In this talk, we study the isometry group \( \textnormal{Iso}(\mathbb{U}_\Delta) \) of the \( \Delta \)-metric Urysohn space \( \mathbb{U}_\Delta \) equipped with the pointwise convergence topology for a countable distance set \( \Delta \) with \( \inf\Delta=0 \). We showed that \( \textnormal{Iso}(\mathbb{U}_\Delta) \) is a Lévy group, so it is extremely amenable. Moreover, we can choose the Lévy family such that its increasing union is isomorphic to Hall's group. This generalizes the results that \( \textnormal{Iso}(\mathbb{U}) \) is Lévy by Pestov and \( \textnormal{Iso}(\mathbb{U}_\Delta) \) contains a dense subgroup which is isomorphic to Hall's group by Etedadialiabadi, Gao, Le Maître and Melleray. Then we will discuss its analogy for the continuous logic case. It is an ongoing project with Su Gao and Víctor Hugo Yañez.

I will describe an application of model theory to algebraic differential equations obtained by Jim Freitag, Remi Jaoui, and myself, a few years back. I will try to communicate what model-theoretic techniques were brought to bear on the problem, and why.

Given a Gromov hyperbolic space equipped with an action of a group by isometries, one can study the orbit equivalence relation of the induced action of the group on the Gromov boundary of the space. Marquis and Sabok proved that the action of hyperbolic groups on their Gromov boundaries turns out to have the property that the orbits can be arranged into lines in a consistent manner, a property known as hyperfiniteness. We show that (infinitely presented) graphical small cancellation groups exhibit a similar phenomenon, inducing hyperfinite orbit equivalence relations on the boundaries of their natural hyperbolic Cayley graphs. This is joint work with Damian Osajda and Koichi Oyakawa.

Endowed with the Chabauty topology, the space of subgroups Sub(G) of any infinite countable group G is a closed subset of the Cantor space, on which G acts by conjugation. The perfect kernel of G is the largest closed subset of Sub(G) without isolated points. It is invariant by conjugation.
In this talk, we will see how the action of a group G on an oriented tree T can give information on the perfect kernel of G, and on the dynamics induced by the action by conjugation on it. In 2023, Azuelos and Gaboriau studied the case where the stabilizers "vanish", i.e. there exists an edge path of T whose stabilizer is finite. They proved that the closure of the G-invariant subset Sub_|•\T|∞(G), which consists in the set of subgroups of G acting on T with infinitely many orbits of edges, is included in the perfect kernel and contains a dense orbit.
Generalizing results obtained by Carderi, Gaboriau, Le Maître and Stalder, we will study the space of subgroups of generalized Baumslag-Solitar groups, i.e. groups acting cocompactly on an oriented tree with infinite cyclic edge and vertex stabilizers. These are typical examples of non vanishing stabilizers.
After proving that the perfect kernel of such non amenable group exactly consists in Sub_|•\T|∞(G), we show that this leads to very different dynamics on it. In particular, we show the existence of an infinite countable G-invariant partition of the perfect kernel such that:
- one piece of the decomposition is closed, and all the other ones are open (and closed iff G is virtually the direct product of a free group with ℤ);
- there exists a dense orbit in each of these pieces.

In 1992, Hrushovski proved that for any finite graph G, there is another finite graph G' into which G embeds and such that every isomorphism between subgraphs of G extends to an automophism of G'. A class of finite structures has the Hrushovski property (or Extension Property for Partial Automorphisms or, simply, EPPA) if it has this property which Hrushovski proved for graphs. Later in the 1990s, Herwig and Lascar showed a connection between the Hrushovski property for certain classes of finite structures and the work of Ribes and Zalesskii on topologies for finitely generated free groups. In this talk, I will discuss this connection between the Hrushovski property and topology.

In this talk, we look at the ingredients for the proof of the measure equivalence classification of Baumslag–Solitar groups. Namely, we study type III equivalence relations, invariants associated them, such as the Krieger flow, and techniques for sliding edges in type III trees. Though a continuation of a talk in the GGT seminar, this talk is self-contained.

I will give a hat puzzle in which multiple players guess the colors of 3 hats on their head (out of an infinite stack) and show the solution.

