Piotr Bizoń, Niels Martin Møller
| 16:15 | Piotr Bizoń | Resonant dynamics in spatially confined Hamiltonian systems The long-time behavior of nonlinear dispersive waves subject to spatial confinement can be very rich and complex because, in contrast to unbounded domains, waves cannot escape to infinity and keep self-interacting for all times. If, in addition, the linear spectrum around the ground state is fully resonant, then the nonlinearity can produce significant effects for arbitrarily small perturbations. The weak field dynamics of such systems can be approximated by solutions of the corresponding infinite-dimensional time-averaged Hamiltonian systems which govern resonant interactions between the modes and a major mathematical challenge is to describe the energy transfer between the modes. I will discuss some recent progress in understanding this problem, emphasizing universal features of resonant dynamics arising in very different physical contexts such as, for instance, dynamics of Bose-Einstein condensates (modeled by the nonlinear Schroedinger equation with a trapping potential) or a weakly turbulent behavior of small perturbations of the anti-de Sitter spacetime (modeled by the Einstein equations with negative cosmological constant). |
| 17:45 | Niels Martin Møller | Translating Solitons and (Bi-)Halfspace Theorems for Minimal Surfaces I will present new results on the classification problem for complete self-translating hypersurfaces for the mean curvature flow. Such surfaces show up as singularity models in the flow (along with other types of solitons, e.g. the self-shrinkers), and have been studied since the first examples were found by Mullins in 1956. Examples from gluing constructions show that one cannot easily classify such solitons - nor can one classify their projections to one dimension lower, nor their convex hulls. But if one does both of these "forgetful" operations, the list becomes very short, coinciding with (and implying) the one given by Hoffman-Meeks in 1989 for minimal submanifolds: All of R^n , halfspaces, slabs, hyperplanes and convex compacts in R^n. This also implies several of the known obstructions to existence, e.g. for convex translating solitons. This is joint work with Francesco Chini (U Copenhagen). |
