Seminars 2004-2005

Enstrophy dissipation and irregular transport in two-dimensional incompressible flows.

Speaker: A. Mazzucato (Department of Mathematics Penn State University)
Time: July 17, 14:30
Place: room B

We consider the problem of enstrophy dissipation for two-dimensional incompressible flows at high Reynolds number. We discuss two notions of enstrophy defects associated to approximate solution sequences to the Euler equations obtained by mollification and by vanishing viscosity. These notions were originally introduced by G. Eyink in order to reconcile the Kraichnan-Batchelor theory of 2D turbulence with properties of weak solutions to 2D Euler. We show that, if the initial enstrophy is finite, the total enstrophy is conserved and in the vanishing viscosity limit a well-defined viscous enstrophy defect exists. If the initial vorticity belongs to certain logarithmic refinements of L^2, then an exact transport equation holds for the corresponding enstrophy density. For rougher data in a Besov space, for which the initial enstrophy is infinite, the enstrophy defects depend upon the approximating sequence and we produce examples of solutions where true dissipation occurs. This is joint work with Helena and Milton Lopes.

Modeling multicellular systems using many-body theory: with application to streak formation in the chick embryo

Speaker: T. Newman (Arizona State University)
Time: June 28, 17:00
Place: room B

Robust multicellular structures and processes are crucial for the construction and function of higher organisms. In these systems large numbers (thousands to millions) of cells communicate through short and long-ranged interactions in order to achieve large-scale coherence. In this talk I will present a general framework for describing these systems. The framework is similar in spirit to many-body theory, which has been so successful in understanding coherent structures in physical systems. The framework provides useful analytical insights into the behavior of multi-cellular systems, and also allows the construction of optimized off-lattice computer algorithms. I will illustrate the latter in the context of early stages of chick embryogenesis, in particular the dynamics of primitive streak formation.

Variational convergence of Ginzburg-Landau with supercritical growth

Speaker: N. Desenzani (SISSA)
Time: June 21, 15:00
Place: room B

Rate of convergence to equilibrium for the Boltzmann equation

Speaker: C. Mouhot (École normale supérieure UMPA - Unité de mathématiques pures et appliquées Lyon)
Time: June 16, 14:30
Place: room B

A canonical Cartan connection for nonholonomic rank two distributions: nilpotent geometry approach

Speaker: B. Dubrov (Belarussian State University, Minsk)
Time: June 9, 14:30
Place: room B

This is a second part of the seminar devoted to the geometry of rank 2 distributions. It is devoted to an alternative approach to the construction of the canonical frame using the methods of nilpotent differential geometry. We give a brief review of geometric structures on filtered manifolds and show how any $(2,n)$-distribution of maximal class defines such structure. Further, we proof that there is a canonical Cartan connection associated with this structure. As an example, we apply this theory to the equivalence problem of underdetermined ODEs $z'=f(x,y,\dots,y^{(n)})$, which appears to be the same as the divergence equivalence of Lagrangians $\int f(x,y,\dots,y^{(n)}) dx$. In particular, we proof that a non-degenerate Lagrangian is divergence equivalent to the most symmetric one $\int (y^{(n)})^2 dx$ if and only if all invariants of the Jacobi curves for the corresponding $(2,n)$-distribution vanish identically.

A canonical frame for nonholonomic rank two distributions: Jacobi curves approach

Speaker: I. Zelenko (SISSA)
Time: June 8, 14:30
Place: room B

The talk is devoted to the local equivalence problem for rank 2 distributions on an $n$-dimensional manifold (or shortly $(2,n)$-distributions) and it is based on the joint work with Boris Doubrov. In 1910 for maximally nonholomomic $(2,5)$-distributions E. Cartan constructed the canonical coframe and found the most symmetric case. We solve the analogous problems for $(2,n)$-distributions with $n>5$, using the theory of Jacobi curves. First we make a kind of symplectification of the problem by lifting the distribution to the annihilator of its square (which is a subset of the cotangent bundle) and by studying the dynamics of this lift under the flow of abnormal extremals. After this simplectification one can treat the equivalence problem by two different approaches: the approach, coming from the theory of Jacobi curves, developed in our previous works with A. Agrachev, and the approach, coming from the nilpotent geometry, developed by N. Tanaka and T. Morimoto. In my talk I want to describe the first approach. The projective structure on each abnormal extremal plays the crucial role in our constructions. The canonical frame can be constructed on (2n-1)-dimensional manifold, which is a principal bundle over annihilator of the square of the distribution with the structural group of all Mobius transformations, preserving 0.

