A 2015 example of Tao showed that the pointwise ergodic theorem for nonabelian free groups can fail for \(L^1\) functions. This is in contrast to the situation for \(L^p\), \(p > 1\). We will present joint work with Lewis Bowen showing that this endpoint phenomenon also occurs for \(\operatorname{PSL}_2(\mathbb R)\). The proof involves analyzing the geodesic flow on an appropriately constructed hyperbolic surface.
In Ramsey theory, ultrafilters often play an instrumental role. By using nonstandard models of the integers, one can replace those third-order objects (ultrafilters are families of subsets) by simple points. In this talk, we present a nonstandard technique that is grounded on the above observation, and show its use in proving some new results in Ramsey theory of Diophantine equations.
We describe the conversion of a ladder system on \(\omega_n\) to a nonzero element of \(\operatorname H^n(\omega_n)\) (computed with respect to a constant sheaf on \(\omega_n\)). In the case of \(n = 1\), this process of conversion is, simply, Todorčević's technique of walks on the countable ordinals - one which, "[d]espite its simplicity [...] can be used to derive virtually all known other structures that have been defined so far on \(\omega_1\)" (Walks on Ordinals, 19). For higher \(n\), this conversion plays a more prospective role, pointing to dramatically underexplored \(\mathsf{ZFC}\) structures on higher \(\omega_n\). Time permitting, we'll outline two questions in this area that we think of as next.
On torsion Grigorchuk groups we construct random walks of finite entropy and power-law tail decay with non-trivial Poisson boundary. Such random walks provide near optimal volume lower estimates for these groups. Joint with Anna Erschler.
I will discuss a notion of co-induction for invariant random subgroups and show how it can be used to construct new collections of continuum many non-atomic, weakly mixing invariant random subgroups for various classes of groups. I will also discuss an application to the problem of continuity of the co-induction operation on pmp actions. This is joint work with Vibeke Quorning.
It is well-known that every non-Archimedean Polish group is Borel isomorphic to the automorphism group of a countable structure, and analogously, that every Polish group is Borel isomorphic to the automorphism group of a separable metric structure. We will present a generalization of this result: every open locally Polish groupoid admits a full and faithful Borel functor to the groupoid of metric \(\mathcal L\)-structures on the Urysohn sphere, for some countable metric language \(\mathcal L\). This partially answers a question of Lupini. We will also discuss the analogous result in the non-Archimedean case.
Many are familiar with the classic result due to Mathias that there are no analytic infinite maximal almost disjoint families, where almost disjoint means "having finite intersection". In this talk, I will discuss what happens if we replace "having finite intersection" with "having intersection in a (fixed) ideal \(\mathcal I\)", where \(\mathcal I\) is a Borel ideal on \(\omega\). It turns out that for a large class of ideals it is possible to prove an analogue of Mathias' theorem, while for other ideals there are definable infinite mad families. No characterization of when one or the other situation arises seems to be known at this time.
A topological dynamical system (i.e. a group acting by homeomorphisms on a compact topological space) is said to be proximal if for any two points \(p\) and \(q\), we can simultaneously push them together, i.e. there is a sequence \(g_n\) such that \(\lim g_n p = \lim g_n q\). In his paper introducing the concept of proximality, Glasner noted that whenever \(\mathbb Z\) acts proximally, that action will have a fixed point. He termed groups with this fixed point property "strongly amenable" and showed that non-amenable groups are not strongly amenable and virtually nilpotent groups are strongly amenable. In this talk, I will discuss recent work precisely characterizing which (countable) groups are strongly amenable. In particular I will show a group is strongly amenable if and only if it has no infinite conjugacy class (ICC) groups as factors. This, as a corollary, proves that the only finite generated strongly amenable groups are virtually nilpotent. The proof technique is to show that, for a special class of symbolical systems, a generic (i.e. comeager) action is proximal.
In a recent Annals paper, Conley and Miller showed that any basis for the countable Borel equivalence relations strictly above \(E_0\) in measure reducibility is uncountable. This is the fourth in a series of talks where we will provide an overview of this result.
In a recent Annals paper, Conley and Miller showed that any basis for the countable Borel equivalence relations strictly above \(E_0\) in measure reducibility is uncountable. This is the third in a series of talks where we will provide an overview of this result.
In a recent Annals paper, Conley and Miller showed that any basis for the countable Borel equivalence relations strictly above \(E_0\) in measure reducibility is uncountable. This is the second in a series of talks where we will provide an overview of this result.
In a recent Annals paper, Conley and Miller showed that any basis for the countable Borel equivalence relations strictly above \(E_0\) in measure reducibility is uncountable. This is the first in a series of talks where we will provide an overview of this result.
