Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

I will begin by reviewing the simplest nontrivial case of a Lagrangian Grassmannian manifold, namely the three-dimensional L(2,4), stressing its remarkable isomorphism with the Lie quadric Q^3, mentioning also its "meta-symplectic counterpart", the four-dimensional L(2,5), which is a rather unexplored object. In spite of the low dimensionality of the objects involved, even at this level, it is possible to formulate rather tricky questions. Then I will switch to the PDE side, examining the standard framework for 2nd order 2D (nonlinear) PDEs based on the Lagrangian bundle of a 5D contact manifold (perhaps better known as "2nd order jet space"), with a particular emphasis on the geometric formulation of Cauchy problems, which, in turn, needs the notion of a characteristic Cahuchy surface. Coming back to the Lagrangian Grassimannians, I will show that the PDEs correspond, in fact, to their hypersurfaces, and that the presence of a lot of characteristics is a well-known phenomenon in Algebraic Geometry known as a "ruling". The advantage of such a bridge between the two disciplines is that is allows to recast known results in a more transparent way, and to formulate new ones as well, especially in the meta-symplectic context, i.e., that of 3rd order 2D (nonlinear) PDEs. As an example, I will demonstrate that 2D parabolic Monge-Ampere equations correspond to the so-called hyperplane sections of Q^3. To conclude with, I will bring in the dual variety of the generic Lagrangian Grassmannian L(n,2n), as well as its singular loci, which are in correspondence with remarkable classes of PDEs, like, e.g., the linearisable ones. In such a framework, a still open conjecture by E. V. Ferapontov, about the integrability of n-dimensional PDEs, may become more easy to work out. The content of this seminar is based on the project "GEOGRAL", which I'm going to carry out at IMPAN with a Marie Skłodowska-Curie fellowship, commencing September 1st. Due to the highly multidisciplinary character of GEOGRAL, collaborative efforts are mandatory, and I hope that this small introduction will stimulate the attention of local personnel potentially interested in joining in.