–869.
an ergoregion, where the Killing field T is space-like. 吀e challenge
[7] S. Alexakis, A. D. Ionescu, and S. Klainerman, Uniqueness of smooth station-
of answering 儀estion 1 in suitable generality remains. ary black holes in vacuum: small perturbations of the Kerr spaces, Comm. Math.
H
Phys. 299 (2010), no. 1, 89–127.
Minimal surfaces in 3
and boundary regularity [8] S. Alexakis and R. Mazzeo, Renormalized area and properly embedded mini-
mal surfaces in hyperbolic 3-manifolds, Comm. Math. Phys. 297 (2010), no. 3,
吀e work on this topic concerns minimal surfaces in hyperbolic 621–651.
H
3-space 3 , with a boundary at infinity. 吀e study of these was ini- [9] , Complete Willmore surfaces in H 3 with bounded energy: boundary
tiated by M. Anderson in the 1980s [10], who solved the analogue of regularity and bubbling, available at arXiv:1204.4955.
the Plateau problem for such surfaces, with a boundary at infinity. [10] M. T. Anderson, Complete minimal varieties in hyperbolic space, Invent. Math.
69 (1982), no. 3, 477–494.
An interesting notion regarding these surfaces was that of the renor- [11] B. Carter, An axy-symmetric black hole has only two degrees of freedom, Phys.
malized area, introduced by Graham and Wi琀en in [16]: although Rev. Le琀. 26 (1971), 331–333.
the area of any such minimal surface is necessarily infinite, one can [12] P. T. Chruściel and J. L. Costa, On uniqueness of stationary vacuum black holes,
Astérisque 321 (2008), 195–265.
nonetheless perform a Hadamard regularization and obtain a well-
defined notion of renormalized area Ren.Area[Y ] of such a surface (continued on page 12)
BULLETIN CRM–6crm.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 forma-
by Frithjof Lutscher (O琀awa) and Frédéric Guichard (McGill) tion to collective motion and collective decision-making in animal
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
Simon A. Levin tenure of the chair, Prof. Levin equilibria and even heteroclinic cycles where large shi昀s in vege-
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 doc-
ecological systems umented emerging property in ecological and social systems. Such
2. Evolutionary perspectives on discounting, public goods and col- collective behaviour emerging from individual decisions can explain
lective behaviour cooperation and adaptive response to predation in social organisms.
3. 吀e challenge of sustainability and the promise of mathematics Nabet et al. (2009) represented individual decision within groups as
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 criti-
cal transitions in physical systems can inform the analysis of similar
Predicting the dynamics and pa琀erns found in ecological systems
phenomena in ecological systems, while raising new challenges for
o昀en fails when we consider populations, communities and ecosys-
theory. 吀is first lecture made clear that the growing recognition that
tems as fundamental units to understanding persistence, species di-
ecosystems’ properties emerge from the collective behaviour of in-
versity, and ecosystem productivity. Instead, important questions in
dividuals is associated with a major shi昀 in management strategies:
basic and applied ecology alike involve complex adaptive systems,
local interactions and feedbacks give rise to macroscopic properties
in which localized interactions among individual agents give rise
that undergo strong fluctuations and sudden shi昀s. We are part of,
to emergent pa琀erns that feed back to affect individual behaviour.
and have to manage, very dynamic ecosystems.
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 criti-
cal transitions, and the ecological and evolutionary conflicts between Ecological and economic systems are alike in that individual agents
levels of organization. compete for limited resources, evolve their behaviours in response
BULLETIN CRM–7crm.math.ca
to interactions with others, and form exploitative as well as coop- serve every species of tree, but we need to preserve trees as a source
erative interactions as a result. In these complex adaptive systems, of building material, wildlife habitat, oxygen generation and carbon
macroscopic properties like the flow pa琀erns of resources (such as storage, among others.
