Blanchard introduced the concepts of Uniform Positive Entropy (UPE) and Complete Positive Entropy (CPE) as topological analogues of K-automorphism. He showed that UPE implies CPE, and that the converse is false. A flurry of recent activity studies the relationship between these two notions. For example, one can assign a countable ordinal which measures how complicated a CPE system is. Recently, Barbieri and García-Ramos constructed Cantor CPE systems at every level of CPE. Westrick showed that natural rank associated to CPE systems is actually a \(\Pi^1_1\)-rank. More importantly, she showed that the collection of CPE \(\mathbb Z^2\)-SFT's is a \(\Pi^1_1\)-complete set. In this talk, we discuss some results, where UPE and CPE coincide and others where we show that the complexity of certain classes of CPE systems is \(\Pi^1_1\)-complete. This is joint work with García-Ramos.
It is a nice exercise in combinatorics to show that every \(2d\)-regular finite graph arises as a Schreier graph of the free group \(F_d\). I will present generalizations of this fact to a measurable setting, as well as some examples showing the limitations. I will formulate these results using both the language of unimodular random networks and that of (p.m.p.) graphings, which are two sides of the same coin. Partially joint work with Ferenc Bencs and Aranka Hrušková.
A well-known result of Giordano-Putnam-Skau asserts that two minimal homeomorphisms of the Cantor space which have the same invariant Borel probability measures are orbit equivalent. I will present a new, rather elementary, proof of that fact, based on a strengthening of a 1979 theorem of Krieger concerning minimal actions of certain locally finite groups on the Cantor space. No familarity with topological dynamics will be assumed.
This is joint work with Simon Robert (Lyon).
We are interested in generalizations of some theorems of ergodic theory to the Borel context, particularly for natural spaces of tilings, colorings and Hamilton paths. The work combines dynamical properties of actions of \(\mathbb Z^d\) with the finite combinatorics of the \(\mathbb Z^d\) lattice. This is joint work with Nishant Chandgotia.
Let \(F\) be a nonabelian free group. We show that, for any two nontrivial finite groups, the natural actions of the wreath product groups \(A\wr F\) and \(B\wr F\), on \(A^F\) and \(B^F\) respectively, are orbit equivalent. On the other hand, we show that these actions are not even stably orbit equivalent if \(F\) is replaced with any ICC sofic group with property (T), and \(A\) and \(B\) have different cardinalities. This is joint work with Konrad Wrobel.
Working in the framework of Borel reducibility, we analyze the complexity of the natural counterparts in terms of quasi-orders of the well-known relations of equivalence for arcs and knots. It turns out that this problem is related to the study of convex embeddability between countable linear orders (and of its analogue for circular orders), which is a topic of independent interest. This is work in progress, joint with Iannella, Kulikov and Marcone.
I will present a dichotomy for equivalence relations on Polish spaces that can be expressed as countable unions of smooth Borel subequivalence relations. It can be seen as an extension of Kechris-Louveau's dichotomy for hypersmooth Borel equivalence relations. A generalization of our dichotomy, for equivalence relations that can be expressed as countable unions of Borel equivalence relations belonging to certain fixed classes, will also be presented. This is a joint work with Benjamin Miller.
A countable group is highly transitive if it admits an embedding in the permutation group of the integers with dense image. I will present a joint work with Pierre Fima, Soyoung Moon and Yves Stalder where we show that a large class of groups acting on trees are highly transitive, which yields a characterization of high transitivity for groups admitting a minimal faithful action of general type on a tree thanks to the work of Le Boudec and Matte Bon. Our proof is new even for the free group on two generators and I will give a detailed overview in this very particular case, showing that the generic transitive action of the free group on two generators is highly transitive.
In this talk I will discuss the structure of the relation of continuous reducibility on affine varieties. If time permits, I will also present some results on polynomial reducibility. The results are joint work with C. Massaza.
The group \(\operatorname{PL}_o I\) of piecewise linear orientation preserving homeomorphisms of the unit interval, equipped with composition, has a rich array of finitely generated subgroups. A basic question one can ask is when one of these groups embeds into another. One group which seems to play a particularly important role in this quasi-order is Richard Thompson's group \(F\). For instance it is conjectured that every finitely generated subgroup of \(\operatorname{PL}_o I\) either contains a copy of \(F\) or else embeds into \(F\). I will describe a general dynamical criterion for when a subgroup of \(\operatorname{PL}_o I\) does not embed into \(F\) which covers all known examples. This is joint work with James Hyde.
Analogues of the infinite Ramsey Theorem to infinite structures have been studied since the 1930s, when Sierpiński gave a coloring of pairs of rationals into two colors such that, in any subset of the rationals forming a dense linear order, both colors persist. In the 1970s Galvin showed that two is the optimum number for pairs of rationals, while Erdős, Hajnal and Pósa extended Sierpiński's result to colorings of edges in the Rado graph. The next several decades saw a steady advance of results for other structures, a pinnacle of which was the 2006 work of Laflamme, Sauer, and Vuksanović, characterizing the exact number of colors for unavoidable colorings of finite graphs inside the Rado graph, and for similar Fraïssé structures with finitely many binary relations, including the generic tournament. This exact number is called the "big Ramsey degree", a term coined by Kechris, Pestov, and Todorčević.
