In this talk, the relationship between integrable systems and invariant curve flows is studied. It is shown that many integrable systems including the well-known integrable equations and Camassa-Holm type equations arise from the non-stretching invariant curve flows in Klein geometries. The geometrical formulations to some properties of integrable systems are also given.

I will introduce a parabolic flow of almost K\"ahler structures,
providing an approach to constructing canonical geometric structures on symplectic manifolds. I will exhibit this flow as one of a family of parabolic flows of almost Hermitian structures, generalizing my previous work on parabolic flows of Hermitian metrics. I will exhibit a long time existence obstruction for solutions to this flow by showing certain smoothing estimates for the curvature and torsion. Finally I will discuss the limiting objects as well as some open problems related to the symplectic
curvature flow.

The integral homology of a compact Hermitian symmetric spaces (CHSS) is generated by the homology classes of its Schubert varieties. Most Schubert varieties are singular. In 1961 Borel and Haefliger asked: when can the homology class [X] of a singular Schubert variety be represented by a smooth subvariety Y of the CHSS?

Remarkably, the subvarieties Y with [Y] = [X] are integrals of a (linear Pfaffian) differential system. I will discuss recent work with Dennis The in which we give a complete list of those Schubert varieties X for which there exists a first-order obstruction to the existence of a smooth Y. This extends (independent) work of M. Walters, R. Bryant and J. Hong.

The sine qua non of our analysis is a new characterization of the Schubert varieties by a non-negative integer and a marked Dynkin diagram. The description generalizes the well-known characterization of the smooth Schubert varieties by subdiagrams of the Dynkin diagram associated to the CHSS.

I will illustrate the talk with examples.