Maslov index on symplectic manifolds infinitesimal Lagrangian manifolds

This paper is a summary of the report at the conference “Semiclassical analysis and nonlocal elliptic problems-2023”. The definition of the Maslov index of a Lagrangian manifold as a class of one-dimensional cohomologies on it gave rise to numerous works generalizing the concepts of the Maslov index. In the works by V. I. Arnold, V. A. Vassiliev and their followers, the theory of Lagrangian bordisms was developed and characteristic classes of Lagrangian submanifolds were constructed on its basis. But there is another approach to describing the Maslov classes of Lagrangian submanifolds, presented in the works by V. V. Trofimov and A. T. Fomenko from a categorical point of view, which served as the source of this report. Inspired by the works by V. V. Trofimov and A. T. Fomenko, we introduce the concept of the so-called infinitesimal Lagrangian manifolds, which, in our opinion, allow us to describe the characteristic classes of Lagrangian manifolds with maximum completeness and calculate the Maslov index for almost any Lagrangian manifold. The question that interests us is the following: when does the Maslov index defined on an individual Lagrangian manifold as a one-dimensional cohomology class become the image of some one-dimensional cohomology class of the total space of the bundle of Lagrangian Grassmannians? An answer is given for various classes of bundles of Lagrangian Grassmannians.