In this talk, we will provide a brief overview of the area of big Ramsey degrees infinite structures. Then we will present recent joint work with Coulson and Patel, showing that free amalgamation classes, in which any forbidden substructures are \(3\)-irreducible, have big Ramsey degrees which are simply characterized. These results extend to certain strong amalgamation classes as well, extending the results of Laflamme, Sauer, and Vuksanović. This is in contrast to the more complex characterization of big Ramsey degrees for binary relational free amalgamation classes with forbidden \(2\)-irreducible substructures, obtained in joint work of the speaker with Balko, Chodounský, Hubička, Konečný, Vena, and Zucker. The work with Coulson and Patel develops coding trees of quantifier-free \(1\)-types and uses forcing to do an unbounded search for monochromatic finite objects. Furthermore, we work with skew subtrees with branching degree two which still code the Fraïssé limit. This allows for more ease when working with relations of arity greater than two, and also allows us to give the first proof of exact big Ramsey degrees bypassing the standard method of "envelopes". It also sets the stage for current work of the speaker on infinite-dimensional Ramsey theory, in the vein of Galvin-Příkrý, for Fraïssé limits of free amalgamation classes in which any forbidden substructures are \(3\)-irreducible.
A definable pair of disjoint non-\(\mathrm{OD}\) sets of reals (hence, indiscernible sets) exists in the Sacks and \(E_0\)-large generic extensions of the constructible universe \(L\). More specifically, if \(a\) is a real either Sacks generic or \(E_0\) generic over \(L\), then it is true in \(L[a]\) that: there is a \(\Pi^1_2\) equivalence relation \(Q\) on the set \(U\), of all nonconstructible reals, with exactly two equivalence classes, and both those classes are non-\(\mathrm{OD}\) sets. This is joint work with Ali Enayat.
There are deep connections between model theory of the infinitary logic \(\mathcal{L}_{\omega_1 \omega}\) and descriptive set theory: Scott analysis, the López-Escobar theorem or the Suzuki theorem are well known examples of this phenomenon. In this talk, I would like to present results of an ongoing research devoted to generalizing these connections to the setting of continuous infinitary logic and Polish metric structures. In particular, I will discuss a continuous counterpart of a theorem of Hjorth and Kechris characterizing essential countability of the isomorphism relation on a given Borel class of countable structures. As an application, I will give a short model-theoretic proof of a result of Kechris saying that orbit equivalence relations induced by continuous actions of locally compact Polish groups are essentially countable. This is joint work with Andreas Hallbäck and Todor Tsankov.
In this talk we shall discuss some anticlassification results for orderable groups. First, we introduce the space of Archimedean orderings \(\operatorname{Ar}(G)\) for a given countable orderable group \(G\). We prove that the equivalence relation induced by the natural action of \(\operatorname{GL}_2(\mathbb{Q})\) on \(\operatorname{Ar}(\mathbb{Q}^2)\) is not concretely classifiable. Then we shall discuss the complexity of the isomorphism relation for countable ordered Archimedean groups. In particular, we show that its potential class is not \(\mathbf{\Pi}^0_3\). This topological constraint prevents classifying ordered Archimedean groups using countable subsets of reals. Our proofs combine classical results on Archimedean groups, the theory of Borel equivalence relations, and analyzing definable sets in the basic Cohen model and other models of Zermelo-Fraenkel set theory without choice. This is joint work with Dave Marker, Luca Motto Ros, and Assaf Shani.
We discuss our result with András Máthé and Jonathan Noel that a disk in the plane can be partitioned into finitely many Jordan measurable pieces that can be moved by translations to form a partition of a square.
We discuss two equivalence relations for diffeomorphisms of finite dimensional smooth manifolds. The first is measure isomorphism between ergodic Lebesgue measure preserving diffeomorphisms of the \(2\)-torus. The second is topological conjugacy for diffeomorphisms of smooth manifolds. In both cases we show that the equivalence relation is unclassifiable. For topological conjugacy, for dimension \(\ge 2\) we can embed \(E_0\), while for dimension \(\ge 5\) we can embed graph isomorphism.
We finish by showing that the equivalence relations are \(\Pi^0_1\)-hard: for measure preserving diffeomorphisms of the torus we exhibit a primitive recursive association of \(\Pi^0_1\)-statements with diffeomorphisms of the torus so that for each \(\phi\),
We consider interesting descriptive set-theoretic problems emerging from theoretical economics. First, we investigate a certain two-player game coming from gambling theory. Then, as a by-product, we obtain a novel game that characterises the Baire class 1 functions. We mention how games can be used to define new natural ranks on the Baire class 1 functions. Finally, we determine the exact complexity of the so-called value of the above game, which turns out to be a less well-known class, namely analytic-inductive.
The three problems referred to in the title originate in operator algebras, quantum information theory, and complexity theory respectively. Recently we established the complexity-theoretic equality \(\mathsf{MIP}^* = \mathsf{RE}\). This equality implies that the membership problem for certain quantum correlation sets is undecidable. Due to prior work by many others, the result implies a negative answer to Tsirelson's problem (quantum information) as well as Connes' embedding problem (von Neumann algebras) and equivalent problems in operator algebras such as Kirchberg's QWEP (C*-algebras). It leaves open the famous question about the existence of a non-hyperlinear group.
