The first part of the project is concerned with a classification of the orbit equivalence relation E coming from the conjugation action of the group of all diffeomorphisms of class C2 on itself. A well-known example given by Arnold shows that there exist C2 diffeomorphisms of the circle with equal rotation numbers, which are not conjugate by any smooth mapping. This raises a natural question as to how complicated relation E is. Methods coming from Borel reducibility theory will be used to estimate lower and upper bounds for complexity of E. In particular, the following problems will be studied. Is E essentially more complicated than the identity relation? Is D reducible to an equivalence relation with countable equivalence classes? Can D be classified by the isomorphism relation on a class of countable models? The second part of the project is a continuation of a line of research initiated by Gao and Kechris. It is devoted to studying Polish ultrametric spaces, that is, metric spaces satisfying a strong version of the triangle inequality, and their isometry groups. A structure theorem, proved by the executioner of the project, representing each separable ultrametric space as a 'bundle' with an ultrametric base and with homogeneous fibers will be further investigated. Its detailed study and analysis of the limit behavior of involved quotient maps will be used to characterize Polish ultrametric spaces and their isometry groups. This will provide an answer to a question posed by Gao nad Kechris. The implementation of the project will allow the executioner of the project to develop a solid research portfolio in a lively developing field of mathematics, contributing in this way to their lasting reintegration, and to European scientific excellence.
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