Networks and systems biology: Inferring gene network models from experimental data

Speaker: M. Bansal (TIGEM - Telethon Institute of Genetics and Medicine Napoli)
Time: May 25, 16:00
Place: room B

On the Strong Maximum Principle

Speaker: A. Cellina (Dipartimento di Matematica e Applicazioni - Universita di Milano Bicocca)
Time: May 25, 14:30
Place: room B

On Existence of Variations

Speaker: A. Cellina (Dipartimento di Matematica e Applicazioni - Universita di Milano Bicocca)
Time: May 24, 14:30
Place: room B

Quasistatic crack growth of brittle thin films

Speaker: J.-F. Babadjian (LPMTM, Université Paris Nord)
Time: May 18, 16:00
Place: room B

This talk is concerned with the quasistatic crack evolution of brittle homogeneous thin films obeying to Griffith's law. The starting point is a three-dimensional cylinder whose thickness becomes arbitrarily small. The existence of a quasistatic evolution in the sense of Francfort-Marigo has been proved by Dal Maso, Francfort and Toader and one seek to know how it behaves when the thickness tends to zero. Firstly the static problem will be presented through a Gamma-convergence analysis with a surface energy which does not provide weak compactness in the space of Special Functions of Bounded Variation. It will be shown that the limit two-dimensional surface energy remains of Griffith type whereas the bulk energy follows the one obtained by Le Dret-Raoult. Then the asymptotic analysis of the quasistatic evolution will be presented in the case of bounded solutions. In particular, it will be shown that it converges to a quasistat ic evolution associated to the relaxed two-dimensional model.

Lie theoretical methods for control systems and applications

Speaker: F. S. Leite (Departamento de Matemática Faculdade de Ciências e Tecnologia Universidade de Coimbra Portugal)
Time: May 17, 11:00
Place: room B

This course will focus on the rich interplay between analysis, algebra, and geometry for which Lie groups are especially suited. This interplay is also the essence of geometric control theory, where a control system is seen as a family of vector fields and even the most basic theoretical tool of the geometric view point is the Lie bracket. Basics of Lie theory with appear in parallel with notions in control systems. Emphasis will be put on semi-simple Lie algebras and classical Lie groups, due to their role in physics and engineering applications. Following lectures: May 18 (14:30), May 24, 25, 31 and June 1, 8, 9 (always 11:00)

Algebraic Approach to Homogeneous Approximation of Control Systems

Speaker: S. Yu Ignatovich (Kharkov National University, Ukraine)
Time: April 20, 14:30
Place: room B

We consider the steering problem for the system $\dot x=a(t,x)+ub(t,x)$ to the equilibrium ($x=0$). The initial point $x(0)$ can be expressed as a Volterra-type series of the form $x(0)=\sum v_{m_1\ldots m_k}\xi_{m_1\ldots m_k}$ where $v_{m_1\ldots m_k}$ are constant vectors (depending on $a$ and $b$ and their derivatives at $t=0$, $x=0$) and $\xi_{m_1\ldots m_k}$ are iterated integrals generated by $\int_0^\theta \tau^{m}u(\tau)d\tau$, $m=0,1,2,\ldots$. This control system generates the certain right ideal structure in the algebra of iterated integrals. Besides, the grading structure naturally arises if some restriction on the control (for example, $|u(t)|\le1$) is fixed. The mentioned right ideal is constructed (in accordance with the graduation) by analysis of linear dependence of values at $t=0$, $x=0$ of elements of Lie algebra of vector fields generated by $a$ and $b$. The homogeneous approximation of the initial control system is expressed explicitly in algebraic terms as orthogonal projections of the Lie elements of the algebra of iterated integrals on the orthogonal complement to the ideal. Reference: G.M.Sklyar and S.Yu.Ignatovich, Approximation of time-optimal control problems via nonlinear power moment min-problems, SIAM J. on Control and Optimiz., 42(2003), 1325-1346.

