We now have a huge list of “nonlinear integrable systems.” Most of them are more or less related to so called “soliton” phenomena, thereby referred to as “soliton equations.” In the last decade, various soliton equations came to be reorganized in a unified framework, the Kadomtsev-Petviashvili (KP) hierarchy and its family [Sa-Sa] [Se-Wi]. Beside these soliton equations, there are a few excepti...

The real vector space of non-oriented graphs is known to carry a differential graded Lie algebra structure. Cocycles in the Kontsevich graph complex, expressed using formal sums on $n$ vertices and $2n-2$ edges, induce -- under orientation mapping infinitesimal symmetries classical Poisson structures arbitrary finite-dimensional affine manifolds. Willwacher has stated existence nontrivial cocyc...

We study the question of integrability Einstein deformations and relate it to desingularization metrics. Our main application is a negative answer long-standing whether or not every 4-orbifold (which an metric space in synthetic sense) limit smooth 4-manifolds. more precisely show that spherical hyperbolic 4-orbifolds with simplest singularities cannot be Gromov-Hausdorff limits 4-metrics witho...

We present proofs of two classical theorems. The first one, due to Darboux and Sauer, states that infinitesimal rigidity is a projective invariant; the other one establishes relations (infinitesimal Pogorelov maps) between the infinitesimal motions of a Euclidean framework and of its hyperbolic and spherical images. The arguments use the static formulation of infinitesimal rigidity. The duality...