Work

The work “Geometrical and analytical methods of investigation of sets of complex and hypercomplex spaces and their maps” is dedicated to solving some important problems of Complex and Hypercomplex Analysis, Theory of generalized Convex sets. The main results are described in 25 leading papers. Among them there are: solved a number of problems associated with the problem of V.M. Dubinin of the maximum of product of inner radii of n non-overlapping domains containing points of unit disk and the power γ of the inner radius of a domain containing the origin; proposed an approach that allows to transfer the most of the results of geometric function theory to the case of arbitrary multidimensional complex spaces; the dimensional analogues of the known theorems of geometrical theory of analytical functions are obtained; the generalization on the case of the domains of arbitrary connectivity and essential specification with a description of extremal configurations of the known inequality of G.V. Kuzmina on inner radii of non-overlapping simply connected domains with respect to points on the unit circle is obtained; the sufficient conditions for multivalued includings in Euclidean spaces are proved; the criteria about affinity of mappings of real multi-dimensional space which are strictly invariant at the set of vertices of rectangular parallelepipeds and at the set of vertices of suspentions is obtained; the hypercomplex analogue of the theorem about half-affinity of the transformation of multidimensional complex space which is homeomorphic and strictly invariant at linearly convex compacts is proved; the problems related to Mizel-Zamfirescu problem are solved; the necessary and sufficient analytical conditions of left (right) locally linear convexity of domains with smooth boundary of multy-dimensional quaternion space and of multy-dimensional space over generalized quaternion algebra are obtained; based on the research of rat brain conducted in the Center for Molecular Biology and Neuroscience and Institute of Basic Medical Sciences, University of Oslo, Norway, the model of dependence of pontine nuclei structure upon the cerebral cortex of rat with the help of power function of complex variable is built. The applicants have 84 publications, including 25 based on the work.