**Geometric Realizations of Completely integrable Systems
**

**Abstract:**

An M-Geometric realization of a PDE evolution is an evolution of curves on
a manifold M such that: a) the curve flow is invariant under the action of a
certain group G, b) the flow induced on the differential G-invariants of the
curves (or curvatures) is the original PDE. The best known example is the NLS
equation whose geometric realization in R^3, invariant under the Euclidean
group, is a version of the Vortex Filament flow, as it was pointed out by
Hasimoto.

In this talk we describe the geometric background of completely integrable systems from the point of view of Invariant Theory. We show how group-based moving frames and Cartan approach to Geometry can be effectively used to find geometric realizations of completely integrable system in different homogeneous spaces. From this point of view we will describe the Poisson Geometry of biHamiltonian systems, connections to AKNS spectral representations and more. Some not so well known examples (like a geometric realization of a decoupled system of KdV equations in the Lagrangian Grassmannian under the symplectic group, and its connection to the projective geometry of Lagrangian planes) will be presented. The talk will be partially accessible for those with little background in differential geometry.