International Conference
ALGEBRAIC METHODS IN DYNAMICAL SYSTEMS
Barcelona – Spain, 04-08/february/2008
SCIENTIFIC PROGRAM
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Schedule
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Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
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9:00 – 10:00 |
Registration |
J.F. Viaud |
R. Pérez-Marco |
J. Mozo |
C. Simó |
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Registration |
Coffee Break |
Coffee Break |
Coffee Break |
Coffee Break |
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10:30 – 11:30 |
B. Malgrange |
J. Kovacic |
M. Van der Put |
J. Cano |
M. Ayoul |
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Coffee Break |
Coffee Break |
Coffee Break |
Coffee Break |
Coffee Break |
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12:00 – 13:00 |
A. Granier |
D. Blázquez |
J. Roques |
G. Duval - A. Maciejewski |
P. Vanhaecke |
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13:15 – 14:15 |
Registration |
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G. Casale |
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M. Przybylska |
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Lunch |
Lunch |
Lunch |
Lunch |
Lunch |
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15:30 – 16:30 |
H. Umemura |
D. Bertrand |
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J.A. Weil |
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Coffee Break |
Coffee Break |
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Coffee Break |
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17:00 – 18:00 |
M. Singer |
J. Muñoz |
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S. Simón |
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Note:
Every
participant of AMDS should keep in mind the following keynotes.
Abstracts
Michael AYOUL (U.
Talk: GALOISIAN OBSTRUCTIONS TO INTEGRABILITY OF
NON-HAMILTONIAN SYSTEMS
Abstract. In this talk,
we will see how Morales-Ramis-Simó theorem on differential Galois groups of
higher variational equations can be translated in another context than the
usual symplectic one. We will need a new definition of complete integrability, inspired by Liouville's integrability, and we
will see how galoisian obstructions to integrability arise in a non-Hamiltonian
frame.
David BLAZQUEZ-SANZ (U. Politècnica de Catalunya)
Talk: CERTAIN ALGEBRAIC ASPECTS OF LIE-VESSIOT
EQUATIONS
Abstract. We explore Lie's
characterization of differential equations admitting a superposition principle.
Lie stated [Lie 1893] that a differential equation admits a superposition
formula if and only if it has finite dimensional Lie-Vessiot-Guldberg algebra.
In recent research [CGM 2006] Cariñena, Grabobski and Marmo proposed a new
version of this theorem. It is argued that original Lie's proof is local, and
then the general statement is not proven. We would see that the global version
of Lie's superposition theorem is also true: A differential equation admits a
superposition formula if and only if its Lie-Guldberg-Vessiot algebra is
spanned by fundamental fields of a Lie group action. Superposition formulas can
be deduced from the Lie group action and reciprocally.
Following Vessiot [Ve 1894] we
develop a Galois theory for such equations in algebraic groups. This theory is
a generalization of Picard-Vessiot theory, but we obtain more general strongly
normal extensions. As indicated by Kovacic [Ko 2006], the logarithmic
derivative is the dictionary between differential equations and differential
field extensions. We develop an algebraic version of Lie reduction method for
differential equations in Lie groups. We find that Lie's reduction method is a
geometric mechanism behind most relevants results in differential algebra:
Kolchin's reduction theorem, Liouville theorem on the Riccati equation, and
Morales-Ramis theorem are revisited in this frame. We give some generalization
of those results.
This work is in
collaboration with Juan J. Morales-Ruiz.
REFERENCES
[Lie 1893] S. Lie,
Sur les équations différentielles ordinaries, qui
possèddent des systemes fondamentaux d'integrales, Compt. Rend. Acad. Sci.
Paris, T CXVI (1893), pp. 11233-1235.
[Ve 1894] E. Vessiot,
Sur les systèmes d'équations différentielles du
premier ordre qui ont des systèmes fondamentaux d'intégrales. Annales de la
faculté des sciences de Toulouse Sér. 1, 8 no. 3 (1894), p. H1--H33.
[CGM
2006] J. F. Cariñena, J. Grabowski, G. Marmo,
Superposition rules, Lie
theorem, and Partial Differential equations, Arxiv: Math-Ph. 0610013
[Kov 2006] Jerald J.
