I will define a class of equivalence relations called Polishable equivalence relations that lies between the class of orbit equivalence relations of Polish group actions and the class of idealistic equivalence relations of Kechris and Louveau. I will present a Scott analysis for such equivalence relations. I will compare this analysis with the Scott analysis for isomorphism equivalence relations from continuous model theory and with versions of the Scott analysis for (certain) orbit equivalence relations of Polish group actions. As a tool in the proofs, I will introduce transfinite filtrations from one topology to another, a new notion of interpolation between topologies that may be of independent interest.
We describe two variants of the classical Ramsey relation \(\mu\to(\lambda)^2_\kappa\) which we denote by the arrows \(\to_{rc}\) and \(\to_{wc}\). Positive relations follow for each from the corresponding relation \(\mu\to(\lambda)^2_\kappa\); these variants are, however, mild weakenings of the Ramsey arrow in the sense that consistently, the infinite partition relations associated to these three arrows entirely coincide. Whether or how these arrows may consistently differ is the more interesting question, one necessarily involving large cardinal considerations and meaningfully beginning at the cardinal \(\aleph_2\). Each of these arrows in fact has its origin in combinatorics operative in homology computations in forcing iterations of length at least \(\omega_2\). This is a motivation we'll briefly describe, along with some recent results and the central open question remaining. For concreteness, we close this abstract with a definition: a graph \((V,E)\) is highly connected if it remains connected after the deletion of fewer than \(|V|\) vertices. Write \(\mu\to_{hc}(\lambda)^2_\kappa\) if any coloring of the edges of the complete graph on \(\mu\) in \(\kappa\) many colors contains a monochromatic highly connected subgraph of size \(\lambda\). For which \(\mu\), \(\lambda\), and \(\kappa\) does this relation hold?
I will give an overview of the application of nonstandard methods to the study of partition regularity of Diophantine equations. I will the explain how these methods can be used to generalize the classical Rado criterion for linear equations to obtain natural necessary conditions for arbitrary Diophantine equations, which are also sufficient for certain degree \(2\) equations. This is joint work with Jordan M. Barrett and Joel Moreira.
We give a Baire category characterization of when a subset of a Polish space is \(\mathbf\Sigma^0_{n+2}\)-hard for \(n > 0\). Our proof uses a priority argument, and Antonio Montalbán's true stages machinery. We apply this characterization to the decomposability conjecture; the problem of describing when a function is a union of countably many continuous functions defined on \(\mathbf\Pi^0_n\) sets.
The concept of disjointness of dynamical systems (both topological and measure-theoretic) was introduced by Furstenberg in the '60s and has since then become a fundamental tool in dynamics. In this talk, I will discuss disjointness of topological systems of discrete groups. More precisely, generalizing a theorem of Furstenberg (who proved the result for the group of integers), we show that for any discrete group \(G\), the Bernoulli shift \(2^G\) is disjoint from any minimal dynamical system. This result, together with techniques of Furstenberg, some tools from the theory of strongly irreducible subshifts, and Baire category methods, allows us to answer several open questions in topological dynamics: we solve the so-called "Ellis problem" for discrete groups and characterize the underlying topological space for the universal minimal flow of discrete groups. This is joint work with Eli Glasner, Benjamin Weiss, and Andy Zucker.
We discuss the notion of weak containment and weak equivalence for pmp actions of countable groups and its relation with invariant random subgroups.
We generalize the theorem of Þórisson, characterizing when two measures agree on invariant sets, to the setting of cardinal algebras.
Given a countable Borel equivalence relation on a standard Borel space \(X\), when do two measures agree on the invariant sets? By a result of Þórisson, if \(G\) is a countable group generating the equivalence relation, then this property holds iff there exists a measure on \(G\times X\) whose pushforwards under the projection and the action are the original measures. We will investigate this and other equivalent formulations of this property.
Given a measure-preserving equivalence relation \(E\), there is a Polish space \(S(E)\) of subequivalence relations, which admits a natural action of the full group \([E]\). Does \(S(E)\) have a dense orbit? We will present results due to François Le Maître which show that the answer is yes when \(E\) is the hyperfinite ergodic equivalence relation, and that the answer is no when \(E\) is induced by a measure-preserving action of a property (T) group.