Quasi-periodic solutions of non linear wave equation

Speaker: P. Baldi (SISSA)
Time: April 06, 14:30
Place: room B

Some remarks on the regularity of partial differential equations on Lie groups

Speaker: S. Polidoro (Dipartimento di Matematica Università di Bologna)
Time: April 05, 14:30
Place: room B

We present some fundamental results from the theory of elliptic and parabolic partial differential equations and their extensions to a class of differential equations with non-negative characteristic form. We discuss some mean value formulas, maximum principle and Harnack type inequalities, and we show that Harnack inequalities, combined with some underlying homogeneous Lie group structures, give accurate asymptotic estimates of the fundamental solution of the corresponding differential operators.

Problemi di tipo Dubins su superfici a curvatura non costante

Speaker: M. Sigalotti (INRIA Sophia Antipolis)
Time: March 22, 15:00
Place: Aula Seminario (Dipartimento di Matematica e Informatica dell'Università di Trieste)

Invariants of Legendrian knots. Combinatorial version of Chekanov theorem and one parametrical Morse theory

Speaker: P. Pushkar (Independent University of Moscow)
Time: March 21, 16:00
Place: room B

Suppose that a Legendrian knot in the standard contact space of one jets of functions on the real line can be defined by a generating family. Applying to the generating family a one-parametric version of Morse theory (we are studing generic bifurcations of Morse complexes), we construct certain combinatorial structures on the wavefront of the Legendrian knot. These structures provide new isotopy invariants of Legendrian knots.

Instability of finite difference schemes for hyperbolic systems of conservation laws

Speaker: P. Baiti (Dipartimento di Matematica e Informatica, Università di Udine)
Time: March 16, 14:30
Place: room B

For strictly hyperbolic systems of conservation laws in one space dimension, the Cauchy problem is well posed, within a class of functions having small total variation. However, when solutions with shocks are computed by means of a finite difference scheme, the total variation can become arbitrarily large. As a consequence, convergence of numerical schemes cannot be proved by establishing a priori BV bounds or uniform $\L^1$ stability estimates. In the seminar we discuss this instability by presenting some results in this direction.

Applications of nilpotent geometry to subriemannian geometry

Speaker: T. Morimoto (Department of Mathematics, Nara Women's University, Japan)
Time: March 9, 16:30
Place: room B

A filtered manifold is a differentiable manifold $M$ equipped with a filtration ${\{{F}^p \}}_{p \in \Bbb Z}$ consisting of subbundles of the tangent bundle $TM$ of $M$ such that the following conditions are satisfied: i) $ {F}^p \supset {F}^{p+1}$, ii) ${F}^0 = 0, \quad \bigcup_{p \in \Bbb Z} {F}^p = TM $, and iii) $[\underline{{F}}^p, \underline{{F}}^q] \subset \underline{{F}}^{p+q}$ for all $ p, q \in \Bbb Z$,where $\underline{{F}}^p$ denotes the sheaf of the germs of sections of${F}^p$. To each point $x$ of a filtered manifold $(M, F)$, there is associated a nilpotent graded Lie algebra (called the symbol algebra of $(M,F)$ at $x$): $gr F_x = \bigoplus gr_p F_x$, where $ gr_p F_x = F_x^p / F_x^{p+1}$. Geometry and analysis on filtered manifolds based on their associated nilpotent graded Lie algebras are called nilpotent geometry and analysis and have been developed systematically (see T.\,Morimoto, Lie algebras, geometric structures and differential equations on filtered manifolds, Advance Studies in Pure Mathematics 37 (2002), pp.\,205--252). Applying them to subriemannian geometry, we will prove the following two theorems: Theorem 1. To each subriemannian manifold $(M, D, g)$ whose distribution $D$ is bracket-generating and whose subriemannian symbol is isomorphic to a constant one $(\mathfrak m ,\sigma)$, one can canonically associate a Cartan connection $(P, M, G_0, \omega)$ on a principal fiber bundle $P$ on $M$, where the structure group $G_0$ is the automorphism group of $(\mathfrak m ,\sigma)$. \end{theorem} Theorem 2.The automorphism group of a subriemannian manifold $(M, D, g)$ whose distribution $D$ is bracket-generating is a finite dimensional Lie group.

