Southern California Analysis and PDE Conference Day 1

A geometric realization of an integrable system is an evolution of curves on a manifold M, invariant under the action of a group G, and such that it becomes the integrable system when the action of G is mod out. The best known geometric realization is the Vortex filament flow (VF), a flow of curves in Euclidean space which is invariant under the Euclidean group. The VF equation becomes the nonlinear Shrodinger equation when written in terms of the natural curvatures of the flow - Hasimoto proved this way the integrability of VF -. In this talk I will review what is known about the classical geometry of curves in homogeneous spaces and its relation to different types of integrable systems. In particular we will talk about how one can reduce different Hamiltonian structures to the space of curvatures (Euclidean, projective, conformal, etc) and how those reductions indicates the existence of biHamiltonian (integrable) systems. We will also describe how curvatures of Schwarzian type for curves in homogeneous spaces usually describe evolutions of KdV type. I will present background material so the talk should be accessible, at least in part, to different audiences.

The shadowing problem is related to the following question: under which condition, for any pseudotrajectory (approximate trajectory) of a vector field there exists a close trajectory? We study $C^1$-interiors of sets of vector fields with various shadowing properties. In the case of discrete dynamical systems generated by diffeomorphisms, such interiors were proved to coincide with the set of structurally stable diffeomorphisms for most general shadowing properties.

We prove that the $C^1$-interior of the set of vector fields with Oriented shadowing property contains not only structurally stable vector fields. Also, we have found additional assumptions under which the $C^1$-interiors of sets of vector fields with Lipschitz, Oriented and Orbit shadowing properties contain only structurally stable vector fields.

Some of these results were obtained together with my advisor S.Yu.Pilyugin.