In the talk, I will explain the characterization \(\mathsf{MIP}^* = \mathsf{RE}\) and motivate it by describing its connection to the study of nonlocality in quantum information, Tsirelson's problem, and operator algebras. I will mention some proof ideas, which draw from the theory of probabilistic checking in complexity theory and approximate stability in group theory.
The main result is joint work with Ji, Natarajan, Wright and Yuen available as arXiv:2001.04383.
Generalizing a Urysohn-like extension property for Hall's countable universal locally finite group, we define a concept of omnigenous groups and prove some results about such groups. One of the main results is that any countable omnigenous locally finite group can be embedded as a dense subgroup of the isometry group of the Urysohn space for all \(\Delta\)-metric spaces, for any countable distance value set \(\Delta\). This implies a conjecture of Vershik from 2008. I will also talk about the current progress on the converse problem, namely to characterize all countable (locally finite) dense subgroups of the isometry groups of Urysohn spaces. This is joint work with Mahmood Etedadialiabadi, Francois Le Maître, and Julien Melleray.
We present a method for showing that a given \(\mathbf\Pi^0_3\) subset of a Polish space is in fact \(\mathbf\Pi^0_3\)-complete. This is motivated by some questions from V. Nestoridis about the sequential spaces \(\ell^p\) and more generally about families of \(F\)-spaces \((X_i)_{i \in (I, \preceq)}\) that form \(\subseteq\)-chains, where \(\preceq\) is a linear ordering.
The intersection \(\cap_{p > a} \ell^p\) is known to be a \(\mathbf\Pi^0_3\) subset of \(\ell^q\) for all \(a, q\) with \(0 \le a < q < \infty\) (Nestoridis). We show that it is in fact a \(\mathbf\Pi^0_3\)-complete set. It turns out the proof can be generalized to the context of Polish spaces with no additional structure like linearity. This gives a method for showing \(\mathbf\Pi^0_3\)-completeness and in fact there are strong indications that it also gives a characterization of the latter property.
The first interesting case of a non-trivial, metrizable universal minimal flow (UMF) of a Polish group was computed by Pestov who proved that the UMF of the homeomorphism group of the circle is the circle itself. This naturally led to the question whether a similar result is true for homeomorphism groups of other manifolds (or more general topological spaces). A few years later, Uspenskij proved that the action of a group on its UMF is never 3-transitive, thus giving a negative answer to the question for a vast collection of topological spaces. Still, the question of metrizability of their UMFs remained open and he asked specifically whether the UMF of the homeomorphism group of the Hilbert cube is metrizable. We give a negative answer to this question for the Hilbert cube and all closed manifolds of dimension at least 2, thus showing that metrizability of the UMF of a homeomorphism group is essentially a one-dimensional phenomenon. This is joint work with Yonatan Gutman and Andy Zucker.
Christensen's Haar null ideal is a well-behaved generalization of Haar null sets to groups, which admit no Haar measure. We show that in the group \(\mathbb Z^\omega\), every Haar positive (that is, non-Haar null) analytic set contains a Haar positive closed set. Using this result, we determine the exact Wadge class of the family of Haar null closed subsets of \(\mathbb Z^\omega\).
I will present an introduction from the perspective of Borel complexity theory to the classification problem for extension of C*-algebras, its motivations from operator theory, and its connections with homological algebra.
In this, the second of a three-part series of talks, we describe a "definable Čech cohomology theory" strictly refining its classical counterpart. As applications, we show that, in strong contrast to its classical counterpart, this definable cohomology theory provides complete homotopy invariants for mapping telescopes of \(d\)-tori and of \(d\)-spheres; we also show that it provides an equivariant homotopy classification of maps from mapping telescopes of \(d\)-tori to spheres, a problem raised in the \(d = 1\) case by Borsuk and Eilenberg in 1936. These results build on those of the first talk. They entail, for example, an analysis of the phantom maps from a locally compact Polish space \(X\) to a polyhedron \(P\); instrumental in that analysis is the definable \(\operatorname{lim}^1\) functor. They entail more generally an analysis of the homotopy relation on the space of maps from \(X\) to \(P\), and we will begin by describing a category particularly germane for this analysis. Time permitting, we will conclude with some discussion and application of a related construction, namely that of the definable homotopy groups of a locally compact Polish space \(X\).
This is joint work with Martino Lupini and Aristotelis Panagiotopoulos.
This is the first talk in a three-part series in which we illustrate how classical invariants of homological algebra and algebraic topology can be enriched with additional descriptive set-theoretic information.
In the first talk we will focus on the "definable enrichment" of the first derived functors of \(\operatorname{Hom}(-,-)\) and \(\operatorname{lim}(-)\). We will show that the resulting "definable \(\operatorname{Ext}(B,F)\)" for pairs of countable abelian groups \(B, F\); and the "definable \(\operatorname{lim}^1(A)\)" for towers \(A\) of Polish abelian groups substantially refine their purely algebraic counterparts. In the process, we will develop an Ulam stability framework for quotients of Polish groups \(G\) by Polishable subgroups \(H\) and we will provide several rigidity results in the case where the ambient Polish group \(G\) is abelian and non-archimedean. A special case of our rigidity results answers a question of Kanovei and Reeken regarding quotients of the \(p\)-adic groups.