Differential equations and conformal structures.

Speaker: P. Nurowski (Institute of Theoretical Physics, Warsaw University)
Time: February 2, 16:00
Place: room B

Minicourse (approx. 5 lectures): In these series of lectures we show how conformal structures of various dimensions and signatures appear in the theory of ordinary differential equations. In 1905 Karl Wuenschmann and, recently, Ezra T. Newman observed that certain contact equivalent classes of 3rd order ordinary differential equations are in one-to-one correspondence with 3-dimensional Lorentzian conformal geometries. In the lecture we describe the Wuenschmann-Newman result and show other examples of differential equations considered modulo various transformations of variables which are equivalent to conformal geometries. In particular, we discuss: 1) a relation between 3rd order ODEs considered modulo point transformations of variables and 3-dimensional Lorentzian Weyl and Einstein-Weyl geometries 2) a relation between 2nd order ODEs considered modulo point transformations and 4-dimensional split-signature conformal geometries with Cartan normal conformal connection reducible to an sl(3,R) connection 3) a relation between 2nd order Monge equations and (+,+,-,-,-) signature conformal geometries with Cartan normal conformal connection reducible to a connection with values in the Lie algebra of the noncompact form of the exceptional group G_2 The Cartan method of equivalence, which is the main technique to obtain the above results, will be discussed at the first lecture, so that a person not familiar with this approach be able to follow the course. Following lectures in February 4, 18, 21 and 23 (14:30)

Invariants of plane fronts (after Chekanov and Pushkar).

Speaker: S. Duzhin (Steklov Mathem. Inst., St. Petersburg)
Time: January 26, 14:30
Place: room B

A front is a plane curve whose singularities are only transverse self-intersections and semi-cubic cusps and whose tangent is nowhere vertical. Closed fronts are plane projections of Legendrian knots, i.e. knots in R^3 lying in the standard contact distribution du-pdq=0. Two Legendrian knots are said to be Legendrian equivalent, if one can be deformed to another by a continuous path in the space of Legendrian knots. Apart from the usual topological type of the knot in R^3, Legendrian knots have two additional integer invariants, the Maslov number and the Bennequin number. Initially, there was the conjecture that the concidence of these two numbers as well as the usual topological equivalence is sufficient for the Legendrian equivalence. This conjecture was disproved by Yu.Chekanov who invented a rather complicated additional invariant of Legendrian knots (the cohomology ring of a specially constructed differential graded algebra). After only mentioning this fact, we will pass to a more detailed description of a much more elementary construction due to P.Pushkar ("Pushkar's fences").

An overview on the orbital transfer problem

Speaker: R. Dujol (IRIT-ENSEEIHT, Toulouse, FRANCE)
Time: January 19, 14:30
Place: room B

The orbital transfer problem deals with a satellite motion along Keplerian orbits. This talk is concerned with the work done by the Optimal Control Group of the IRIT laboratory about this topic. Results due to the application of control theory as well as results from numerical computations will be adressed.

Singularities of Implicit differential Equations

Speaker: A. Remizov (M.V. Lomonosov Moscow State University, Moscow, Russian Federation)
Time: December 14, 11:00
Place: room B

We consider inplicit differential equations (IDE): F(z,y,p)=0, p=dy/dx, where x belongs to R^1, y,p,f belongs to R^n. Singular points of IDE are points (x,y,p) of the (2n+1)-dimensional space, where F=0 and det(dF/dp)=0, i.e. in such a point it is impossible to reduce IDE to an explicit equation p=(f(x,y). The first part of the lecture is devoted to the case n=1. This is a well-studied problem, the basic construction was invented by H.Poincare`. In this lecture some results of J.Bruce, L.Dara, A.Davydov, F.Tari will be presented. The second part of of the lecture is devoted to the case n>1. This is not so well understood problem (except of the case of quasilinear systems), and we are going to present some recent results.

