Explore a 49-minute mathematics lecture from the Erwin Schrödinger International Institute's Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" that delves into nonlinear Grassmannians and their relationship with coadjoint orbits of diffeomorphism groups. Learn how manifolds of submanifolds describe various coadjoint orbits, including those of all diffeomorphisms, volume preserving diffeomorphisms, symplectomorphisms, and contact diffeomorphisms. Discover how different geometric structures require specific augmentations of nonlinear Grassmannians with additional submanifold data. Examine the smooth structure of these augmented nonlinear Grassmannians and understand their practical application in constructing singular solutions of the geodesic equation on diffeomorphism groups. The lecture presents collaborative research conducted with Cornelia Vizman.