This is joint work with Jeffrey Bergfalk and Martino Lupini.
In the theory of unitary group representations, the following theorem of Elmar Thoma from the early 1960s is fundamental: A countable discrete group is "type I" if and only if it has an abelian finite index subgroup. By way of a celebrated theorem of Glimm from the same period, a group being "type I" is equivalent to saying that the irreducible unitary representations of the group admits a smooth classification in the familiar sense of Borel reducibility, and in fact they are all finite-dimensional in this case. Glimm's theorem, and later work by Hjorth, Farah and Thomas, implies that if a group is not type I, then it is quite hard to classify the irreducible unitary representations.
In this talk I will give an overview of the descriptive set-theoretic perspective on the classification of irreducible representations, and I will discuss a new proof of Thoma's theorem due to F.E. Tonti and the speaker.
We show that for every ordinal \(\alpha \in [1, \omega_1)\), there is a closed set \(F \subset 2^\omega \times \omega^\omega\) such that for every \(x \in 2^\omega\), the section \(\{y\in \omega^\omega : (x,y) \in F\}\) is a two-point set and \(F\) cannot be covered by countably many graphs \(B(n) \subset 2^\omega \times \omega^\omega\) of functions of the variable \(x \in 2^\omega\) such that each \(B(n)\) is in the additive Borel class \(\mathbf\Sigma^0_\alpha\). This rules out the possibility to have a quantitative version of the Luzin-Novikov theorem. The construction is a modification of the method of Harrington who invented it to show that there exists a countable \(\Pi^0_1\) set in \(\omega^\omega\) containing a non-arithmetic singleton. By another application of the same method, we get closed sets excluding a quantitative version of the Saint Raymond theorem on Borel sets with \(\sigma\)-compact sections. (Joint work with P. Holický)
In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation \(T\), one takes averages of a given integrable function over the intervals \(\{x, T(x), T^2(x), \ldots, T^n(x)\}\) in front of the point \(x\). We prove a "backward" ergodic theorem for a countable-to-one pmp \(T\), where the averages are taken over subtrees of the graph of \(T\) that are rooted at \(x\) and lie behind \(x\) (in the direction of \(T^{-1}\)). Surprisingly, this theorem yields forward ergodic theorems for countable groups, in particular, for pmp actions of finitely generated groups, where the averages are taken along set-theoretic (but backward) trees on the generating set. This strengthens Bufetov's theorem from 2000, which was the leading result in this vein. This is joint work with Jenna Zomback.
Beyond the Baire space, recursively presented metric spaces are structures which serve as a setting for effective descriptive set theory. Motivated by the classical distinction between a complete separable metric space and its corresponding Polish space topological structure, we will explore the notions and issues involved in moving from a recursively presented metric space to its effective Polish space structure. We will survey different approaches to these issues, in particular work by Moschovakis on recursive frames and work by Louveau on effective topology, and prove some original results which clarify some foundational problems in the area.
A locale is, informally, a topological space without an underlying set of points, with only an abstract lattice of "open sets". Various results in the literature suggest that locale theory behaves in many ways like a generalization of descriptive set theory with countability restrictions removed. This talk will introduce locale theory from a descriptive set-theoretic point of view, and survey some known and new results which are common to both contexts. In particular, we will introduce the "\(\infty\)-Borel hierarchy" of a locale, and sketch the existence of "\(\sigma\)-analytic, non-\(\infty\)-Borel sets".
The talk will be an outline of the book we published with Paul Larson recently. In particular, I will show how amalgamation problems in algebra naturally appear in consistency results for the choiceless set theory \(\mathsf{ZF} + \mathsf{DC}\), and how they can be stratified from a set-theoretic point of view.
Martin's conjecture is an attempt to make precise the idea that the only natural functions on the Turing degrees are the constant functions, the identity, and transfinite iterates of the Turing jump. The conjecture is typically divided into two parts. Very roughly, the first part states that every natural function on the Turing degrees is either eventually constant or eventually increasing and the second part states that the natural functions which are increasing form a well-order under eventual domination, where the successor operation in this well-order is the Turing jump.
In joint work with Benny Siskind, we prove part 1 of Martin's conjecture for a class of functions that we call measure-preserving. This has a couple of consequences. First, it allows us to connect part 1 of Martin's conjecture to the structure of ultrafilters on the Turing degrees. Second, we also show that every order-preserving function on the Turing degrees is either eventually constant or measure-preserving and therefore part 1 of Martin's conjecture holds for order-preserving functions. This complements a result of Slaman and Steel from the 1980s showing that part 2 of Martin's conjecture holds for order-preserving Borel functions.