At the dawn of the millenium, Whyte completed the quasi-isometry classification of the Baumslag-Solitar groups. The analogous problem of their measure equivalence classification, however, has only seen partial progress. Of note, a decade ago, Kida proved that under certain mixing conditions, no measure equivalence coupling can exist between some of these groups, e.g. BS(2,3) and BS(5,7). In this talk, we conclude the measure equivalence classification and discuss some of the techniques used.

The Frankl–Rödl matching theorem says that, in any sparse enough regular hypergraph of large enough degree, there's matching that covers almost all of the vertices. It was one of the first applications of Rödl's (now ubiquitous) nibble method. In this talk, I will prove a measurable version of the Frankl–Rödl theorem using a measurable version of the nibble method and some results about weak containment for hypergraphs.

A permutation model is a type of model of set theory with atoms that in most cases does not satisfy the axiom of choice. In this talk, we will define what a dynamical ideal is, explain how one can be used to construct a permutation model, and then explore some connections between dynamical properties of the dynamical ideal with fragments of choice which hold in the corresponding permutation model. Along the way, we will look at examples and nonexamples of dynamical ideals which satisfy these properties. This talk is based on joint work with Jindrich Zapletal.

A fractional perfect matching is the linear programming analog of a perfect matching, where we allow edges to take on values in the interval [0,1] instead of just {0,1}. A compactness argument shows that any locally finite hyperfinite graphing that admits a perfect matching will admit a measurable fractional perfect matching. However, in an upcoming paper by Bernshteyn and Weilacher, they construct a polynomial growth Borel forest on a Polish space that has no Borel fractional perfect matching, even after throwing away an invariant meager set. In contrast to this result, we will show that if a Borel graph has components that are quasi-transitive and amenable, then if it admits a perfect matching it will admit a Borel fractional perfect matching.

Paradoxical sets appear in many areas of mathematics. These are sets that are constructed using the Axiom of Choice and are somehow counterintuitive; examples include the Vitali set and the construction in the Banach–Tarski paradox. The main question of my PhD thesis is whether some fragments of the Axiom of Choice can be recovered from the existence of certain paradoxical sets. I will present some negative answers to this question and introduce the basic ingredients of Set Theory along the way, focusing on my favorite example of a paradoxical set: a partition of ℝ​​³ into unit circles. 

A greedy algorithm argument demonstrates that any undirected graph of degree bounded by d has chromatic number at most d + 1. This upper bound is sharp; the obvious obstructions are odd cycles and complete graphs. In 1941, Brooks proved that these obstructions are the only ones: Any undirected graph of degree bounded by d not containing odd cycles or complete graphs admits a proper d-coloring. The determinacy argument of Marks demonstrates that there is no Borel analogue of Brooks's theorem. However, in 2016, Conley, Marks, and Tucker-Drob proved a measurable version of Brooks's theorem.
  In the 1980s, combinatorialists introduced a coloring notion for directed graphs known as dicoloring. Work of Harutyunyan and Mohar resulted in a Brooks-like characterization of the directed graphs of maximum degree bounded by d that admit d-dicolorings. In this talk, we present and sketch a proof of a measurable version of this theorem. We explain also why the theorem has no Borel analogue.

The aim of this talk is first to briefly describe a natural way of measuring simplicity/complexity of classification problems by using Invariant Descriptive Set Theory and then mainly to discuss recent applications in the context of topological dynamics. We mainly deal with the classification of transitive maps on the interval, on the Cantor set and on the Hilbert cube with respect to the topological conjugacy relation. At the end, we provide some attempts to identify the complexity of classification of minimal maps.

A theorem of Downarowicz states that for every compact metric Choquet simplex C, there exists a Toeplitz symbolic subshift whose simplex of invariant measures is affinely homeomorphic to C. A theorem of Sabok states that the relation of affine homeomorphism between compact metric Choquet simplices is a complete orbit equivalence relation. Naturally, if two (Toeplitz or not) subshifts are isomorphic, then their simplices of invariant measures are affinely homeomorphic. One could hope that in some restricted class of subshifts, a converse statement holds, which could lead to a nontrivial lower bound on the conjugacy relation of minimal subshifts.
 Reexamining the proof of Downarowicz, we note that the subshifts he constructs belong to the class of generalized Oxtoby subshifts. Contrary to our expectations, we then prove that the isomorphism relation for generalized Oxtoby subshifts is hyperfinite. We employ a method similar to Kaya’s.
 The lesson here is, perhaps, that the possibility of realizing all Choquet simplices demonstrated by Downarowicz has little to do with the Borel complexity of the isomorphism relation of subshifts.

