Relations between Toda and KdV-type hierarchies

In this talk we present various Lie algebra decompositions in the $\mathbb{Z}\times \mathbb{Z}$-matrices and the pseudo differential operators that each give rise to systems of compatible Lax equations. All these systems can also be formulated in zero curvature form and possess appropriate, associated Cauchy problems. The construction of solutions of these hierarchies requires infinite dimensional flag varieties and various bundles over them.
Moreover, these solutions can be expressed in determinants of operators arising in this geometric picture. This furnishes a uniform picture that leads to the relations mentioned in the title.