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Classifying the conjugacy relation of the group of C2 diffeomorphisms of the unit circle, and characterizing isometry groups of separable ultrametric spaces

Final Report Summary - REDUCE (Classifying the conjugacy relation of the group of C2 diffeomorphisms of the unit circle, and characterizing isometry groups of separable ultrametric spaces.)

From the point of view of scientific objectives, the overall realization of the project considerably exceeded its initial objectives, even though some of the goals have not been completed. The project proposal consisted of two fairly independent parts, the second of which has been completely fulfilled, and even pushed much further compared to initial expectations and hopes. Moreover, new research directions revealed themselves in the course of the realization of the project, which led to interesting and unexpected results, as well as new scientific collaborations. On the other hand, the research devoted to the first part of the project objectives has been initiated only in the last months of the project realization. Nevertheless, the researcher has good reasons to believe that it will soon lead to new results.

The results of the project have been announced in the form of ten seminar and conference talks, and six papers.

During the realization of the project, the researcher's host organization, that is, the Institute of Mathematics of the Polish Academy of Sciences, offered Dr Malicki a full employment contract, including requirements contained in Annex III of Grant Agreement.

As a part of his activities, Dr Malicki participated in weekly seminars held at the host organization and Warsaw University, where he gave talks on his research progress, and topics related to the project. He also participated in the Insitute's Phd student seminar, giving talks aimed at broadening students' knowledge about contemporary research directions in mathematics.

Dr Malicki fully benefited from the reintegration period, established new collaborations links with Polish and international mathematicians, and broadened his knowledge and research interests. Among other activities, he initiated a collaboration with prof Winfried Just, Ohio University, USA, that led to a fruitful reserach in the field of mathematical biology. He was also a participant to the seminar on Bioinformatics and Mathematical Biology held at the Department of Mathematics of Warsaw University.

Main scientific results of the realization of the project:
1. The researcher studied full isometry groups of Polish ultrametric spaces. He defined a separable variant the unrestricted generalized wreath product, and proved that it has natural uniqueness and universality properties. Then he isolated a class of Polish spaces, called W-spaces, and proved that isometry groups of W-spaces can be described using this construction. Also, he obtained a complete characterization of those W-spaces, whose isometry groups have uncountable strong cofinality.
2. The researcher studied full automorphism groups of countable rooted trees in connection with two group properties: uncountable strong cofinality, and ample generics. He characterized automorphism groups of countable rooted trees with these properties in terms of algebraic closures of finite subsets of a tree.

He also gave a new proof of the theorem that for every countable rooted tree, its full automorphism group has property (FA').