To every transformation of an uncountable set one can naturally associate a group of transformations which is "full" in some sense. The notion of full group was defined by H.A. Dye in 1959. These groups turned out to be of extreme importance in various areas of dynamical systems, especially in the theory of orbit equivalence in measurable, Borel, and Cantor dynamics.
The second talk will be mostly focused on old and new results related to topological and algebraic properties of full groups in measurable, Borel and Cantor dynamics. In particular, we will discuss simplicity and connectedness of full groups, amenability, and the structure of dense subgroups. All necessary definitions will be given.