Ultracoproducts of $G$-flows
Given a topological group $G$, a $G$-flow is a continuous action of $G$ on a compact Hausdorff space $X$. This talk will discuss a notion of ultracoproduct for $G$-flows, which arise from considering ultraproducts of commutative $G$-$C^∗$-algebras by Gelfand duality. We apply the construction to develop an understanding of the properties of various classes of subflows of a flow, i.e. minimal, topologically transitive, etc. For groups which are locally Roelcke precompact, ultracoproducts of $G$-flows lead to a well-behaved notion of weak containment for a wide class of $G$-flows, and in particular for all $G$-flows when $G$ is locally compact. In ongoing joint work with Gianluca Basso, we apply ultracoproducts of $G$-flows to achieve a new characterization of those Polish groups $G$ with the property that every minimal flow has a comeager orbit.