Using the work of Hutchcroft and Nachmias on indistinguishability of the wired uniform spanning forest, R. Tucker-Drob recently proved a powerful theorem: any probability measure-preserving locally countable ergodic Borel graph admits an ergodic hyperfinite subgraph. We give a completely different and self-contained proof of this theorem, which also goes through for a quasi-invariant probability measure, yielding a generalization of Tucker-Drob's theorem: any locally countable ergodic Borel graph on a standard probability space admits an ergodic hyperfinite subgraph. Our proof uses new tools whose combination yields an alternative way of exploiting nonamenability. In the first talk, we will state the main result and discuss one or two gadgets involved in the proof. In the second talk, we will finish the discussion of the new tools and give a sketch of our proof.
Using the work of Hutchcroft and Nachmias on indistinguishability of the wired uniform spanning forest, R. Tucker-Drob recently proved a powerful theorem: any probability measure-preserving locally countable ergodic Borel graph admits an ergodic hyperfinite subgraph. We give a completely different and self-contained proof of this theorem, which also goes through for a quasi-invariant probability measure, yielding a generalization of Tucker-Drob's theorem: any locally countable ergodic Borel graph on a standard probability space admits an ergodic hyperfinite subgraph. Our proof uses new tools whose combination yields an alternative way of exploiting nonamenability. In the first talk, we will state the main result and discuss one or two gadgets involved in the proof. In the second talk, we will finish the discussion of the new tools and give a sketch of our proof.
We provide a direct proof of a recent theorem of Ben-Yaacov, Melleray and Tsankov. If \(G\) is a Polish group and \(X\) is a minimal metrizable \(G\)-flow with all orbits meager, we use \(X\) to produce a new \(G\)-flow \(S_G(X)\) which is minimal and non-metrizable.
Duality theory provides a general framework for strong completeness theorems yielding an exact correspondence between syntax and semantics for various fragments of logic. This talk will survey some known results in this framework, and then present a result relating the Cartesian fragment of ultrametric continuous logic to continuous locally presentable categories. The result treats structures on partial ultrametric spaces, on which predicates may be treated equivalently either as subspaces or as their \([0, 1]\)-valued indicator functions.
I will describe some work in progress regarding the interplay between representations and topological actions.
We will describe a new approach to convergence theorems in ergodic theory, which treats amenable groups and non-amenable groups on an equal footing. In particular, this approach gives a natural generalization of the von Neumann, Birkhoff, and Shannon-McMillan-Breiman theorems to the case of negatively curved groups. We will focus in the talk on the case of free non-abelian groups where the arguments are direct and easily accessible. Based on joint work with Lewis Bowen, and on joint work with Felix Pogorzelski.
In 2012, Palamourdas claimed that the graph \(G\) generated by \(n\) commuting Borel functions on a standard Borel space had Borel chromatic number (BCN) at most \(2n + 1\). In fact, his proof assumed that the functions have no fixed points. In joint work with Palamourdas, we revisit the problem and prove that if every vertex is connected to a fixed point, then there is an increasing filtration \(\{X_i\}\) of the Polish space for which \(G|X_i\) has BCN at most \(2n\) for every \(i\).
In the talk, we will discuss dynamical properties of the action of Thompson's group \(F\) on the unit interval, using which we will show that the group \(F\) admits no non-trivial characters. Implications for the structure of invariant random subgroups will be also discussed.
We give a completely constructive solution to Tarski's circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If \(k > 0\) and \(A,B \subset \mathbb R^k\) are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than \(k\), then \(A\) and \(B\) are equidecomposable by translations using Borel pieces. This answers a question of Wagon. Our proof uses ideas from the study of flows in graphs, and a recent result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the hyperfiniteness of free Borel actions of \(\mathbb Z^d\). This is joint work with Spencer Unger.
The question of constructing hyperdegrees with some given properties appears naturally not only in recursion theory but also in effective descriptive set theory. We are particularly concerned with the hyperdegrees, which appear in the body of some certain categories of recursive trees (Kleene trees, Spector-Gandy trees). It turns out that a compactness-type result by Kreisel is an important tool in this study. In this talk, we give some applications of the latter and we conclude with an open problem.
