Mar 22, 14h:00 - Boris Khesin (University of Toronto)

The pentagram map on polygons in the projective plane was introduced by R.Schwartz in 1992 and is by now one of the most popular and classical discrete integrable systems. We survey definitions and integrability properties of the pentagram maps on generic plane polygons and their generalizations to higher dimensions. In particular, we define long-diagonal pentagram maps on polygons in , encompassing all known integrable cases. We also describe the corresponding continuous limit of such pentagram maps: in dimension d it turns out to be the -equation of the KdV hierarchy, generalizing the Boussinesq equation in 2D. This is a joint work with F.Soloviev and A.Izosimov.

Mar 22, 16h:00 - Valerio Assenza (IMPA)

Magnetic flows are the toy models for the motion of a charged particle over a Riemannian manifold under the influence of a magnetic force. These dynamics appear in several physical and mathematical contexts and they had been investigating with different approaches in the last four decades. From a geometrical point of view, to every magnetic system we associate an operator called magnetic curvature. Such an operator encodes the classical Riemannian curvature together with terms of perturbation due to the magnetic interaction, and it carries relevant properties in terms of magnetic dynamics. For instance, in this talk we bring into the discussion of magnetic flows a theorem by E.Hopf (refined by Green): if a geodesic flow is without conjugate points, then the total scalar curvature is non positive and equal to zero if and only if the metric is flat. The generalization of this result leads naturally to investigate the flatness in a magnetic sense. This is part of a joint work with James Marshall Reber and Ivo Terek.