Kovacic,
Geometric Characterization
of Strongly
Daniel BERTRAND (U. Paris VI)
Talk: ALGEBRAIC $D$-GROUPS AS GALOIS GROUPS AND
THE GEOMETRIC LINDEMANN-WEIERSTRASS THEOREM
Abstract. Let $G$ be a
semi-abelian variety over the differential field $(K = {\bf C}(z)^{alg},
\partial = d/dz$). The universal extension $\tilde G$ of $G$ admits a canonical
structure of algebraic $D$-group over $K$, that is, a unique extension of
$\partial$ to a derivation on $K(G)$ respecting the group structure. Let
$\partial \ell n_{\tilde G} $ be the corresponding logarithmic derivative on
$\tilde G$, and let $\partial_{L\tilde G}$ be the Gauss-Manin connection on its
Lie algebra $L\tilde G$. For $(x,y) \in
L{\tilde G} \times {\tilde G}$, the analytic relation $y = exp_{\tilde G}(x)$
can be expressed in terms of differential algebra by the differential
equation $\partial \ell n_{\tilde
G} y = \partial_{L\tilde G} x$. Under
various hypotheses on $x \in L\tilde G(K)$ and on the constant part of $ G$,
we show that $y$ then generates over $K$ a field of
maximal transcendence degree $dim{\tilde G}$. We shall give two proofs of this
result, based on model theory on the one hand,
and on Pillay's differential Galois theory on the other hand, but both
finally relying on Manin's kernel theorem, as reexpressed by Coleman and Chai.
(Joint work with Anand Pillay).
José CANO (U.
Talk: THE SPACE OF GENERALIZED
FORMAL POWER SERIES SOLUTION OF AN ODE
Abstract. We prove that
the set of truncations of generalized power series solutions of an ordinary
differential equations is contained in a semi-algebraic set of dimension bounded
by twice the order of the differential equation.
Guy CASALE (U.
TALK: INTEGRABILITY OF
DISCRETE HAMITONIAN AND DIFFERENCE GALOIS THEORY
Abstract. We study the
relationship between the integrability of rational symplectic maps and difference
Galois theory. Using the Malgrange groupoid, we give a Galoisian condition, of
Morales-Ramis type, ensuring the non-integrability of a rational symplectic map
in the non-commutative sense (Mischenko-Fomenko). As a particular case, we
obtain a complete discrete analogue of Morales-Ramis Theorems for
non-integrability in the sense of Liouville. This a joint work with julien
Roques.
Guillaume DUVAL (U.
Talk:
Abstract. Let
$H(q;p)=\frac{1}{2}\sum_{}^{}p_i^2+V(q)$ where $V(q)$ is an homogeneous
potential of degree $k\neq 0$. Morales and Ramis and others developed a
differential Galois theory for the integrability of Hamiltonian systems. Later
in [1], Morales and Ramis applied this theory to the Hamiltonian system
associated to $H(q;p)$. To that purpose, they studied some variational
equations of the form
\begin{equation}\label{eqvar}
\eta’’=−z(t)^{k−2}V’’(c)\eta,
\end{equation}
where $V′′(c)$
is the Hessian matrix of $V$ at some « Darboux point » (i.e. $c=V(c)$).
Assuming that $V’’(c)$ is diagonalizable, Morales and Ramis find necessary
conditions for the integrability of the Hamiltonian system associated to $H$ in
terms of the degree of $V$ and of the eigenvalues of $V′′(c)$.
Those values are the only ones for which the connected component of the Galois
group of \eqref{eqvar} is Abelian.
By considering
Bibliography
[1] Juan J. Morales-Ruiz and
Jean Pierre Ramis. A note on the non-integrability of some Hamiltonian systems
with a homogeneous potential. Methods Appl. Anal., 8 (1):113–120, 2001.
Anne GRANIER
(U.
Talk: A GALOIS GROUPOID FOR Q-DIFFERENCE EQUATIONS
Abstract. In
his paper "Le groupoïde de Galois d'un feuilletage" (2001), B.