A Cayley diagram for a Cayley graph \(G = \operatorname{Cay}(\Gamma, E)\) is an edge labelling of \(G\) with generators from \(E\) so that a path is labelled with a relation in \(\Gamma\) if and only if it is a cycle. I will show how \(\operatorname{Aut}(G)\)-f.i.i.d. Cayley diagrams can be used to lift \(\Gamma\)-f.i.i.d. solutions of local combinatorial problems to \(\operatorname{Aut}(G)\)-f.i.i.d. solutions. And, I will investigate which graphs admit \(\operatorname{Aut}(G)\)-f.i.i.d. Cayley diagrams, answering a question of Weilacher on approximate chromatic numbers in the process.
Although the Borel structure of \(\mathbb R/\mathbb Q\) is rather pathological in comparison to that of \(\mathbb R\), many theorems about the latter can be generalized to the former, and even to far more general quotient spaces. I will give a survey of several such theorems.
For a topological group \(G\), a \(G\)-flow is a continuous action of \(G\) on a compact Hausdorff space \(X\); we call \(X\) the phase space of the \(G\)-flow. A \(G\)-flow on \(X\) is minimal if \(X\) has no closed non-trivial invariant subset. The universal minimal \(G\)-flow, \(M(G)\), has every minimal \(G\)-flow as a quotient and it is unique up to isomorphism. We show that whenever we have a short exact sequence \[0\to K\to G\to H\to 0\] of topological groups with the image of \(K\) a compact normal subgroup of \(G\), then the phase space of \(M(G)\) is homeomorphic to the product of the phase space of \(M(H)\) with \(K\). For instance, if \(G\) is a Polish, non-Archimedean group, and the image of \(K\) is open in \(G\), then \(H\) is a countable discrete group. The phase space of \(M(H)\) is homeomorphic to \(\operatorname{Gl}(2^{2^{\aleph_0}})\), the Stone space of the completion of the free Boolean algebra on \(2^{\aleph_0}\) generators by Balcar-Błaszczyk and Glasner-Tsankov-Weiss-Zucker. Therefore, the phase space of \(M(G)\) is homeomorphic to \(K\times \operatorname{Gl}(2^{2^{\aleph_0}})\). When the sequence splits, that is, \(G\cong H\ltimes K\), then the homeomorphism witnesses an isomorphism of flows, recovering a result of Kechris and Sokić.
The Ki-Linton theorem asserts that the set of base \(b\) normal numbers is a \(\mathbf\Pi^0_3\)-complete set. The base \(b\) normal numbers can be viewed as the set of generic points for an associated dynamical system. This leads to the question of the complexity of the set of generic points for other numeration/dynamical systems, for example continued fractions, \(\beta\)-expansions, Lüroth expansions to name a few. We prove a general result which covers all of these cases, and involves a well-known property in dynamics, a form of the specification property. We then consider differences of these sets. Motivated by the descriptive set theory arguments, we are able to show that the set of continued fraction normal but not base \(b\) normal numbers is a complete \(D_2(\mathbf\Pi^0_3)\) set. Previously, the best known result was that this set was non-empty (due to Vandehey), and this assumed the generalized Riemann hypothesis. The first part of the work is joint with Airey, Kwietniak and Mance, and the second part with Mance and Vandehey.
This talk will compute the relations between the cardinality of some sets under determinacy. Woodin showed under \(\mathsf{ZF}\), dependent choice, and real determinacy that the set of countable sequences of countable ordinals does not inject into the set of \(\omega\)-sequences of countable ordinals and in fact does not even inject into the class of \(\omega\)-sequences of ordinals. This argument passes through a set called \(S_1\) and uses \(\mathsf{AD}^+\) techniques involving \(\infty\)-Borel codes, inner models of choice, and forcing arguments. In this talk, we will show a continuity result for functions from the set of sequences of countable ordinals of a fixed countable length into \(\omega_1\). This continuity result will be used to show in just \(\mathsf{AD}\) that the set of \(\omega\)-sequences of countable ordinals has strictly smaller cardinality than the set of countable-length sequences of countable ordinals. Then under \(\mathsf{AD}\) and dependent choice for the reals, this result along with category arguments, generic coding, and a bounding result of Steel will be used to show that the set of countable sequences of countable ordinals does not inject into the class of \(\omega\)-sequences of ordinals. These arguments are combinatorial and are more adaptable to the analogous questions for \(\omega_2\) and the odd projective ordinals. This is joint work with Stephen Jackson and Nam Trang.
A Polish module is a topological module whose underlying topology is Polish. In this talk, I will discuss some very recent work (joint with Forte Shinko) where we study when uncountable Polish modules continuously inject into one another and the pre-order induced by these injections. In particular we will show that, for a wide class of rings, there are countably many minimal elements in this pre-order. As an application, we will construct a countable family of uncountable abelian Polish groups, at least one of which embeds into any other uncountable abelian Polish group.
Topological dynamics of Polish groups has interesting aspects not present in dynamics of locally compact groups. For example, there exist Polish groups whose all continuous actions on compact spaces have fixed points. Groups of this type, called extremely amenable, were first constructed by Herer and Christensen using certain submeasures. Later, Gromov and Milman made a connection between extreme amenability and the concentration of measure phenomenon from probability theory.