Systems Biology

Speaker: V. Torre (SISSA)
Time: December 10, 11:00
Place: room B

Riccati equation for two fields of extremals

Speaker: M. Zelikin (Steklov Mathematical Institute, Department of Differential Equations)
Time: December 6, 11:00
Place: room B

Nonconvex variational problems and minimizing Young measures

Speaker: G. Dolzmann (University of Maryland, Mathematics Department)
Time: December 1, 16:30
Place: room B

ften fail to be weakly lower semicontinuous because the energy densities $f$ are not quasiconvex in the sense of Morrey. In this talk we analyse properties of minimizing Young measures generated by minimizing sequences for these variational integrals. We prove that moments of order $q>p$ exist if the integrand is sufficiently close to the $p$-Dirichlet energy at infinity. Analogous results hold true for solutions of systems of PDE that are close to the Euler-Lagrange equations for the $p$-Dirichlet energy. A counterexample related to the one-well problem in two dimensions shows that one cannot expect in general $L^\infty$ estimates, i.e., that the support of the minimizing Young measure is uniformly bounded. This is joint work with Jan Kristensen (Edinburgh) and Kewei Zhang (Sussex).

Generalization of Wilczynski invariants to nonlinear ODEs

Speaker: B. Doubrov (Belarussian State University, Minsk)
Time: December 1, 14:30
Place: room B

E.Wilczynski described in [1] all invariants of linear ODEs $y^{(n+1)}+p_{n}(x)y^{(n)}+\dots + p_0(x)y = 0$ considered up to the pseudogroup of transformations $(x,y)\mapsto (\lambda(x),\mu(x)y)$. These invariants can also be interpreted as differential invariants of non-parametrized curves in $n$-dimensional projective space. Using the universal linearization of ordinary differential equations, we show how these invariants can be generalized to become invariants of non-linear ODEs viewed up to contact transformations. Next, we discuss the role of these invariants in the canonical coframe for ordinary differential equations (see [2]). In particular, we show that if all these invariants vanish, then all other invariants of the canonical coframe become first integrals of the original equation. Further, using the technique developed in [3], we show how to constuct ODEs with vanishing Wilczynski invariants and present several non-trivial examples. References [1] E.J. Wilczynski, Projective differential geometry of curves and ruled surfaces, Leipzig, Teubner, 1905. [2] B. Doubrov, B. Komrakov, T. Morimoto, Equivalence of holonomic differential equations, Lobachevsky Journal of Mathematics, v.3, 1999, pp.39-71. [3] R. Bryant, Two exotic holonomies in dimension four, path geometries, and twistor theory}, Proc. Symp.\Pure Math. 53 (1991), pp. 33-88.

From discrete to Continuum via Gamma-convergence: an introduction

Speaker: M. Cicalese (SISSA)
Time: November 30, 16:30
Place: room B

Cycle of Seminars: In this series of seminars we will discuss the relation between discrete and continuous variational problems by Gamma-convergence methods. Following seminars in December 3 (14:30), 7, 9, 15, 16 (always 16:30).

Existence of solutions for a nonlinear Schroedinger equation with singular potential

Speaker: M. Badiale (Math Department, University of Torino)
Time: November 17, 16:00
Place: room B

Flows, momentum, coordinate invariance, and necessary conditions for optimality of curves

Speaker: H. Sussmann (Rutgers University, U.S.A.)
Time: November 17, 14:30
Place: ICTP Seminar Room