Malgrange defined, for non linear differential equations, an object which
generalizes the Galois group of linear differential equations, namely the
Galois groupoid of a foliation. In this talk, we introduce a definition for the
Galois groupoid of a q-difference system. We then show the consistency of the
definition proving that for some class of linear systems, the Galois group as described
by J. Sauloy in "Galois theory of fuchsian q-difference equations"
(2003), can be recovered from the Galois groupoid.
Jerald KOVACIC
(CUNNY)
Talk: STRONGLY
Abstract. The differential Galois
theory of strongly normal extensions is ripe for study. It has been neglected, possibly because
Kolchin used his own axiomatic definition of algebraic group. Instead, we use
differential schemes, another area ripe for study.
We
start with a sketch of Picard-Vessiot theory emphasizing its connection with
tensor products.
After
defining strongly normal extensions, we show that an approach similar to that
used for Picard-Vessiot theory also works for strongly normal extensions. However we must replace differential rings
with differential schemes. This is not
surprising as the Galois group is a group scheme that is not necessarily
affine.
Strongly
normal extensions are abundant; every connected group scheme is the Galois
group of some strongly normal extension. And there is a ``factory'' to produce
them - the logarithmic derivative. Yet explicit examples are difficult to find.
There has been some work on characterizing the type of equation needed, but
much more is needed.
Bernard MALGRANGE (U. Joseph
Fourier)
Talk: INTEGRABILITY AND NON-LINEAR GALOIS THEORY
Abstract.
I
will try to explain how non-linear Galois theory is related, in the case of
Liouville integrable systems, with objects of algebraic geometry as;
Gauss-Manin connection,extensions of abelian varieties, and isomonodromic deformations of flat rank one
bundles.
Jorge MOZO
(U. Valladolid)
Talk: MONOMIAL SUMMABILITY AND LINEAR PERTURBED
DIFFERENTIAL EQUATIONS
Abstract. We obtain a monomially
summable factorisation of a linear system of doubly singular linear
differential equations, i.e., singular perturbation of a linear system with an
irregular singularity. We will discuss some examples, further developments,
extensions and possible future applications of these results, in order to study
differential Galois theory of perturbed systems. It is a joint work with M.
Canalis-Durand and R. Schäfke.
Jesús MUÑOZ
(U. Salamanca)
Talk: STRUCTURE OF TIME IN
LAGRANGIAN MECHANICS
Abstract. Classically
time is kept fixed for infinitesimal variations in problems in mechanics.
Apparently, there appears to be no mathematical justification in the literature
for this standard procedure. This can be explained canonically by unveiling the
intrinsic mathematical structure of time in Lagrangian mechanics. Moreover,
this structure also offers a general method to deal with inertial forces.
Ricardo PEREZ-MARCO (CNRS - U. Paris XIII)
Talk: ELEMENTS OF
LOG-RIEMANN SURFACE THEORY
Abstract. Log-Riemann
surfaces, and in particular transalgebraic curves, generalize the classical
notion of algebraic curve allowing infinite ramification points. We present
some results from the geometric and the algebraic theory of log-Riemann
surfaces, in particular those in connection with differential algebra and
transalgebraic theories.
Maria PRZYBYLSKA (
Talk: INTEGRABILITY
OF HOMOGENEOUS POTENTIALS - RECENT RESULTS AND PROBLEMS
Abstract. The subject of this talk is integrability of
homogeneous polynomial potentials with more than two degrees of freedom.
Homogeneity implies the existence of straight-line type particular solutions
defined by Darboux points. Morales-Ruiz and Ramis have shown that the analysis
of a differential Galois group of variational equations along these solutions
yields strong integrability obstructions expressed by means of restrictions on
eigenvalues of Hessian calculated at a Darboux point. It will be presented how
using multidimensional residues technique one can find some universal relations
between eigenvalues of the Hessian calculated at all Darboux points that give
additional integrability obstructions. It appears also that equilibria of the
potential and their degenerations have the influence on these relations. The
relations appear the starting point to the classification programme of
homogeneous integrable potentials and the progress for the case of three
degrees of freedom will be recapitulated.