There exist examples of compact Hausdorff spaces with no non-trivial homeomorphisms, and in particular with no non-trivial group actions on them. In the opposite extreme, there are many spaces that have a huge number of homeomorphisms. Since C*-algebras are noncommutative generalizations of spaces, we may ask whether a given (separable) C*-algebra has "many" automorphisms, or, more generally, "many" actions of a given group. In this context, we must of course identify actions up to a reasonable notion of equivalence, since otherwise most C*-algebras will have uncountably many actions. We are not only interested in knowing "how many" actions there are, but also determining the Borel complexity of the relation of equivalence between them. Questions of this sort have been studied in the context of von Neumann algebras, and particularly the hyperfinite \(\mathrm{II}_1\)-factor \(\mathcal R\). In this setting, the combination of a famous theorem of Ocneanu with a recent result by Brothier-Vaes shows that there is a dichotomy for the cardinality of the set of outer actions of a given group on \(\mathcal R\): for an amenable group, there is a unique one, while for nonamenable groups there are uncountably many.
In the context of C*-algebras, one may want to replace the hyperfinite \(\mathrm{II}_1\)-factor with a UHF-algebra of infinite type. In this setting, far less is known, and all the uniqueness results that are available so far work only for abelian groups (but not even all of them). In this talk, I will report on some recent joint work with Martino Lupini, where we establish the existence of uncountably many, non equivalent "free" actions of a given group with property (T) on a UHF-algebra. In fact, we show that the relation of equivalence for these actions is a complete analytic set.
This work is part of a project to understand the complexity of the equivalence relation of isomorphism of minimal flows of countable groups (and, more specifically, minimal subshifts) in the framework of Borel reducibility. We have two results concerning Toeplitz subshifts: one on the complexity of isomorphism of Toeplitz \(\mathbb Z\)-subshifts with separated holes and one where the acting group is non-amenable (and residually finite). Those results only scratch the surface of the general problem and many interesting open questions remain. This is joint work with Marcin Sabok.
This is a general talk on locally compact quantum (lcq) groups aimed primarily for non-experts. I will motivate the axiomatic definition of lcq groups due to Kustermans and Vaes, and present some examples. I will then introduce the concepts of actions of lcq groups, noncommutative Poisson boundaries, closed and open quantum subgroups, unitary representations and their induction.
I will discuss recent joint work with C. Rosendal on the large-scale, or coarse, structure of the groups of self-homeomorphisms of manifolds. Classical geometric group theory is the study of the large-scale geometry of finitely generated discrete groups, or compactly generated topological groups. Rosendal recently developed a means of extending this to a more general class of Polish groups. I will explain some of this framework, which Polish groups are known to admit a well-defined coarse structure, and why groups of homeomorphisms are an important class of examples.
One of the most interesting results of Borel graph combinatorics is the \(G_0\) dichotomy, i.e. the fact that an analytic graph has uncountable Borel chromatic number if and only if it contains a Borel homomorphic image of a graph called \(G_0\). It was conjectured that an analogous statement could be true for graphs of infinite Borel chromatic number, or at least for some well-behaving subclass of the class of infinitely chromatic graphs. Using descriptive set-theoretic methods, we produce examples showing that some versions of these statements are false.
I will present a short conceptual proof of Hjorth's turbulence theorem in terms of a suitable open game. I will then explain how these ideas can be modified to prove a new dynamical criterion that provides an obstruction to classification by orbits of a CLI group action. This is joint work with Aristotelis Panagiotopoulos.
We discuss a recent paper of Pequignot in which he answers a question of Kechris, Solecki, and Todorčević concerning graphs generated by a single Borel function. It is shown that the shift graph on the space of infinite subsets of natural numbers does not admit a Borel homomorphism into every such graph with infinite Borel chromatic number. Notably, only definability arguments are used and no counterexample is produced.
