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Friday, April 29, 2022

**Abstract:** The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation $T$ has emerged. This picture included substantial evidence that pointed to these groups being all topologically isomorphic to a single group, namely, $L^0$---the topological group of all Lebesgue measurable functions from $[0,1]$ to the circle. In fact, Glasner and Weiss asked if this was the case. We will describe the background touched on above, including the relevant definitions and the connections with Descriptive Set Theory. Further, we will indicate a proof of the following theorem that answers the Glasner--Weiss question in the negative: for a generic measure preserving transformation $T$, the closed group generated by $T$ is {\bf not} topologically isomorphic to $L^0$. The proof rests on an analysis of unitary representations of the non-locally compact group $L^0$. PS: There will be a reception at 3 (common room)