Julien ROQUES (ENS, Paris)
Talk: GALOIS GROUPS OF
Q-HYPERGEOMETRIC EQUATIONS
Abstract. In this talk, I
will focus my attention on the calculation of the (difference) Galois groups of
q-hypergeometric equations.
Carles SIMO
(U. de Barcelona)
Talk: Beyond
non-integrability
Abstract. As soon as
non-integrability is found for a given family of systems the problem of
understanding the dynamics is fully open. Some methods and simple examples will
be presented to face this question.
Sergi SIMON (U.
Talk:
THE MEROMORPHIC NON--INTEGRABILITY OF THE SWINGING ATWOOD'S MACHINE
Abstract. The Swinging Atwood's
Machine (SAM) is a compound mechanism comprising a pulley and a pendulum and
linking two point masses, one of them allowed to swing in a plane. This coupled
oscillator, basically a variation of the well-known Atwood Machine introduced
in the late eighteenth century, exhibits an astonishingly complex behaviour
despite its simple physical description.
We prove the non--integrability
of the Hamiltonian system modeling SAM, both in the general case and in the
case of neglected pulley masses, using basic results from the incipient
framework of Ziglin-Morales-Ramis theory. The subsequent recollection of basics
in Analytical Mechanics and Differential Algebra, as well as a number of
alternative or partial proofs of the same basic result, will be assembled into
a comprehensive survey of the different methods, whether algebraic or
semi-analytical, of detecting chaotic behaviour in general potential systems.
There is more to this talk
than a mere survey, though; aside from said status as a paradigm for
methodology, our present studies on SAM will also shed light on further
advances and new research in the lines of Ziglin-Morales-Ramis theory.
This is a joint work with
Juan J. Morales-Ruiz, Olivier Pujol, José-Phillippe Pérez, Jean-Pierre Ramis,
Carles Simó and Jacques-Arthur Weil"
Michael SINGER (NCSU)
Talk: DIFFERENTIAL GALOIS
THEORY OF DIFFERENCE EQUATIONS
Abstract. I will develop
a Galois theory of linear difference equations where the Galois group are
linear differential groups that is,
groups of matrices whose entries satisfy a fixed set of polynomial differential
equations. These groups measure the differential dependence among solutions of
linear difference equations.
I will discuss how this
theory can be used to reprove Hoelder's Theorem that the Gamma function
satisfies no differential polynomial equation as well as new results concerning
differential dependence of solutions of higher order difference equations, such
as families of q-hypergeometric equations. I will also discuss the inverse
problem for this Galois theory and the meaning of this Galois theory for
parameterized families of difference equations.
This is joint work with
Charlotte Hardouin.
Hiroshi
UMEMURA (U.
Talk:
ON A GENERAL GALOIS THEORY OF DIFFERENCE EQUATIONS
Abstract. We proposed in
Marius
VAN DER PUT (U.
Talk:
MODULI SPACES OF SINGULAR DIFFERENTIAL EQUATIONS AND THE PAINLEVÉ EQUATIONS
Abstract. This talk is
concerned with joint work with Masa-Hiko Saito (
Pol VANHAECKE (U.
Talk: THE GEOMETRY OF
LAURENT SOLUTIONS
Abstract. The Laurent
solutions to a complex integrable system encode information about its
monodromy, are decisive for the possibility of completing the integrable flows
on a partial compactification of the phase space and they yield key information
about the geometrical structure of the complex invariant manifolds. Besides
reveiling the beauty of algebraic completely integrable systems, these
geometrical facts about the Laurent solutions are actually very important for
proving algebraic integrability and for applications to algebraic geometry. I
will try to explain the main ideas and illustrate them on simple, but
non-trivial, examples.
Jean François VIAUD (U.
Rochelle)
Talk: A GUIDED TOUR THROUGH GALOIS' THEORIES.
Abstract. I will present some common
aspects of Galois theories, using Grothendieck's point of view. In particular,
these general aspects are reusable to get (I hope so) a Galois theory for
foliations.
Jacques-Arthur WEIL (U. Limoges)
Talk: PRACTICAL APPROACHES
TO HIGHER VARIATIONAL EQUATIONS IN THE MORALES-RAMIS-SIMO THEORY
Abstract.