Classical Helly's theorem says that given finitely many convex sets in \(\mathbb R^d\), if any \(d + 1\) of them have a non-empty intersection, then the whole family has a non-empty intersection. A more recent fractional version of Katchalski and Liu says that if the assumption holds for the \(\alpha\)-fraction of all \(d + 1\) tuples, then the conclusion holds for a \(\beta\)-fraction of the whole family (and \(\beta\) depends only on \(\alpha\) and \(d\)). Matoušek had demonstrated that this fractional Helly property, or FHP, also holds for families of finite VC-dimension (e.g. for semialgebraic sets of bounded description complexity). We study FHP and its variants in the model-theoretic setting: for definable families of sets relative to certain finitely additive measures, and its relation to Shelah's classification. In particular, FHP holds for any definable family of subsets of the field of \(p\)-adics relatively to the additive Haar measure - for a fixed \(p\)! We report some progress on what can be done uniformly in \(p\). We show that families of definable sets of bounded complexity in finite fields satisfy fractional Helly and that an Ax-Kochen style analysis can be carried out to establish FHP in the ultraproducts of \(p\)-adics, at least for measures concentrated on finite sets.
To each topological group \(G\) we can associate two canonical \(G\)-ambits: one is the greatest ambit, and the other is the enveloping semigroup of the universal minimal flow of \(G\). Ellis asked if these two ambits coincide. Pestov observed that the existence of extremely amenable groups implies that these ambits can be different, and conjectured that for "most" groups, these ambits do in fact differ. However, even in some very simple cases, like countable discrete groups, the question is open. In this talk, we consider the Polish group \(S_\infty\) and show that the two ambits are different by studying the semigroup operation inside the greatest ambit. This is joint work with Dana Bartošová.
Let \(M\) be any Fraïssé limit which has no algebraicity, such as the random graph, the rational numbers with their ordering, the rational Urysohn space, etc. We will provide necessary and sufficient conditions in terms of \(\operatorname{Age}(M)\) for the following to hold: "for a generic substructure \(A\) of \(M\), every automorphism of \(A\) extends to a full automorphism of \(M\)." We will also discuss extensions of the main result to uncountable metric Fraïssé structures such as the Urysohn metric space.
Mathias proved in 1969 that there are no infinite analytic maximal almost disjoint families of subsets of \(\omega\). His proof is essentially Ramsey-theoretical. A few years ago I found a "classical" proof of this theorem which uses a tree/derivative argument. But in this talk, I will give yet another proof, this time one that is closer in spirit to Mathias original proof, but avoids the heavy Ramsey-theoretical stuff by instead using a bit of garden-variety invariant descriptive set theory.
This will be a continuing seminar surveying the subject of weak containment of measure-preserving group actions.
Multiplier algebras of Nevanlinna-Pick spaces form an important class of non-selfadjoint operator algebras of functions. They can be concretely realized as algebras of functions on analytic varieties in a complex ball. The study of the classification problem for these algebras was initiated by Davidson, Ramsey and Shalit, and has attracted considerable attention over the last few years.
I will talk about joint work with Martino Lupini in which we investigate this classification problem from the point of view of Borel complexity theory.
This will be a continuing seminar surveying the subject of weak containment of measure-preserving group actions.
This will be a continuing seminar surveying the subject of weak containment of measure-preserving group actions.
I will illustrate an application of simple ideas from percolation to the study of invariant random subgroups (IRS). We use percolation to construct a continuum of IRSs on groups, for which we show that Furstenberg entropy varies continuously. For example, we obtain a short proof to the solution of the Furstenberg realization problem for the lamplighter group. Joint work with Ariel Yadin.
A Polish metric structure consists of a bounded Polish metric space together with real-valued Lipschitz predicates. We develop analogs of the Scott analysis and the López-Escobar theorem in this setting. The appropriate language is infinitary continuous logic.
We show that every Polish metric structure has a continuous Scott sentence. Restricting the logic to Lipschitz formulas yields the Gromov-Hausdorff distance between Polish structures. We also show that every isometry-invariant bounded Borel function on Polish metric structures (e.g., the diameter) is given by a sentence in infinitary continuous logic.
Joint work with Itaï Ben Yaacov, Michal Doucha, and Todor Tsankov.
This will be a continuing seminar surveying the subject of weak containment of measure-preserving group actions.