Given a separable complete metric space, it is possible to see it as a first-order metric structure in a countable language and hence we can study its automorphism group. The aim of the talk will be to provide a criterion to determine whether the automorphism group of a first-order metric structure satisfies the automatic continuity property, meaning that every homomorphism between such a group and a separable topological group is automatically continuous. This is a joint work with Christian Rosendal.

We prove that for any generating set S​​ of Γ=ℤⁿ​​, the continuous edge chromatic number of the Schreier graph of the Bernoulli shift action G=F(S,2^Γ)​​ is χ'_c(G)=χ'(G)+1​​. In particular, for the standard generating set, the continuous edge chromatic number of F(2^ℤn)​​ is 2n+1​​.

Countable Borel equivalence relations (CBERs) have been extensively studied within the framework of descriptive set theory, and a powerful way of analyzing them is by understanding what countable structures one can uniformly assign to each equivalence class in a Borel way. For instance, a graphing of a CBER E is a Borel graph whose connected components are exactly the E-classes, and a CBER is treeable if it admits an acyclic graphing, called a treeing. Treeable CBERs are analogous to free groups, so, naturally, results stating treeability of certain classes of CBERs are difficult to obtain. A recent such result is by Chen, Poulin, Tao, and Tserunyan, stating that if a CBER E admits a locally-finite graphing whose components are quasi-trees, then E is treeable. Their proof uses geometric-group-theoretic ideas and Stone duality, which we will present. Time permitting, we will also discuss a generalization of this result (which they proved as well), that precisely identifies the structure underlying all treeable CBERs. This is an expository talk based on work by Ruiyuan Chen, Antoine Poulin, Ran Tao, and Anush Tserunyan.

It is well-known that a countable Borel equivalence relation is hyperfinite if and only if it admits a Borel class-wise ℤ-ordering. In this talk we first consider a concept of compatibility between two Borel class-wise ℤ-orderings for a hyperfinite Borel equivalence relation. We show a dichotomy theorem which identifies a canonical obstacle for two such orderings to be incompatible. In the second part of the talk, we show that if a hyperfinite-over-hyperfinite equivalence relation admits a self-compatible Borel class-wise ℤ²-ordering, then it is hyperfinite. This is joint work with Ming Xiao.

Given sets X, Y and P ⊆ X × Y with proj_X(P) = X, a uniformization of P is a function f : X → Y such that (x, f(x)) ∈ P for all x ∈ X. If now E is an equivalence relation on X, we say that P is E-invariant if x₁ E x₂ ⇒ P_x1 = P_x2, where P_x = {y : (x, y) ∈ P} is the x-section of P. In this case, an E-invariant uniformization is a uniformization f such that x₁ E x₂ ⇒ f(x₁) = f(x₂).
 Consider now the situation where X, Y are Polish spaces and P is a Borel subset of X × Y. In this case, standard results in descriptive set theory provide conditions which imply the existence of Borel uniformizations. These fall mainly into two categories: "small section" and "large section" uniformization results.
 Suppose now that E is a Borel equivalence relation on X, P is E-invariant, and P has "small" or "large" sections. In this talk, we address the following question: When does there exist a Borel E-invariant uniformization of P?
 We show that for a fixed E, every such P admits a Borel E-invariant uniformization iff E is smooth. Moreover, we compute the minimal definable complexity of counterexamples when E is not smooth. Our counterexamples use category, measure, and Ramsey-theoretic methods.
 We then consider "local" dichotomies for such pairs (E, P). We give two new proofs of a dichotomy of Miller in the case where P has countable sections, the first using Miller’s (G₀, H₀) dichotomy and Lecomte’s ℵ₀-dimensional G₀ dichotomy, and the second using a new ℵ₀-dimensional analogue of the (G₀, H₀) dichotomy. We also prove anti-dichotomy results for the "large section" case and discuss the "K_σ section" case, which is still open.
 This is joint work with Alexander Kechris.

