Work

The investigation of algebraic and topological structures by methods of group theory and computer algebra is an important field of study in mathematics that has a number of applications. In particular, the research project contains the study of such algebraic structures as nearrings, groups, cohomologies of groups, symmetry groups. Using GAP we plan to construct and investigate some classes of nearrings with identity with a view of classification of such models. The theory of cohomologies of groups was one of the origins of the homological algebra. It was also related to the theory of group extensions and projective representations, where cohomologies arise as factor sets. This theory is widely used in topology, number theory, algebraic geometry. One of the research directions is the study of the symmetry group (homeotopy group is an analogue of the mapping class group) of the Morse function stabilizer and the automorphism group of the Kronrod–Reeb graph defined for this function. Each diffeomorphism of stabilizer of a function induces an automorphism of the Kronrod–Reeb graph, that is, a homomorphism between these groups is defined. The kernel of this homomorphism is a subgroup of stabilizer consisting of diffeomorphisms that induce the identity map on a graph, that is, diffeomorphisms that leave invariant every regular connected component of every level set of the function. The idea of the study is to consider the kernel of this diffeomorphism and to establish an isomorphism between the homeotopy group of stabilizer and some subgroup of automorphisms of the Kronrod–Reeb graph.