Ever since Lagrange proved the invariance of the Euler-Lagrange equation under arbitrary nonlinear coordinate changes, and recognized the enormous importance of this observation, the search for coordinate-invariant conditions for optimality of curves, and for coordinate-free formulations of these conditions, has been one of the driving forces leading to new and better results. In this talk we will first show how this search logically leads from the Euler-Lagrange conditions to two different possible Hamiltonian formulations. One of them represents the path that was actually followed, while the other one, the path not taken (and dramatically missed by giants such as Weierstrass and Caratheodory), could have led rather quickly to the discovery of more general conditions, and to the control point of view, in which the momentum is no longer defined as the gradient of the Lagrangian with respect to the velocity. We will then pursue the coordinate-invariant path, showing how it leads naturally to manifestly coordinate-free conditions involving Lie brackets and, in our most recent work, to conditions expressed in terms of flows rather than vector fields, and generalized rather than ordinary derivatives. These conditions apply to systems that are non-smooth in the sense of "non-smooth analysis" (i.e., involving locally Lipschitz vector fields) and also to many systems that are even less smooth than that, such as the reflected brachistochrone problem, which is Holder-1/2 but not Lipschitz.

On some equations arising in Nonlinear Optics

Speaker: V. Felli (University of Milano Bicocca - Dipartimento di Matematica e Applicazioni)
Time: November 16, 16:00
Place: room B

The seminar will be an overview of some equations arising in Nonlinear Optics. More precisely we will describe the TE-mode equation, the Master-Mode Locking equation and the Swift-Hohenberg equation, and present some results by Stuart-Zhou, Kapitula-Kutz-Sandstede and Akhmediev concerning the aforementioned equations.

Multiple solutions for a supercritical elliptic equation

Speaker: M. Badiale (Math Department, University of Torino)
Time: November 16, 14:30
Place: room B

Mathematical Virology: Mathematical models for the structure and assembly of viruses based on group theory and tiling theory.

Speaker: R. Twarock (City University of London)
Time: November 10, 16:00
Place: room A

Concepts from group theory and tiling theory are used to construct mathematical models for viral capsids, i.e. shells formed from proteins that encapsulate the viral genome. In particular, the surface lattices of viral capsids with overall icosahedral symmetry are classified. This approach leads to Viral Tiling Theory, which describes the locations of the protein subunits and inter-subunit bonds in viral capsids. It predicts the structure of viral capsids which have fallen out of the remit of any previous theory, and in particular solves a long-standing puzzle concerning the structure of cancer-causing viruses like papillomavirus. From a mathematical point of view, this involves the classification of the local symmetries of certain classes of spherical tilings with overall icosahedral symmetry. This talk is targeted at a mathematical audience and background knowledge in biology is not required.

A Representation of Quantum Operations on Two Q-bits in the Geometric Algebra of a Six-Dimensional Euclidean Vector Space

Speaker: T. F. Havel (MIT)
Time: November 10, 14:30
Place: room B

We extend recent work by Khaneja, Whaley and others on the Lie algebra of the unitary group for two Q-bits to more general quantum operations, using a well-known Lie algebra isomorphism between su(4) and so(6) to construct a purely Euclidean model that is amenable to analysis by geometric algebra methods.

On feedback classification of small-dimensional affine control systems with two inputs and obstacles for their flatness

Speaker: I. Zelenko (SISSA)
Time: November 4, 14:30
Place: room B

First we will discuss how to determine intrinsically the number of parameters in a classification problem, containing functional moduli. For this we analyze the Poincare series of moduli numbers of spaces of jets. Secondly, we show how the problem of feedback-equivalence of affine systems with two inputs in state space of dimension $4$ and $5$ can be reduced to the same problem for affine systems with one input, which was treated before. In order to make this reduction we distinguish the subsystem of our control system, consisting of the directions of all extremals in dimension $4$ and all abnormal extremals in dimension $5$ of the time optimal problem, defined by the original control system. Finally, for generic four-dimensional affine system with two inputs we construct feedback invariants, which are obstacles for their $1$-flatness. Namely, the system is $1$-flat if an only if these invariants vanish. The construction is based on the use of the coordinates of the Engel normal form for rank 2 distribution, corresponding to the affine system, and on the existence of the canonical projective structure on each abnormal extremal path of this distribution. Some results of the talk were obtained in collaboration with A. Agrachev and J.-B. Pomet.

