Caltech Logic Seminar
Time: Wednesday, September 23, 12:00 – 1:00pm Pacific time (21:00 CEST)
Speaker: Dana Bartošová, University of Florida
Title: On phase spaces of universal minimal flows of groups with compact normal subgroups
Abstract: For a topological group G, a G-flow is a continuous action of G on a compact Hausdorff space X; we call X the phase space of the G-flow. A G-flow on X is minimal if X has no closed non-trivial invariant subset. The universal minimal G-flow, M(G), has every minimal G-flow as a quotient and it is unique up to isomorphism. We show that whenever we have a short exact sequence 0→K→G→H→0 of topological groups with the image of K a compact normal subgroup of G, then the phase space of M(G) is homeomorphic to the product of the phase space of M(H) with K. For instance, if G is a Polish, non-Archimedean group, and the image of K is open in G, then H is a countable discrete group. The phase space of M(H) is homeomorphic to Gl(22ℵ0), the Stone space of the completion of the free Boolean algebra on 2ℵ0 generators by Balcar-Błaszczyk and Glasner-Tsankov-Weiss-Zucker. Therefore, the phase space of M(G) is homeomorphic to K×Gl(22ℵ0). When the sequence splits, that is, G≅H⋉K, then the homeomorphism witnesses an isomorphism of flows, recovering a result of Kechris and Sokić.
Information: Online talk https://caltech.zoom.us/j/99296122790?pwd=bUN4RS94RVYrTEtGTGhqTHRJbm9nZz09
Southern Illinois University Logic Seminar
Time: Thursday, September 24, 1pm US Central Daylight Time (20:00 CEST)
Speaker: Arno Pauly, Swansea University
Title: How computability-theoretic degree structures and topological spaces are related
Abstract: We can generalize Turing reducibility to points in a large class of topological spaces. The point degree spectrum of a space is the collection of the degrees of its points. This is always a collection of Medvedev degrees, and it turns out that topological properties of the space are closely related to what degrees occur in it. For example, a Polish space has only Turing degrees iff it is countably dimensional. This connection can be used to bring topological techniques to bear on problems from computability theory and vice versa. The talk is based on joint work with Takayuki Kihara and Keng Meng Ng (https://arxiv.org/abs/1405.6866 and https://arxiv.org/abs/1904.04107).
Information: The seminar will take place virtually via zoom.
CUNY Set Theory Seminar
Time: Friday, September 25, 11:00 New York time (17:00 CEST)
Speaker: Ralf Schindler, University of Münster
Title: Martin’s Maximum^++ implies the P_max axiom (*)
Abstract: Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and ‘consistent’ needs to mean ‘consistent in a strong sense.’ It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. This is joint work with D. Asperó building upon earlier work of R. Jensen and (ultimately) Keisler’s ‘consistency properties’.
Information: The seminar will take place virtually. Please email Victoria Gitman (email@example.com) for the meeting id.