I will describe the above developments. In this context, I will present a new concentration of measure theorem inspired by geometric ideas related to the Loomis-Whitney theorem. I will describe the dynamical consequences, in the spirit of Gromov and Milman, of our concentration of measure theorem. These consequences generalize the Herer-Christensen result mentioned above as well as related results of Glasner and Pestov. All this depends on a new geometric classification of submeasures, which I will outline.
This is a joint work with F. Martin Schneider.
We study the descriptive complexity of Borel binary relations, compared with the notion of continuous reducibility. In a first part, we present some important tools interesting for themselves. The second part is devoted to Borel equivalence relations. The last part, more recent, is about the generalization of some of the previous results to relations which are not necessarily equivalence relations. The case of graphs is particularily nice. Also, there is a surprising exception with the classes of rank two, related to topological Ramsey theory on the space of rational numbers.
We will discuss measurable cohomology of groups with Kazhdan's property (T). This will lead us to some cohomological characterizations of property (T), together with implications towards (the absence of) various lifting properties of these groups. In particular, we will construct "genuine" approximate homomorphisms into matrix algebras for a large class of property (T) groups. This is based on a recent joint work with Adrian Ioana and Matthew Wiersma.
A proper coloring of a graph is called equitable if every color class has (approximately) the same number of vertices. In the finite setting, the celebrated Hajnal-Szemerédi theorem establishes the existence of equitable \((d + 1)\)-colorings, where \(d\) is a bound on the vertex degrees. We discuss the existence of equitable \((d + 1)\)-colorings in the measure-theoretic and purely Borel contexts. Time permitting, we also discuss measure-theoretic analogs of recent work of Kostochka-Nakprasit on the existence of equitable \(d\)-colorings for graphs of low average degree. This is joint work with Anton Bernshteyn.
Arens-Eells spaces (aka Lipschitz free spaces or transportation cost spaces) give rise to interesting examples of Banach spaces and provide analytic techniques within Banach space geometry. But they are also of importance as a tool for analysing objects outside Banach space theory using functional analytical techniques. I will present two such uses. The first is to abstract harmonic analysis where Arens-Eells spaces can be used to provide a very simple conceptual proof of a recent characterisation of amenability of topological groups due to F. M. Schneider and A. Thom. The second application is to the geometric study of topological groups, namely, to establish the Gromov correspondence between coarse equivalence and topological couplings in the widest possible setting.
In 2005, Kechris, Pestov and Todorčević exhibited a correspondence between combinatorial properties of structures and dynamical properties of their automorphism groups. In 2012, Angel, Kechris and Lyons used this correspondence to show the unique ergodicity of all the actions of some groups. In this talk, I will give an overview of the aforementioned results and discuss recent work generalizing results of Angel, Kechris and Lyons.
Cost is a \([1, \infty)\)-valued measure-isomorphism invariant of aperiodic equivalence relations defined by Gilbert Levitt and heavily studied by Damien Gaboriau. For a large class of equivalence relations, including amenable, the cost is \(1\). Yoshikata Kida and Robin Tucker-Drob recently defined the notion of an inner amenable equivalence relation as an analog of inner amenability in the setting of groups. We show inner amenable equivalence relations also have cost \(1\). This is joint work with Robin Tucker-Drob.
We introduce a new class of jump operators on Borel equivalence relations, associated to countable groups. For each countable group \(\Gamma\), we define the \(\Gamma\)-jump of an equivalence relation \(E\) and produce an analysis of these jumps analogous to the situation of the Friedman-Stanley jump with respect to actions of \(S_\infty\). In particular, we show that for many (but not all) groups the \(\Gamma\)-jump of \(E\) is strictly above \(E\) and iterates of the \(\Gamma\)-jump produce a hierarchy of equivalence relations cofinal in terms of potential Borel complexity. We also produce new examples of equivalence relations strictly between \(E_0^\omega\) and \(F_2\), and give an application to the complexity of the isomorphism problem for countable scattered linear orders. This is joint work with Sam Coskey.
I will discuss a measurable version of the Hall marriage theorem for actions of abelian groups. In particular, it implies that for free measure-preserving actions of such groups, if two equidistributed measurable sets are equidecomposable, then they are equidecomposable using measurable pieces. The latter generalizes a recent result of Grabowski, Máthé and Pikhurko on the measurable circle squaring and confirms a special case of a conjecture of Gardner. This is joint work with Tomasz Cieśla.
We construct Borel graphs which settle several questions in descriptive graph combinatorics. These include "Can the Baire measurable chromatic number of a locally finite Borel graph exceed the usual chromatic number by more than one?" and "Can marked groups with isomorphic Cayley graphs have Borel chromatic numbers for their shift graphs which differ by more than one?". As a result, we completely determine the set of pairs \((x,y)\) such that there exists a locally finite Borel graph with usual chromatic number \(x\) and Baire measurable chromatic number \(y\).
The von Neumann-Day problem has two parts: Is every amenable group elementary amenable? Is every group not containing a nonabelian free group (so-called small groups) amenable? Both problems have been settled in the negative. In this talk, I will present joint work with S. Kunnawalkam Elayavalli where we show that both problems have a generic negative solution. More specifically, we present natural Polish spaces of countable amenable groups and countable small groups and prove that the set of nonamenable small groups is comeager in the space of small groups and the set of non-elementary amenable groups is comeager in the space of amenable groups. We also discuss the analogous problem for groups satisfying laws and relate it to the well-known open question of whether or not every amenable group satisfying a nontrivial law is uniformly amenable. Time permitting, we will also discuss the question of when an amenable group can have the same first-order theory as a nonamenable group.