In this talk, we discuss structurability as a concept connecting the classification of countable Borel equivalence relations (CBERs) to interpretations in countable first order logic (L_{\omega_1\omega}). A CBER is an equivalence relation E on a standard Borel space X such that E is a Borel subset of X^2, and every E-class is countable. Important classes of CBERs are often characterized in terms of the types of structures that can be erected in a uniform Borel manner across the E-classes. More precisely, given a theory T, we say that a CBER E on X is T-structurable if there is a Borel structure M on X such that M is the disjoint union of the restrictions M|C to each E-class C, and each M|C is a model of T. For a fixed theory T, the class of T-structurable CBERs is called elementary. Examples of elementary classes include smooth CBERs (structurable by the theory of a singleton), hyperfinite CBERs (structurable by the theory of transitive ℤ-actions), and treeable CBERs (structurable by the theory of connected acyclic graphs).
  Elementary classes of CBERs can be compared by studying interpretations between their theories, where an interpretation from T to T' is a definable recipe for building models of T from models of T'. For example, hyperfiniteness implies treeability, and there is a corresponding interpretation defining a tree from a transitive ℤ-action. On the other hand, smoothness also implies treeability, but there is no interpretation defining a tree from a singleton. In general, to define a T'-structuring from a T-structuring of E, we may need to use some "extra structure” that all CBERs have but is not encoded in T. We axiomatize this extra structure with a theory T* describing a countable separating family and a countable covering family of functions, like those obtained from the Luzin-Novikov theorem. This theory does the trick: T-structurability implies T'-structurability if and only if there is an interpretation from T' to T \cup T*.

The Frattini subgroup of a group G is the intersection of all maximal subgroups of G. Equivalently, it can be defined as the set of all non-generating elements of G. The study of Frattini subgroups has a long history, and in particular, it was observed by many that the Frattini subgroup of groups with “hyperbolic-like” geometry is often small in a suitable sense. In this talk I’ll give a gentle introduction to the topic, and discuss results on Frattini subgroups of various generalizations of hyperbolic groups, from a joint work with Denis Osin and Kate Rybak.

We verify a conjecture of Vershik by showing that Hall’s universal countable locally finite group can be embedded as a dense subgroup in the isometry group of the Urysohn space and in the automorphism group of the random graph. In fact, we show the same for all automorphism groups of known infinite ultraextensive spaces. Generalizing a Urysohn-like extension property for Hall's group, we introduce a notion of "omnigenous groups" and show that every locally finite omnigenous group can be embedded as a dense subgroup in the isometry groups of various Urysohn spaces. Finally, we give a characterization of the isomorphism type of the isometry group of the Urysohn Δ-metric spaces in terms of the distance value set Δ.
This is joint work with Su Gao, Francois Le Maître, and Julien Melleray.

In a recent-ish paper, Conley and Miller studied the notions of measure hyperfiniteness and measure reducibility of countable Borel equivalence relations, variants of the usual notions of hyperfiniteness and Borel reducibility. In their paper, they asked whether there is a "measure successor of E₀"—i.e. a countable Borel equivalence relation E such that E is not measure reducible to E₀ and any F which is measure reducible to E is either equivalent to E or measure reducible to E₀. In ongoing work, Jan Grebik and I have isolated a combinatorial condition on Borel group actions which implies that the associated orbit equivalence relation is a measure successor of E₀. We have also found several examples of group actions which are plausible candidates for satisfying this condition. The key notion is a property of Borel graphs that we call "lossless expansion" after a similar property which is studied in computer science and finite combinatorics. I will explain the context for Conley and Miller's question, the condition that Grebik and I have isolated and its connections to computer science and discuss some of the candidate examples we have identified.

