Recursion Theory/Descriptive Set Theory Seminar - Fall 2025

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Time: Fridays 11:00-12:30
Location: 736 Evans

Topic: Generalized Fraïssé theory, and generic structures, actions, and applications. In a classic 2007 paper, Kechris and Rosendal used Fraïssé theory to understand the dynamics of the Polish group of automorphisms of a countable homogeneous structure. In the last couple decades, there have been some generalizations of Fraïssé theory beyond the classical setting of countable collections of finitely generated structures. Particularly, Irwin and Solecki's Projective Fraïssé limits, and more recently Doucha, Melleray, and Tsankov's topological Fraïssé classes. These generalizations are likewise being applied to understand more general settings of generic spaces, structures, and actions. For example, current topics of interest in group theory (whether there are generic isomorphism classes in various spaces of countable groups), topological dynamics (what spaces of subshifts have comeager conjugacy classes), Borel reducibility (what groups have generic actions that are hyperfinite), and operator algebras (is there a generic isomorphism class in the space of II_1 factors, generalizing the negative answer to Connes embedding). We will discuss the classical theory and then its generalizations and applications.

Tentative schedule:

• Sep 5: Introduction (Andrew)
• Sep 12: Intro to Fraïssé limits (Atticus)
• Sep 19: Josh Frish: Minimal Subdynamics: Descriptive ideas about Dynamical Questions.
  Let Γ be a countably infinite discrete group. A Γ-flow X (i.e., a nonempty compact Hausdorff space equipped with a continuous action of Γ) is called S-minimal for a subset S⊆Γ if the partial orbit S⋅x is dense for every point x∈X. (When S=Γ, we recover the usual notion of minimality.) Despite the simplicity of the definition, given a group Γ, finding an S-minimal dynamical system is typically quite difficult (in particular even when Γ is the free group and S is a subgroup it was not previously known). In this talk, I will discuss a very recent result on how to construct S-minimal systems for any countable collection of infinite subsets simultaneously. Although the problem is purely dynamical, the techniques make heavy use of recent ideas from descriptive set theory. Indeed, once the main result is established, we can return to derive some non-obvious, purely Borel, corollaries. This is joint work with Anton Bernshteyn
• Sep 26, 26, Oct 3, Oct 17: Fraïssé theory and conjugacy classes in Polish groups and their actions (Francesca, Jad, Esme) Kechris and Rosendal (2007) Turbulence, amalgamation, and generic automorphisms of homogeneous structures
• Oct 10: No seminar -- Maryland conference
• Oct 24, 31: Intro to topological Fraïssé limits, and their applications to subshifts (Daniel, Katalin) Doucha, Melleray, Tsankov (2025) Dense and comeager conjugacy classes in zero-dimensional dynamics
• Nov 7: The space of subshifts of a countable group (Francesca): Frish, Kechris, Shinko, Vidnyanszky (2025) Realizations of countable Borel equivalence relations
• Nov 14: topological realizations of countable Borel equivalence relations (TBA): Frish, Kechris, Shinko, Vidnyanszky (2025) Realizations of countable Borel equivalence relations
• Nov 21: Generic II_1 factors and Connes embedding (Alex T.) Goldbring (2021) Enforceable operator algebras
• Nov 28: No seminar -- thanksgiving
• Dec 4: 10:30am - 12pm (note the start that is 30min early!): Intro to projective Fraïssé limits (Felix) Irwin and Solecki (2006) Projective Fraïssé limits and the pseudo-arc
• Dec 11: The Borel complexity of generic actions of the free group (Liza) Iyer, Shinko (2024) Asymptotic dimension and hyperfiniteness of generic Cantor actions