We prove simplicity for the automorphism groups of order and tournament expansions of homogeneous structures like the bounded Urysohn metric space, the random graph or more generally Fraïssé limits of free, transitive, non-trivial amalgamation classes. In particular, we will show that the automorphism group of the linearly ordered random graph is a simple group. This is joint with Filippo Calderoni and Katrin Tent.
A Borel equivalence relation that is induced by an action of a Polish group is essentially countable if it admits a countable complete Borel section. The notion of (\(\sigma\)-)lacunarity strengthens essential countability by requiring the complete section to be uniformly separated within each orbit. Kechris proved that every action of a locally compact Polish group is lacunary. More recently, B. Miller found a \(G_0\)-type dichotomy that characterizes \(\sigma\)-lacunarity for actions of TSI Polish groups. I will show that the notion of \(\sigma\)-lacunarity and essential countability coincide for Borel equivalence relations that are induced by actions of Polish groups. I will discuss some consequences for actions of TSI non-Archimedean Polish groups.
A Polish group \(G\) is tame if for any continuous action of \(G\), the corresponding orbit equivalence relation is Borel. Extending results of Solecki, Ding and Gao showed that if \(G\) is a tame non-Archimedean abelian group, then in fact all actions of \(G\) are potentially \(\mathbf\Pi^0_6\), that is, they are Borel reducible to a \(\mathbf\Pi^0_6\) orbit equivalence relation. They noted that all previously known examples of such actions were in fact potentially \(\mathbf\Pi^0_3\), and conjectured that their upper bound could be improved to \(\mathbf\Pi^0_3\). We refute this by finding an action of a tame non-Archimedean abelian group which is not potentially \(\mathbf\Pi^0_5\). This is joint work with Shaun Allison.
I will show that Dilworth's theorem remains true in the Borel context: for a given natural number \(n\), a Borel quasi-order \(\le\) on a Polish space \(X\) either contains an \((n+1)\)-sized antichain, or \(X\) can be covered by \(n\) Borel chains. I will also discuss a generalization of a related theorem of Harrington, Marker, and Shelah, characterizing the existence of a perfect antichain.
Generalizing and simplifying recent work of Dobrinen, we show that if \(\mathcal L\) is a finite binary relational language and \(\mathcal F\) is a finite set of finite irreducible \(\mathcal L\)-structures, then the class \(\mathcal K = \operatorname{Forb}(\mathcal F)\) has finite big Ramsey degrees.
A pointwise ergodic theorem for the action of a countable group Gamma on a probability space equates ergodicity of the action to its a.e. pointwise combinatorics. One result (joint work with Jon Boretsky) we will discuss is a short, combinatorial proof of the pointwise ergodic theorem for free, probability measure-preserving (pmp) actions of amenable groups along Tempelman Følner sequences, which is a slightly less general version of Lindenstrauss's celebrated theorem for tempered Følner sequences. In fact, we prove that such actions have a certain tiling property, which implies the pointwise ergodic theorem. Another result we will discuss is a similar tiling property in the quasi-pmp setting for the natural boundary actions of the free group on \(n\) generators (\(n < \infty\)), which implies the corresponding pointwise ergodic theorem.
A well-known and long-standing open problem in the theory of Borel equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. Previous progress on this problem has been confined to groups possessing coarse Euclidean geometry and polynomial volume growth (ultimately leading to a positive answer for groups that are either virtually nilpotent or locally nilpotent). In this talk, I will discuss the coarse geometric notion of asymptotic dimension and its recently discovered applications to this problem. Relying upon the framework of asymptotic dimension, it is possible to both significantly simplify the proofs of prior results and uncover the first examples of solvable groups of exponential volume growth all of whose Borel actions generate hyperfinite equivalence relations. This is joint work with Clinton Conley, Steve Jackson, Andrew Marks, and Robin Tucker-Drob.
The Poisson boundary of a random walk is a measure of the space of asymptotic non-trivial events that can occur for a random walk on a group. In this talk, I will give an introduction to the notion of a Poisson boundary for a random walk on a countable group due to Furstenberg. I will define the Poisson boundary, explain the relationship between the Poisson boundary and harmonic functions, and explain the relationship between the Poisson boundary and amenability (due to Kaimanovich and Vershik) and the relationship between the Poisson boundary and the ICC property of groups. This is joint work with Yair Hartman, Omer Tamuz, and Pooya Vahidi Ferdowsi. No prior knowledge of random walks on groups will be assumed.
Descriptive combinatorics is the study of combinatorial problems (such as graph coloring) under additional topological or measure-theoretic regularity restrictions. It turns out that there is a close relationship between descriptive combinatorics and distributed computing, i.e., the area of computer science concerned with problems that can be solved efficiently by a decentralized network of processors. In this talk, I will outline this relationship and present a number of applications.