Complexity of several isomorphism relations coming from the field of dynamical systems has been investigated previously. On the measurable side of things, Hjorth showed that the isomorphism relation for ergodic measure-preserving transformations is not reducible to an S-infinity action. Foreman, Rudolph & Weiss proved that this relation is not Borel. Regarding topological dynamical systems, Gao showed that topological conjugacy of Cantor systems is bireducible to the maximal S-infinity equivalence relation, and asked about the complexity of this relation restricted to minimal Cantor systems. Based on ideas from the Foreman, Rudolph & Weiss work, we prove that topological conjugacy of minimal Cantor systems is not Borel. (Talk based on joint work with Kosma Kasprzak, Dominik Kwietniak, Philipp Kunde, Felipe Garcia-Ramos.)

Consider a stationary ergodic process with discrete state space. The asymptotic equipartition property (AEP) states that most of the probability mass is more or less evenly distributed among sufficiently large blocks of possible outcomes. This property plays an important role for entropy and information theory. The AEP is a consequence of the SMB theorem on pointwise equipartition, which was proved by Shannon, McMillan and Breiman in the 50's and was extended to the setting of amenable groups due to work of Ornstein, Weiss and Lindenstrauss in the 80's, 90's and early 2000's. However, beyond amenable groups only few results on equipartition have been established so far.
 In this talk we shall focus on the case of free groups and explain how one can utilize a suitable mixing condition to obtain pointwise equipartition along even spheres. The proof is based on a generalization of an abstract SMB theorem for cocycles over amenable equivalence relations due to Nevo and Pogorzelski. Using the fact that the free group acts amenably on its boundary, this generalization can be used to establish pointwise equipartition along suitable random subsets in the free group. The final result is then obtained by an integration argument, which relies on the above mentioned mixing condition. Joint work in progress with Felix Pogorzelski.

We study the complexity of determining whether two Archimedean orders on a finitely generated group admit an automorphism sending one to the other. We sketch an argument due to Calderoni, Marker, Motto Ros and Shani that when the number of generators is two, the complexity of the isomorphism relation is hyperfinite. We give anti-classification results for more generator: when the number of generators is bigger than three, we show it is not hyperfinite, using machinery of Zimmer. when it is bigger than four, we show it is not treeable, using machinery of Popa and Vaes.

We will show that the conjugacy relations both of minimal systems and pointed minimal systems are not Borel reducible to any Borel S_∞-action.

Classical results of ergodic theory show that, if a sufficiently nice group G acts sufficiently nicely on a Polish space X, then for all Borel probability measures µ₁, µ₂ on X, the following properties are equivalent: (i) µ₁ and µ₂ agree on the G-invariant σ-algebra I_G, and (ii) there exists a probability measure µ̃ on the product space X × ​X satisfying µ̃ ∘ π_i^-1 = µ_i for i = 1, 2 as well as µ̃(E_G) = 1. In analogy with a fundamental principle of optimal transport theory, we say in this case that the Borel equivalence relation E_G satisfies "strong duality". In this work we pose the question of understanding when a general Borel equivalence relation (not necessarily induced by a group action) satisfies strong duality. We prove that all hypersmooth Borel equivalence relations satisfy strong duality, and we apply this result to determine an exact characterization of the so-called "Brownian germ coupling problem" which has recently been studied in stochastic calculus.

Hayes and I proved in 2023 that every non amenable initially sub amenable group admits two sofic embeddings that are not automorphically conjugate, generalizing a theorem of Elek and Szabo. I will discuss the proof.

Interest in systems of set theory based on intuitionistic logic grew after Bishop’s work on constructive analysis in the late sixties. In 1975, Robin Grayson showed how complete Heyting algebras can be used to construct Heyting-valued models for intuitionistic set theory generalizing Scott-Solovay’s Boolean-valued models for classical set theory.
 Introduced by William C. Powell in a lecture in 1977 under the name of complete Heyting filtered algebra, Powell algebras were mentioned only obscurely in the literature after and were subsequently forgotten. Powell algebra is an abstraction of complete Heyting algebras, and a Powell-valued model VP is a generalization of Heyting-valued models. The main new class of examples of Powell algebras, however, is very different from complete Heyting algebras. It is based on the notion of partial combinatory algebra (pca), an elegant abstract version of Kleene’s formulation of partial recursive functions. The semantics of statements in VP, when the Powell algebra P is based on a pca, can thus be described as a notion of realizability which we call Powell’s realizability. Powell’s realizability when P is constructed from Kleene’s first pca generalizes Kleene’s 1952 number realizability from arithmetic to formulas of set theory and can be used to establish interesting relative consistency results.
 In this talk, I will present the definition of a Powell algebra and show how it generalizes complete Heyting algebras, discuss partial combinatory algebras and show how a pca gives rise to a Powell algebra. Then I will outline the proof of the soundness of the axioms of Intuitionistic Zermelo Fraenkel set theory (IZF) for a general Powell-valued model.