For any Polish space \(X\) it is well-known that the Cantor-Bendixson rank provides a co-analytic rank on \(F_{\aleph_0}(X)\) if and only if \(X\) is sigma-compact. In the case of \(\omega^\omega\) one may recover a co-analytic rank on \(F_{\aleph_0}(\omega^\omega)\) by considering the Cantor-Bendixson rank of the induced trees instead. We shall generalize this idea to arbitrarily Polish spaces and thereby construct a family of co-analytic ranks on \(F_{\aleph_0}(X)\) for any Polish space \(X\). We study the behaviour of this family and compare the obtained ranks to the original Cantor-Bendixson rank. The main results are characterizations of the compact and sigma-compact Polish spaces in terms of this behavior.
We show that a certain ring, constructed by von Neumann and realized as the coordinatization of a continuous geometry, can also be realized as the metric Fraïssé limit of the class of finite-dimensional matrix algebras over a field of scalars, equipped with the rank metric. We show that the automorphism group of this metric structure is extremely amenable, implying (by the metric Kechris-Pestov-Todorčević correspondence) an approximate Ramsey property, which is also proved directly.
We study the complexity of the classification problem for Cartan subalgebras in von Neumann algebras. We will discuss a construction that leads to a family of \(\mathrm{II}_1\) factors whose Cartan subalgebras, up to unitary conjugacy, are not classifiable by countable structures. We do this via establishing a strong dichotomy, depending if the action is strongly ergodic or not, on the complexity of the space of homomorphisms from a given equivalence relation to \(E_0\). We will start with some of the necessary preliminaries, and then outline the proofs of the aforementioned results.
A sequence \((z_n)\) in the unit disc is called an interpolating sequence for \(H^\infty\) if for every bounded sequence of complex values \((w_n)\), there exists a bounded analytic function \(f\) in the disc such that \(f(z_n) = w_n\) for all \(n\). Such sequences were characterized by Lennart Carleson.
I will talk about a generalization of Carleson's theorem to other classes of functions. The proof of this result uses the solution of the Kadison-Singer problem due to Marcus, Spielman and Srivastava. This is joint work with Alexandru Aleman, John McCarthy and Stefan Richter.
In this talk, based on joint work with Daniel Drimbe and Adrian Ioana, we will focus on an operator-algebraic criterion sufficient for deducing measure equivalence of countable groups in the sense of Gromov. In particular, we will give a tool for determining when measure equivalence between \(\Gamma_1 \times \Gamma_2\) and \(\Lambda_1 \times \Lambda_2\) can be upgraded to measure equivalence between the factors, as is the case in a well-known result of Monod and Shalom. The motivation is to bring to bear the power of Sorin Popa's deformation/rigidity theory, but no familiarity with that theory will be assumed.
Let \(\mathcal L\) be a countable language, let \(\mathcal I\) be an isomorphism-type of countable \(\mathcal L\)-structures and let \(a\) be an oracle. We say that \(\mathcal I\) is "\(a\)-strange" if it contains a recursive-in-\(a\) structure and its Scott rank is exactly \(\omega_1^a\). Such structures exist but there are no known natural examples. Theorem (\(\mathsf{AD}\)): If \(C\) is a collection of \(\aleph_1\) isomorphism-types of countable structures, then for a Turing cone of \(a\)'s, no member of \(C\) is \(a\)-strange.
Strong conceptual completeness (SCC) theorems allow the syntax of a logical theory to be canonically recovered from its space of models equipped with suitable structure, and are known for finitary first-order logic (Makkai) and fragments thereof (Gabriel-Ulmer, Lawvere, and others). In this talk, I will present a SCC theorem for \(L_{\omega_1\omega}\): a countable \(L_{\omega_1\omega}\)-theory can be recovered from its standard Borel groupoid of countable models.
Let \(E\) be an analytic equivalence relation which does not have perfectly many equivalence classes. For any oracle \(a\), define \(L(a,E)\) to be the set of \(E\)-equivalence classes which contain an element \(y\) with the property that \(\omega_1^{\langle a, y\rangle} = \omega_1^a\). For a Turing cone of \(a\)'s, \(L(a,E)\) is countable.
We describe a number of related questions at the interface of set theory and homology theory, centering on (1) the additivity of strong homology, and (2) the cohomology of the ordinals. In the first, the question is, at heart: To how general a category of topological spaces may classical homology theory be continuously extended? And in the tension between various potential senses of continuity lie a number of delicate set-theoretic questions. These questions led to the consideration of the Čech cohomology of the ordinals; the surprise was that this is a meaningful thing to consider at all. It very much is, describing or suggesting at once (i) distinctive combinatorial principles associated to the \(n\)-th infinite cardinal, for each \(n\), holding in \(\mathsf{ZFC}\), (ii) rich connections between cofinality and dimension, and (iii) higher-dimensional extensions of the method of minimal walks.
A classical result of Dougherty, Jackson and Kechris implies that the action of the free group on its Gromov boundary induces a hyperfinite equivalence relation. We will discuss a generalization of this result to a wider class of hyperbolic groups. Joint with Jingyin Huang and Marcin Sabok.