Abstract
In this paper we present an overview of the direct relation between differential invariants of projective type for curves in flat semisimple homegenous spaces and PDEs of KdV type. We describe the progress in the proof of a conjectured Theorem stating that for any such space there are geometric evolutions of curves that induce completely integrable evolutions on their invariants of projective-type. The Theorem also states that these evolutions decouple always into either complexly coupled KdV equations (conformai type), decoupled KdV equations (Lagrangian type) or Adler-Gelfand-Dikii generalized KdV types (projective type). The paper also describes the fundamental role of group-based moving frames in this study.
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