Existence of conformal metrics with constant $Q$ curvature

Speaker: A. Malchiodi (SISSA)
Time: October 26, 14:00
Place: ICTP Seminar Room

Given a compact four dimensional manifold, we prove existence of conformal metrics with constant $Q$-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and minimax schemes, jointly with a compactness result for Palais-Smale sequences.

On feedback classification of affine control systems with one input and small-dimensional affine control systems with two inputs

Speaker: I. Zelenko (SISSA)
Time: October 20, 14:30
Place: room E

The present talk is devoted to the local classification of generic affine control systems on $n$-dimensional manifold with one input for any $n\geq 4$ or with two inputs for $n=4$ and $n=5$, up to state-feedback transformations, preserving the affine structure. First we associate with any such system the canonical coordinates. With the help of this coordinates in the case of affine control systems with one input and $n\geq 4$ we obtain that the set of orbits of generic germs of the systems w.r.t. the action of the group of state-feedback transformations, preserving the affine structure, can be parametrized (in $C^\infty$ category for $n=4$ and in real analytic category for $n\geq 5$) by $n-2$ arbitrary functions of $n$ variables and $(n-3)n$ almost arbitrary functions of $n-1$ variables (here the word "almost" means a kind of normalization of these functions on some coordinate subspaces of the canonical coordinates). Further we show how the problem of feedback-equivalence of affine systems with two inputs in state space of dimension $4$ and $5$ can be reduced to the same problem for affine systems with one input. In order to make this reduction we distinguish the subsystem of our control system, consisting of the directions of all extremals in dimension $4$ and all abnormal extremals in dimension $5$ of the time optimal problem, defined by the original control system. The talk is based on the joint work with Prof. A. Agrachev.

Solitary water waves and their stability by minimisation

Speaker: B. Buffoni (Ecole Polytechnique Federale de Luasanne)
Time: October 7, 09:30
Place: room A

A minimization method for a family of quasi-linear problems

Speaker: B. Buffoni (Ecole Polytechnique Federale de Luasanne)
Time: October 6, 15:00
Place: room E

Local controllability of a 1D Schrodinger equation

Speaker: K. Beauchard (Université Paris Sud, Département de Mathématiques)
Time: October 5, 14:30
Place: room E

We consider a non relativistic charged particle in a 1D potential well. This quantum system is subject to a control, which is a uniform electric field. It is represented by a complex probability amplitude solution of a Schrodinger equation. We prove the local controllability of this non linear system, around the ground state. Our proof uses the return method, a Nash-Moser implicit function theorem and moment theory.

Variational formulations of water waves

Speaker: B. Buffoni (Ecole Polytechnique Federale de Luasanne)
Time: October 5, 10:00
Place: room E

Dubins' problem on surfaces of non positive curvature

Speaker: M. Sigalotti (INRIA - Institut National de Recherche en Informatique et en Automatique, Sophia Antipolis, France)
Time: September 29, 14:30
Place: room B

Let $M$ be a complete, connected, two-dimensional Riemannian manifold with non positive Gaussian curvature $K$. We say that $M$ satisfies the {\it unrestricted complete controllability} property for the Dubins' problem (UCC for short) if the following holds: Given any $(p_1,v_1)$ and $(p_2,v_2)$ in $TM$, there exists a curve $\gamma$ in $M$, with arbitrary small geodesic curvature, such that $\gamma$ connects $p_1$ to $p_2$ and, for $i=1,2$, $\dot\gamma$ is equal to $v_i$ at $p_i$. Property (UCC) is equivalent to the complete controllability of a family of control systems of Dubins' type, parameterized by the (arbitrary small) prescribed bound on the geodesic curvature. It is well-known that the Poincar\'e half-plane does not verify property (UCC). During the talk, we will show that UCC) holds if $M$ is of the first kind. Moreover, the converse will be proved to be true if $\sup_M K<0$. Further necessary and sufficient conditions will be discussed when $\sup_M K=0$ and $K$ is bounded from below. Joint work with Yacine Chitour.