Let (X, μ) be a standard probability space. We say a (not necessarily proper) colouring of a graph G ⊆ X² is κ-domatic, for κ a cardinal, if each vertex x ∈ G sees exactly κ many different colours among its neighbours. In 2022, Edward Hou showed that any μ-preserving ω-regular Borel graph G ⊆ X² admits a μ-measurable ω-domatic colouring. We will sketch the proof and use this result to show that there exists a measurable 2-colouring of the Hamming graph on 2^ℕ which is (maximally) unfriendly, i.e. there are vertices x with countably infinite neighbours of a different colour from x.

The aim of this talk is to present and discuss a characterization of measured property (T) in terms of graphing expansion. This characterization, a graphing generalization of the Connes–Weiss Theorem for group property (T), will be the starting point of the talk. As an application, we will outline a proof for stability of measured property (T) under factor maps. This leads to the following question: what new examples of graphings with measured property (T) can we construct? A new example is offered by the Palm equivalence relation of, say, the Poisson point process on a lcsc group with property (T). As a non-example we have graphings with planar connected components.
The talk is partly based on joint work with Łukasz Grabowski and Sam Mellick.

Intuitively, de Finetti's theorem states that if we make a sequence of measurements in a setting where we know it to be irrelevant in which order these measurements are obtained, then these measurements are conditionally independent (independent given some latent random element). To be more precise, here is one version of de Finetti's theorem: Given a sequence of real random variables X₁, X₂, ... whose joint distribution is invariant under permutations of the indices, if we condition each X_i on the exchangeable algebra E obtaining the random variable (X_i | E), then the (X_i | E) are identically and independently distributed.
 It turns out that the assumption that the state space is "nice" (here, the real numbers) is crucial to this theorem. One can ask if this theorem holds for sequences of random elements whose state space is some more general measure space (that is, not just for sequences of real random variables). In this talk, I discuss this question. In particular, I give a characterization of conditionally iid sequences without any assumptions on the state space, as well as a version of de Finetti for sequences whose common distribution is Radon (strengthening a recent theorem due to Alam Irfan).
 This is joint work with Peter Potaptchik (Oxford) and Daniel M. Roy (University of Toronto).

In measured group theory (MGT), one studies group by their actions on finite or sigma-finite measure spaces. The notion of Measure Equivalence (ME), due to Gromov, is very similar to quasi-isometry and holds many powerful invariants. We will survey treeability in the ME context, look at the main obstruction to a strengthening of ME, namely orbit equivalence (OE). We will sketch why free groups of different rank are ME, but not OE. We will then look at these notions in the measure-class preserving context and see how cost is not useful here.

We say that two unitary or orthogonal representations of a finitely generated group G are additive conjugates if they are intertwined by an additive map, which need not be continuous. We associate to each representation of G a topological action that is a complete additive conjugacy invariant: the action of G by group automorphisms on the Bohr compactification of the underlying Hilbert space. Using this construction we show that the property of having almost invariant vectors is an additive conjugacy invariant. As an application we show that G is amenable if and only if there is a nonzero homomorphism from L²(G) into ℝ​​/ℤ that is invariant to the G-action.

Krauthgamer and Lee showed that every connected graph of polynomial growth admits an injective contraction mapping to (ℤⁿ, ||⋅|​​|_∞) for some n ∈ ℕ​​. We strengthen and generalize this result in a number of ways. In particular, answering a question of Papasoglu, we construct coarse embeddings from graphs of polynomial growth to ℤⁿ. Furthermore, we extend these results to Borel graphs. Namely, we show that graphs generated by free Borel actions of ℤⁿ are in a certain sense universal for the class of Borel graphs of polynomial growth. This provides a general method for extending results about ℤⁿ-actions to all Borel graphs of polynomial growth. For example, an immediate consequence of our main result is that all Borel graphs of polynomial growth are hyperfinite, which answers a well-known question in the area. An important tool in our arguments is the notion of Borel asymptotic dimension. Besides, we introduce the notion of Borel asymptotic power dimension and get more results for graphs of polynomial growth. This is joint work with Anton Bernshteyn.

