Hladysh B. Functions with critical points on the boundary of low-dimensional manifolds

The results, obtained in this dissertation, are devoted to the study of local and global classifications of smooth functions with isolated (non-degenerate) critical points on manifolds with the boundary, and also the deformations in general po-sition of such functions are advised. The first considered functions class consists of Morse functions, defined on smooth manifolds with the boundary, all critical points of which belong to the boundary, and correspondent functions restriction are also Morse functions (namely mm-functions). It is obtained for these functions the local presentation about a critical point in the special form of the quadratic polynomial. In the case of simple mm-functions on the surface, we describe all possible atom and f-atoms. In addition, it is constructed the conditions of minimization of singularities number in this functions class and founded the necessary conditions of continuation of simple Morse function, defined on the oriented surface boundary, to the simple optimal mm-function being defined on the whole surface. Another researched functions class includes functions with non-degenerate singularities (either inner or boundary) being defined on the compact oriented surface. It is proved their topological equivalence to m-functions and constructed the combinatorial invariant in the form of equipped Kronrod-Reeb graph for further layer equipped equivalence. Furthermore, we illustrate all simple atoms and deduce the formula for surface gender, using for it the information obtained from the equipped KR-graph. Another investigation direction is deformations in the general position of either one of the above-described functions class or of Morse functions class defined on closed surfaces. These deformations were described in the context of deformations of correspondent Kronrod-Reeb graphs (in the case of Morse function the equipped KR-graph is Reeb graph in the usual sense). In addition to this, we procure the connections between optimality and polarity each of func-tions classes defined on a smooth compact oriented surface with the connected boundary (the boundary allows to be empty). The last considered in the thesis functions class consists of simple functions, all critical points of which are isolated, belong to the boundary, and the sets of function singularities and singularities of function restriction coincide. It is obtained their local topological classification, described atoms, and constructed optimality criterion. Subsequently, we investigate the full topological invariant for such functions being also optimal in the form of the chord diagram of saddle critical point.