We shall discuss the structure of topologically invariant \(\sigma\)-ideals with Borel base on homogeneous Polish spaces. An ideal \(\mathcal I\) of subsets of a topological space \(X\) is called topologically invariant if for each set \(A\in\mathcal I\) and each homeomorphism \(h:X\to X\) we get \(h(A)\in\mathcal I\).
During the lecture I plan to cover the following topics:
(1) classification of topologically invariant \(\sigma\)-ideals on topologically homogeneous zero-dimensional Polish spaces;
(2) extremal (i.e., maximal, largest, smallest) topologically invariant \(\sigma\)-ideals on some "nice" Polish spaces;
(3) cardinal characteristics of topologically invariant \(\sigma\)-ideals on some "nice" Polish spaces and their interplay with the cardinal characteristics of the ideal \(\mathcal M\) of meager sets.