I will present an overview of my work on universal objects in functional analysis, including the Gurarij space, the Poulsen simplex, and their noncommutative analogues. I will then explain how one can compute their universal minimal flows using methods from Ramsey theory and model theory, which is joint work with Bartošová, López-Abad, and Mbombo.
This talk will focus on the interplay between finite combinatorics and ergodic theory. Consider a free ergodic measure-preserving action \(a \colon \Gamma \curvearrowright (X,\mu)\) of a countable group \(\Gamma\) on a standard probability space \((X, \mu)\). Using a symmetric generating set \(S\) for \(\Gamma\), one can associate to \(a\) a graph \(G\) with vertex set \(X\) in which two distinct vertices \(x, y \in X\) are adjacent if and only if they are connected by a group element in \(S\) (thus every connected component of \(G\) is isomorphic to the Cayley graph of \(\Gamma\)). It is natural to wonder how combinatorial properties of \(G\) interact with ergodic-theoretic properties of \(a\). The Lovász Local Lemma, or the LLL for short, is an important tool in graph theory (and combinatorics in general) that is often used to establish the existence of graph colorings satisfying certain "local" constraints. It turns out that if \(\Gamma\) is amenable, then the graph \(G\) satisfies a measurable analog of the LLL if and only if \(a\) has infinite entropy. In this talk, I will outline the ideas behind the proof of the forward direction of this equivalence, at the heart of which lies the connection between Shannon entropy and Kolmogorov complexity.
A topological group \(G\) is profinite if it is compact and totally disconnected. Equivalently, \(G\) is the inverse limit of a system of finite groups carrying the discrete topology. An example is the additive group of the \(2\)-adic integers.
We study profinite groups from a logician's point of view. Lubotzky and Jarden showed that topologically f.g. groups are given by their first-order theory. We raise the question whether a single sentence can suffice. Next, in the setting of computable profinite groups, the Haar measure is computable, so the usual notions of algorithmic randomness can be defined. We consider the strength of randomness necessary for effective versions of "almost everywhere" type theorems to hold (with Fouché). Finally, we consider the complexity of isomorphism using the theory of Borel reducibility in descriptive set theory. For topologically finitely generated profinite groups this complexity is the same as the one of identity for reals. In general, it is the same as the complexity of isomorphism for countable graphs. The latter result can be adapted to locally compact closed subgroups of the group of permutations on \(\mathbb N\) (joint with Kechris and Tent). This result has been obtained independently and by different means in work of Rosendal and Zielinski (arXiv:1610.00370, Oct 2016).
It is an old question whether Thompson's group \(F\) is amenable or not. To prove that \(F\) is not amenable, we can show that \(F\) is not Liouville for any generating measure on \(F\). By looking at its action on the set of dyadic rationals, one can show that for finitely supported generating measures, \(F\) is not Liouville, which could be a first step in proving that \(F\) is not amenable. There is a subclass of amenable groups, called strongly amenable groups. These are the groups for which every proximal action on a compact Hausdorff space has a fixed point. We can modify the usual action of \(F\) on dyadic rationals to prove that \(F\) is not strongly amenable.
Mycielski showed that for every comeager subset of the plane, there is a perfect subset of reals so that the collection of all pairs of non-identical reals from this perfect set is a subset of the original comeager set. A Mycielski type property can be formulated for other equivalence relations besides equality. Holshouser and Jackson showed that \(E_0\) has the Mycielski property. In this talk, I will show that \(E_1\), \(E_2\), and \(E_3\) do not have the Mycielski property. This is joint work with Connor Meehan.
The fractional chromatic number (FCN) of a graph is the solution to the linear relaxation of an integer program that computes the chromatic number (CN). Hence a descriptive set theorist may consider the Borel FCN, a quantity as yet unstudied, by restricting to only Borel colourings. We show that for an acyclic graph with Borel CN \(3\), the Borel FCN may (like the CN) drop to \(2\). We also show that the CN, FCN, and Borel FCN of a Borel graph may be made almost any values desired that follow the obvious inequalities.