Given a measure-preserving equivalence relation \(E\), there is a Polish space \(S(E)\) of subequivalence relations, which admits a natural action of the full group \([E]\). Does \(S(E)\) have a dense orbit? We will present results due to François Le Maître which show that the answer is yes when \(E\) is the hyperfinite ergodic equivalence relation, and that the answer is no when \(E\) is induced by a measure-preserving action of a property (T) group.
We consider the concept of disjointness for topological dynamical systems, introduced by Furstenberg. We show that for every discrete group, every minimal flow is disjoint from the Bernoulli shift. We apply this to give a negative answer to the "Ellis problem" for all such groups. For countable groups, we show in addition that there exists a continuum-sized family of mutually disjoint free minimal systems. In the course of the proof, we also show that every countable ICC group admits a free minimal proximal flow, answering a question of Frisch, Tamuz, and Vahidi Ferdowsi. (Joint work with Eli Glasner, Todor Tsankov, and Benjamin Weiss)
A large part of measured group theory studies structural properties of countable groups that hold "on average". This is made precise by studying the orbit equivalence relations induced by free Borel actions of these groups on probability spaces. In this vein, the cyclic (more generally, amenable) groups correspond to hyperfinite equivalence relations, and the free groups to the treeable ones. In joint work with R. Tucker-Drob, we give a detailed analysis of the structure of hyperfinite subequivalence relations of a treed quasi-measure-preserving equivalence relation, deriving some of analogues of structural properties of cyclic subgroups of a free group. Most importantly, just like every cyclic subgroup is contained in a unique maximal one, we show that every hyperfinite subequivalence relation is contained in a unique maximal one.
A large part of measured group theory studies structural properties of countable groups that hold "on average". This is made precise by studying the orbit equivalence relations induced by free Borel actions of these groups on probability spaces. In this vein, the cyclic (more generally, amenable) groups correspond to hyperfinite equivalence relations, and the free groups to the treeable ones. In joint work with R. Tucker-Drob, we give a detailed analysis of the structure of hyperfinite subequivalence relations of a treed quasi-measure-preserving equivalence relation, deriving some of analogues of structural properties of cyclic subgroups of a free group. Most importantly, just like every cyclic subgroup is contained in a unique maximal one, we show that every hyperfinite subequivalence relation is contained in a unique maximal one.
For a countable Borel equivalence relation \(E\) we consider the weak choice principle "every countable sequence of \(E\)-classes has a choice function". We establish a relationship between ergodicity of the equivalence relations and the study of these choice principles. We will separate these choice principles as follows: if \(E\) is \(F\)-ergodic (with respect to some measure) then there is a model of set theory in which "choice for \(E\) classes" fails yet "choice for \(F\) classes" holds. For example, "choice for \(E_\infty\) classes" is strictly stronger than "choice for \(E_0\) classes".
A key lemma in the proof is the following statement: if \(E\) is \(F\)-ergodic with respect to an \(E\)-quasi-invariant measure \(\mu\) then the countable power of \(E\), \(E^\omega\), is \(F\)-ergodic with respect to the product measure \(\mu^\omega\). The proof relies on ideas from the study of weak choice principles.
In this talk, we will go over some of the basic ideas behind forcing and explain how they were used to construct models of set theory without the axiom of choice. We will then establish the relationship with ergodicity and focus on proving the lemma mentioned above.
We construct pairs of marked groups with isomorphic Cayley graphs but different Borel chromatic numbers for the free parts of their shift graphs. This answers a question of Kechris and Marks. We also show that these graphs have different Baire measurable and measure chromatic numbers, answering analogous versions of the question.
A subset of a Polish group is Roelcke precompact if, given any open subset \(V\) of the group, it can be covered by finitely many sets of the form \(VfV\). Such sets form an ideal and the familiar Roelcke precompact groups are those for which this ideal is improper. A group is said to be locally Roelcke precompact when this ideal countains an open set. Examples of such groups - in addition to all Roelcke precompact groups and all locally compact groups - include the isometry group of the Urysohn metric space and the automorphism group of the countably-regular tree.
All locally Roelcke precompact groups are locally bounded in the sense of the coarse geometry of topological groups developed by C. Rosendal. Indeed, we characterize them as those locally bounded Polish groups for which every coarsely bounded subset is Roelcke precompact. We also characterize them as those groups whose completions with respect to their Roelcke (or lower) uniformities are locally compact. We also assess the conditions under which this locally compact space carries the structure of a semi-topological semigroup extending multiplication in the group.