The defense of the candidate's dissertation of Vladislav Evgenievich Kruglov

On June 21, 2023, Vladislav E. Kruglov defended his PhD thesis on the topic "Modules of topological conjugacy of Ω-stable flow surfaces"; scientific supervisor: Doctor of Physical and Mathematical Sciences, Professor Olga Vitalievna Pochinka

The dissertation work is devoted to the topological classification of various subclasses of the class of Ω-stable flows on surfaces up to topological conjugacy. When solving such a problem, the question inevitably arises about the presence and number of special analytical invariants — moduli of stability or, in other words, moduli of topological conjugacy. It is with such moduli that the most beautiful and important results of the dissertation work are connected.
Within the framework of the study, it is proved that gradient-like flows on surfaces are topologically conjugate if and only if they are topologically equivalent, effective algorithms for recognizing the isomorphism of the main invariants of gradient-like flows on surfaces are constructed; necessary and sufficient conditions for finiteness of the number of moduli for Morse-Smale flows are found, and for Morse-Smale flows with finite number of moduli a complete invariant of topological conjugacy is constructed; for Ω-stable flows a complete classification is constructed in the sense of topological equivalence by means of a combinatorial invariant; it is established that polar flows with saddle points connected to each other by both separatrices have a finite number of moduli.
The Defense Committee voted unanimously to award the degree of Candidate of Mathematical Sciences, highly appreciating the work done. The materials of the dissertation can be found at the link.