A big part of mathematical activity revolves around classification problems. However, not every classification problem has a satisfactory solution, and some classification problems are more complicated than others. Dynamical properties such as generic ergodicity and turbulence are crucial in the development of a rich complexity theory for classification problems. In this talk, we will review some of the existing anti-classification techniques and we will introduce a new obstruction for classification by orbit equivalence relations of TSI Polish groups; a topological group is TSI if it admits a compatible two-sided invariant metric. We will then show that the wreath product of any two non-compact subgroups of \(S_\infty\) admits an action whose orbit equivalence relation is generically ergodic with respect to orbit equivalence relations of TSI group actions.
We review recent developments in which computability-theoretic dimensions of individual points have been used to answer open questions in classical geometric measure theory, questions whose statements do not involve computability theory or logic.
In this talk, we show that any generically \(E_\infty\)-ergodic equivalence relation cannot be Borel reducible to one that is induced by a Borel action of a non-Archimedean TSI Polish group (i.e. a closed subgroup of \(S_\infty\) that has a two-sided invariant metric). We apply this fact to a family of equivalence relations recently studied by Clemens and Coskey, which they prove to be induced by actions of non-Archimedean CLI Polish groups. We also define a family of equivalence relations induced by Borel actions of a non-Archimedean TSI Polish group that are universal for their potential complexity class, mirroring results of Hjorth-Kechris-Louveau on actions of \(S_\infty\). We apply this analysis to the theory of tame abelian groups, extending results of Solecki, Ding-Gao, and Malicki.
Given a countable group \(\Gamma\), there is a compact space of subgroups \(\operatorname{Sub}(\Gamma)\), which is equipped with the \(\Gamma\)-action via conjugation. We study the notion of weak containment on this space, namely when one subgroup is contained in the orbit closure of another, which is related to weak containment of quasi-regular unitary representations of \(\Gamma\). In particular, we will consider necessary and sufficient conditions for the existence of dense orbits.
We show that the category of standard Borel spaces is the free or "universal" category equipped with some familiar set operations of countable arity (e.g., products) obeying some simple compatibility conditions (e.g., products distribute over disjoint unions). In this talk, we will discuss the precise formulation of this result, its connection with the amalgamation property for kappa-complete Boolean algebras, and its proof using methods from categorical logic.
By a flow, we mean a continuous action of a topological group \(G\) on a compact Hausdorff space \(X\). We refer to \(X\) as the phase space of the flow. We are primarily interested in minimal flows, that is, flows with no non-trivial proper closed invariant subset. Among minimal flows, there exists a maximal one called the universal minimal flow, \(M(G)\), which admits a continuous homomorphism onto every minimal flow. When \(G\) is non-Archimedean, that is, it admits a neighbourhood basis of the identity of open subgroups, then \(M(G)\) is \(0\)-dimensional. These are exactly groups of automorphisms of first-order structures with the topology of pointwise convergence. If \(M(G)\) is \(0\)-dimensional, we can think dually in terms of its algebra of clopen subsets. We summarize which algebras are known to appear as phase spaces of universal minimal flows and we pose questions about the unknown.
It is a well-known open problem to determine if every group is sofic. A sofic group \(G\) is said to be flexibly stable if every sofic approximation to \(G\) can converted to a sequence of disjoint unions of Schreier graphs by modifying an asymptotically vanishing proportion of edges. We will discuss a joint result with Lewis Bowen that if \(\operatorname{PSL}_d(\mathbb Z)\) is flexibly stable for some \(d\ge 5\), then there exists a group which is not sofic.
Let \(\Gamma\) be a countable group. The invariant random subgroup of a pmp action of \(\Gamma\) on \(X\) is the measure on the space of subgroups of \(\Gamma\) obtained by pushing forward the measure on \(X\) via the map sending \(x\) to its stabilizer. A result of Elek states that if two pmp actions of \(\Gamma\) have the same invariant random subgroup and one is hyperfinite, then they are strongly equivalent, so in particular they are both hyperfinite. We present a proof due to Giraud.
In this talk, we will go over the proof of the following theorem of Kathryn Mann: if \(\operatorname{Homeo}(M)\) is the group of all homeomorphisms of a compact manifold \(M\), endowed with the compact open topology, then every homomorphism from \(\operatorname{Homeo}(M)\) to any separable topological group is necessarily continuous.
In this talk, we will go over the proof of the following theorem of Kathryn Mann: if \(\operatorname{Homeo}(M)\) is the group of all homeomorphisms of a compact manifold \(M\), endowed with the compact open topology, then every homomorphism from \(\operatorname{Homeo}(M)\) to any separable topological group is necessarily continuous.
Given a countable group \(\Gamma\), the outer automorphism group \(\operatorname{Out}(\Gamma)\) is either countable or of cardinality continuum. A finer and more suitable notion is to consider the Borel complexity of \(\operatorname{Out}(\Gamma)\) as a Borel equivalence relation. We show that in this context, \(\operatorname{Out}(\Gamma)\) is of rather low complexity, namely that it is a hyperfinite Borel equivalence relation. In general, we show that for any Polish group \(G\) and any countable normal subgroup \(\Gamma\), the quotient group \(G/\Gamma\) is hyperfinite. This is joint work with Joshua Frisch.