


 
 Динамические системы, геометрия и теория управления 
 
 международная школасеминар 


  

Курсы лекций: Andrei Agrachev (SISSAISAS, Italy)
Dissipative systems and infinite horizon variational problemsWe are going to discuss the longtime behaviour of mechanical systems with an isotropic friction. Such a system admits a natural least action principle with the timevarying Lagrangian. We study the longtime behaviour in the Euler representation. The Euler representation means that an observer is sitting in the given point of the configuration space and is measuring velocity of the particle passing the point. The "longtime" means that time spent by the particle to arrive to the point at the moment of the measurement is much bigger than the initial velocity of the particle. We are interested in the limiting value of the velocity as the ratio <time>/<absolute value of the initial velocity> tends to infinity. The limit exists at every point and forms a smooth potential vector field that has a natural variational and geometric characterization if the friction is sufficiently strong. Critical value of the friction has a geometric meaning: bigger "the curvature of the system" stronger friction we need to guarantee the existence of the limiting velocity. Another important tool besides the curvature is partially hyperbolic dynamics. The topic is indeed an interplay of Geometry, Dynamics, and Optimal Control. I am going to explain all necessary background (mainly in the expositary style).
 Dmitry Alekseevsky (University of Hull, England)
Conformal Geometry and VisionThe first 2 lectures contain an elementary exposition of conformal geometry of the sphere, (including theory of conformal circles («conformal geodesics») and conformal curves); basic facts of contact and symplectic geometry. Two last lectures are devoted to application of conformal geometry to vision. One lecture gives a short description of basic facts of early vision: basic principles, transformation of information in retina, retinotopic conformal map from retina to LGN and visual V1 cortex. The last lecture gives an exposition of some geometric models: Petitot contact model of V1 cortex, PetitotCittiSarti symplectic model, BressloffCowan spherical model of hypercolumns and some unification of these models.
 JeanPaul Gauthier (LSIS, France)
Motion Planning in the SubRiemannian ContextA subriemannian metric being given, I shall give a series of lectures about the problem of approximating or interpolating a nonhorizontal curve by an horizontal one. These approximations or interpolations are constructed explicitly, with applications to robotics and more generally to the control of nonholonomic kinematic systems.
 Yanqi Qiu (LATP, France)
Determinantal Point ProcessesIn this lecture, I will first recall the materials on the theory of determinantal point processes and then I will mainly focus on the tolerance and rigidity phenomenon in the sense of Ghosh and Peres for several interesting determinantal point processes.
 Stanislav Smirnov (Université de Genève, Switzerland)
TBA
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