nutrients and capital) emerge from large numbers of microscopic in- One of the great challenges when considering ecosystem function is
teractions, and feed back to affect individual behaviours. Contagion the question of scales. How do we relate phenomena across differ-
can lead to critical transitions from one basin of a琀raction to another,
ent spatial and temporal scales? And how robust are small-scale
as for example with eutrophication, desertification, pest outbreaks, results on larger scales? One approach is to begin with a detailed
and market collapses. In both sorts of systems, evolution of one type model (such as a forest simulator model) and then scale up and
or another leads to the differentiation of roles and the emergence of compare to global vegetation models, where world-wide pa琀erns
system organization, but with no guarantee of robustness. It is cru- are tracked rather than individual species. Another example is the
cial to understand how evolutionary forces have shaped individual DARWIN model (http://darwinproject.mit.edu/) that tracks
behaviours in the face of uncertainty. 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 activi-
to how one can design mechanisms for self-reinforcing rules for co- ties and natural causes (wildfire) can trigger the transition between
operation. them (Staver et al., 2011). A similar bistable situation occurs in shal-
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 be-
mi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
mon pool resources and collective movement. transitions occur and whether they can be detected by early warning
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.
tainable future as one of intergenerational equity: can we enjoy 吀e potential that mathematics has to offer to those challenges is in
economic growth without negative consequences for future gener- developing a statistical mechanics framework of ecological commu-
ations? Sustainability, of course, applies to many different areas, for nities and socio-economic systems; describing and detecting emer-
example, the financial sector, energy and natural resources, biologi- gent pa琀erns; finding indicators of critical transitions; and support-
cal and cultural diversity, or ecosystems services. 吀e la琀er is o昀en ing governance of multiscale systems.
particularly difficult to value, so mathematics may provide particu-
吀e mathematics of governance includes optimal control, voting the-
larly useful insights.
ory, collective motion, games and negotiation. All too o昀en, there
When evaluating ecosystems services, one places the focus not on is clear scientific consensus, but action is lacking. In part, this lack
preserving every single species and aspect but rather certain gen- of action stems from missing commitment to the public good. Many
eral functions, particularly those that are of great importance to hu- aspects of how to set and enforce commitment to the public good
mans. For example, we might not be able to preserve every species were discussed in the second lecture in detail. 吀e overall question
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 pre- (continued on page 14)
BULLETIN CRM–8crm.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 confé- que l’analyse des ondes sis-
rence Mathématiques de la planète Terre de la série Simons le 10 avril miques perme琀ait de détecter
à Montréal dans la salle du Cœur des sciences. Pas moins de 400 per- des zones de perturbation des
sonnes s’étaient déplacées pour l’événement. Elle a livré sa splen- ondes de pression (P-ondes) des
dide conférence en français et a enregistré elle-même le doublage tremblements de terre. De telles
en anglais devant une salle vide. Des vidéos français et anglais de la régions avaient été identifiées,
conférence peuvent être écoutés à http://www.videocrm.ca/. qui chevauchaient exactement
les régions avec des îles vol-
Trop souvent, on limite Mathématiques de la planète Terre 2013 aux
caniques isolées, et la tempé-
changements climatiques et au développement durable. La confé-
rature du fond océanique était
rence d’Ingrid Daubechies cadrait au contraire parfaitement sous le
plus grande dans ces régions.
premier thème « Une planète à découvrir ». Elle relatait la coopé-
Mais, comme les panaches sont
ration de la conférencière avec des géophysiciens, et leurs résultats
si petits et la perturbation de la
très récents sur la compréhension du processus de formation des îles
vitesse des P-ondes si faible, le
volcaniques isolées. Sur le fond des océans, les roches les plus ré-
risque d’erreurs est grand dans
centes se retrouvent le long des dorsales où les plaques tectoniques Ingrid Daubechies
la reconstruction numérique de
divergent. L’activité volcanique le long de ces dorsales fait remon-
la structure interne de la Terre, sauf si on utilise des outils suffisam-
ter du magma depuis le manteau, lequel forme de nouvelles roches.
ment performants. C’est là que les ondele琀es sont si utiles. Elles sont
Mais, il existe des îles volcaniques isolées comme Hawaï, Tahiti, les
l’outil parfait pour analyser de petits détails locaux. De plus, l’analyse
Açores, le Cap Vert, etc. Si l’on regarde l’archipel d’Hawaï, les îles
en ondele琀es permet de concentrer toute l’énergie sur ces petites ré-
sont alignées par ordre d’âge décroissant, avec la plus grande île,
gions et de négliger les autres régions.