We recall the definition of (finite) Borel asymptotic dimension, due to Conley, Jackson, Marks, Seward, and Tucker-Drob. We describe the inductive procedure they developed to obtain results for groups admitting certain normal series. We show that it could be sharpened to include a larger class of solvable groups. This is a joint work with Qingyuan Chen, Alon Dogon, and Brandon Seward.

Thompson's theorem states that a non-CLI Polish group admits an action which is unpinned. One corollary of this is that a non-CLI Polish group admits an action which is not essentially countable. We will review the notion of pinned equivalence relations and CLI groups, and present Thompson's proof from a metric structures point of view. Specifically, using the metric Scott analysis, due to Ben Yaacov, Doucha, Nies, and Tsankov, we will present a very streamlined proof of Thompson's theorem.

Consider a family of measure preserving transformations acting on a probability space, which are chosen at random by a stationary ergodic Markov chain. This setting gives rise to a skew product, which defines an instance of a random dynamical system (RDS). Among other contexts skew products of this form arise naturally within the ergodic theory of group actions.
 In many situations it is desirable to know whether the ergodicity of the family of transformations implies the ergodicity of the skew product. In this talk we shall present a characterization of the class of Markov chains, for which this implication holds true. To this end we introduce the notion of strict irreducibility for general Markov chains, which amounts to the absence of a state space decomposition into deterministic sets. This generalizes a concept originally introduced by A. I. Bufetov for finite state Markov chains in order to give a sufficient condition for the above implication to arbitrary state spaces and shows that it is in fact also necessary. The proof uses the framework of ergodic theory for general Markov operators and relies on spectral properties of the Perron-Frobenius operator associated to a skew product of the above type. The talk is based on joint work with Pablo Lummerzheim and Felix Pogorzelski.

Marks and Unger proved a Baire measurable variant of Hall's classical theorem, namely that if a locally finite bipartite Borel graph G satisfies |N(F)| > (1+ε​​) |F| for all finite independent sets F ⊆​​ V(G), for some fixed ε​​ > 0, then G admits a Borel matching on a Borel comeager invariant set. In the non-bipartite context, Tutte's theorem characterizes which finite graphs admit perfect matchings. By using a strategy similar to Marks–Unger, we establish a Baire measurable variant of Tutte's theorem for locally finite Borel graphs. A consequence of this result is the existence of Baire measurable perfect matchings for all Schreier graphs induced by free Borel actions of finitely generated non-amenable groups. This is joint work with Clark Lyons.

Group operations on a fixed countably infinite universe, say ℕ, form a Polish space 𝒢​​ (with the topology inherited from ℕ^{ℕ×​​ℕ}). Thus we can view group properties as isomorphism-invariant subsets of 𝒢​, and it makes sense to ask: what properties are generic (in the sense of Baire category)?
In my talk, I will address this question and if time permits, I may also say a few words about generic properties of compact groups.

A deep result of Gao–Jackson is that orbit equivalence relations induced by Borel actions of countable discrete abelian groups on Polish spaces are hyperfinite. Hjorth asked if indeed any orbit equivalence relation induced by a Borel action of an abelian Polish group on a Polish space, which is also essentially countable, must be essentially hyperfinite. We show that any countable Borel equivalence relation (CBER) which is treeable must be classifiable by an abelian Polish group. As the free part of the Bernoulli shift action of F₂ is a treeable CBER, and not hyperfinite, this answers Hjorth’s question in the negative.
  On the other hand, for certain abelian Polish groups such as ℝ^ω, Hjorth’s question has a positive answer. Indeed, we show that any orbit equivalence relation induced by a Borel action of a countable product of locally compact abelian Polish groups which is also potentially Π⁰₃ must be Borel-reducible to 𝔼​​₀^ω. By a dichotomy result of Hjorth-Kechris, this implies that essentially countable such orbit equivalence relations are hyperfinite. This uses a result of Cotton that locally compact abelian Polish groups yield essentially hyperfinite orbit equivalence relations, as well as the Hjorth analysis of Polish group actions.

