Lychak D. Functions on two-dimensional manifolds

Functions on surfaces are considered in thesis. It is proved that Kronrod-Reeb graph with the additional information (signs) determines a Morse function of a general position on a two-dimensional manifold. It is proved that every Morse vector field with the numeration of saddle points corresponds to a unique Morse function. An fd-graph is constructed. It is proved that the fd-graph determines the atom of a critical level of a smooth function with isolated critical points on a two-dimensional manifold. A complete topological classification of smooth functions with isolated singularities on surfaces is obtained. An equipped Kronrod-Reeb graph is constructed for a smooth function with simple singularities, which belong to the different level lines. It is proved that it determines a smooth function with simple singularities, which belong to the different level lines, up to a smooth equivalence. A global smooth classification of such functions is obtained.