Finite sharply \(2\)- and \(3\)-transitive groups were classified by Zassenhaus in the 1930s and they all arise from field-like structures. It remained an open question whether the same is true for infinite groups. I will explain the background and the answer to this question.
This will be a continuing seminar surveying the subject of weak containment of measure-preserving group actions.
McDuff was the first to provide a family of continuum many pairwise-nonisomorphic separable \(\mathrm{II}_1\) factors. In a recent preprint, Boutonnet, Chifan, and Ioana proved that any ultrapowers of any two distinct McDuff examples are also nonisomorphic. As a result, this shows that McDuff's examples are also pairwise non-elementarily equivalent, thus settling the question of how many first-order theories of \(\mathrm{II}_1\) factors there are. From the model-theoretic point of view, this resolution of the question is not satisfying as we do not see an explicit family of sentences that distinguish the McDuff examples. In this talk, I will present a partial resolution to this problem by discussing the following result: If \(M_\alpha\) and \(M_\beta\) are two of McDuff's examples, where \(\alpha, \beta \in 2^\omega\) are such that \(\alpha|k = \beta|k\) but \(\alpha(k) \not = \beta(k)\), then there must exist a formula of quantifier-complexity at most \(6k + 3\) on which they disagree. The proof uses Ehrenfeucht-Fraïssé games. The talk represents joint work with Bradd Hart.
This will be a continuing seminar surveying the subject of weak containment of measure-preserving group actions.
This will be a continuing seminar surveying the subject of weak containment of measure-preserving group actions.
This will be a continuing seminar surveying the subject of weak containment of measure-preserving group actions.
In this talk, I will show that there is no Borel reduction of an isomorphism relation of a counterexample to Vaught's conjecture to the countable admissible ordinal equivalence relation in set-generic extensions of the constructible universe.
(Joint with François Le Maître) Profinite branch groups are a large, interesting family of infinite compact subgroups of automorphism groups of rooted trees. We demonstrate the equivalence of many automatic continuity type properties for these groups. In particular, we show the equivalence of the small index property, the Bergman property, and the condition that every proper normal subgroup is open; groups enjoying the latter condition are called strongly just infinite. Strongly just infinite profinite branch groups with mild additional restrictions are then shown to additionally satisfy the invariant automatic continuity property and the locally compact automatic continuity property. Examples are then presented. As an application, non-discrete simple locally compact Polish groups with the small index property, the invariant automatic continuity property, and the locally compact automatic continuity property are presented.
This will be a continuing seminar surveying the subject of weak containment of measure-preserving group actions.
The Furstenberg-Poisson boundary is a natural measure-theoretical notion of the boundary of a group. In this talk, we will introduce some background and motivation, define the boundary formally, state some results and applications, and time permitting, discuss some open problems.
This will be a continuing seminar surveying the subject of weak containment of measure-preserving group actions.
This will be a continuing seminar surveying the subject of weak containment of measure-preserving group actions.
I will explain a "local" spectral gap theorem for translation actions of dense subgroups generated by algebraic matrices on arbitrary simple Lie groups. This extends to the non-compact setting works of Bourgain-Gamburd and Benoist-de Saxcé. I will also present several applications to the Banach-Ruziewicz problem, strong ergodicity of actions on homogenous spaces, orbit equivalence rigidity, and spectra of averaging operators on compact groups. This is joint work with Remi Boutonnet and Alireza Salehi-Golsefidy.
This will be a continuing seminar surveying the subject of weak containment of measure-preserving group actions.
Cantor series expansions are a generalization of \(b\)-ary expansions. We explore in depth the number-theoretic and statistical properties of certain sets of numbers arising from their Cantor series expansions. As a direct consequence of our main theorem, we deduce numerous new results as well as strengthen known ones. Examples include normality along polynomial subsequences and computability of normal numbers with respect to these expansions.
This will be a continuing seminar surveying the subject of weak containment of measure-preserving group actions.
The Furstenberg-Poisson boundary is a natural measure-theoretical notion of the boundary of a group. In this talk, we will introduce some background and motivation, define the boundary formally, state some results and applications, and time permitting, discuss some open problems.
This will be a continuing seminar surveying the subject of weak containment of measure-preserving group actions.
This will be a continuing seminar surveying the subject of weak containment of measure-preserving group actions.