KP equations, M-curves and nonnegative Grassmannians, Piotr Grinevich

Title: KP equations, M-curves and nonnegative Grassmannians Abstract: In the theory of KP2 equations there exist two methods to determine real nonsingular multi-soliton solutions (whose geometrical properties are quite intriguing and closely related to Tropical Geometry). The first method is by doing Darboux' transformations in the points of a completely nonnegative Grassmannians. The second method is by degenerating M-curves with properly chosen divisors. Both objects (i.e. nonnegative Grassmanians and M-curves) show up in many different problems in various branches of Mathematics. The purpose of our work was to investigate the way in which these two objects are related with each other. In more formal terms, to find a method to associate a degenerate M-curve with a divisor on it to a point in nonnecgative Grassmannian, so that the corresponding KP2-solutions are the same. Based on joint papers with