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 Dans sa conférence, Ingrid Daubechies a donné un mini-cours sur
volcanique, c’est-à-dire une sorte de cheminée volcanique au travers l’analyse en ondele琀es adaptée aux images digitales composées de
du manteau. Rappelons que la profondeur du manteau est environ la pixels. Une image en tons de gris est simplement une matrice de
moitié du rayon de la Terre. Puisque la plaque tectonique de surface nombres donnant le ton de gris de chaque pixel. À partir de ce琀e ma-
se déplace, ceci expliquerait la formation successive d’îles alignées, trice, on construit quatre matrices plus petites. La première contient
dont la différence d’âge pourrait être calculée par la distance entre lesles moyennes horizontales et verticales des pixels voisins pris deux
îles et la vitesse de la plaque tectonique. Mais il faut rajouter d’autres
à deux, la seconde les moyennes horizontales et les différences ver-
éléments de preuve pour que la conjecture soit définitivement accep- ticales des pixels voisins pris deux à deux, la troisième les moyennes
tée par la communauté scientifique, l’un de ces éléments pouvant verticales et les différences horizontales des pixels voisins pris deux
être de « voir » le panache. Un outil pour explorer l’intérieur de la à deux, et la dernière, les différences horizontales et verticales des
Terre est la télédétection : on envoie des signaux et on analyse les si-pixels voisins pris deux à deux. On itère le processus sur la première
gnaux réfléchis ou réfractés sur les différentes couches dans le sol. La matrice (celle des moyennes horizontales et verticales). Jusque-là, pas
technique est utilisée pour chercher du pétrole. Mais, les panaches de perte d’information. Ingrid Daubechies a expliqué comment les
sont si profonds sous la croûte terrestre que les signaux artificiels ondele琀es perme琀ent de compresser l’information et comment on
ne sont pas assez puissants pour une telle analyse. Les seuls signaux peut extraire les détails fins dans une petite région, tout en ayant
suffisamment puissants pour analyser les détails à une telle profon- compressé beaucoup l’information. L’utilisation des ondele琀es pour
deur sont les ondes sismiques générées par les grands tremblements la reconstruction d’images permet d’éliminer les erreurs de recons-
de terre. L’équipe autour d’Ingrid Daubechies a eu accès à de grandes truction numérique et d’être certain que les zones singulières identi-
bases de données contenant les enregistrements des ondes sismiques fiées dans l’image sont effectivement spéciales. Ingrid Daubechies a
captées par les stations sismiques autour de la planète. montré des images « propres » obtenues grâce aux ondele琀es, dans
lesquelles les régions artificielles ont été enlevées, et elle a pu annon-
Donc, 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 don-
fins 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 Dah-
len et Guust Noleta ont approché Ingrid Daubechies en 2005 pour Le public a posé de nombreuses questions dans la salle de confé-
voir 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–9crm.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 trans-
Master), 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 plan-
field, 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 dissipa-
ity, 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 prob-
lem, 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 en-
ered the whole spectrum from N = 3, and even the restricted 3-body ergy configuration for a N -body system with fixed angular momen-
problem 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 den-
sity distributions, thus leading to a natural “granular mechanics” ex-
In 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 ac-
ary 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 de-
plain 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 sym-
In 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 ac-
existence of a large measure set of stable motions and lower dimen-
cumulate. 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 pre-
Two 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 ex-
plained how a variational approach allows to transform the problem
to a billiard type problem with elastic collisions.
BULLETIN CRM–10You can also read