A countable Borel equivalence relation (CBER) E on a Polish space X is said to be treeable if there is a Borel forest G ⊂​​ X² whose trees are precisely the equivalence classes of said relation. E is quasi-treeable if it has a Borel graphing, each of whose components is quasi-isometric to a tree.
 In joint work with Ruiyuan (Ronnie) Chen, Antoine Poulin and Anush Tserunyan, we show that quasi-treeable CBERs are treeable by giving a construction of a median graph associated to the quasi-treeing, which will be the main focus of this talk.

Answering a question of Tserunyan and Tucker-Drob, we provide examples of behaviors of Radon–Nikodym cocycles for a countable-to-one function in a measure class preserving setting. We provide examples of cocycles arising from the shift on the Baire space, showing that oscillatory behavior is possible, as well as converging to zero in a non-summable way. Our proof that these examples indeed exhibit the desired behavior relies on the Chung–Fuchs theorem for random walks on ℤ. This is joint work with Tasmin Chu.

In various cases, a law (that is, a quantifiers free formula) that holds in a group with high probability, must actually hold for all elements. For instance, a finite group in which the commutator law [x,y]=1 holds with probability at least 5/8, must be abelian. For infinite groups, one needs to work a bit harder to define the probability that a given law holds. One natural way is by sampling a random element uniformly from the r-ball in the Cayley graph and taking r to infinity; another way is by sampling elements using random walks. It was asked by Amir, Blachar, Gerasimova, and Kozma whether a law that holds with probability 1, must actually hold globally, for all elements. In a recent joint work with Be’eri Greenfeld, we give a negative answer to their question.
 In this talk I will give an introduction to probabilistic group laws and present a finitely generated group that satisfies the law x^p=1 with probability 1, but yet admits no group law that holds for all elements.

Van der Corput's Difference Theorem (vdCDT) is a useful tool in the study of multiple ergodic averages. We begin with a review of van der Corput's difference theorem and some of its applications in ergodic theory. We then review the notions of Lebesgue spectrum and singular spectrum for measure preserving ℤ-actions, as well as measure preserving actions of an amenable group. Next, we show how the classical vdCDT produces sequences that have Lebesgue spectrum in a suitably interpreted sense, and that an analogous vdCDT for countable amenable groups produces sequences that correspond to subrepresentations of the left regular representation. As applications we will obtain results about multiple ergodic averages for actions of countable abelian groups with noncommuting transformations.

The Connes Embedding Problem (CEP) had been an open problem in the theory of von Neumann algebras since the 1970′s when, in 2020, it was resolved by Ji, Natarajan, Vidick, Wright, and Yuen who proved that MIP* = RE, a computational complexity result involving quantum entanglement. Thus, the study of the CEP and its resolution brings together the fields of Functional Analysis, Computational Complexity Theory, Mathematical Quantum Physics, and—with the alternative connection between the two results found by Goldbring and Hart—Model Theory.
  Over the summer of 2023, I had the opportunity to engage in an independent reading project on this topic in order to better understand these fields and the result, overseen and mentored by Prof. Sabok and Prof. Panangaden. This talk is an adaptation of my write-up for this project and attempts to explain the CEP and its resolution by MIP* = RE through Model Theory at an elementary level.

In this talk, we continue the series on fixed price for a higher rank semisimple real Lie group G and its lattices. The proof of fixed price for G relies on a particular Poisson-Voronoi random tessellation on the symmetric space of G. The tessellation has deep ties to the group structure of G and geometric properties of the symmetric space. We'll introduce this tessellation, give some motivation for its relation to the group structure, and prove the property that allows fixed price-- any pair of cells in the tessellation shares an unbounded wall. This is joint work with Mikolaj Fraczyk and Sam Mellick.

Cost is an important invariant of probability measure preserving and essentially free actions. A group has fixed price if every such action has the same cost. Whilst usually studied only for countable groups, it can be defined for arbitrary locally compact second countable unimodular groups, and has implications for lattices in such groups (for example, if the group has fixed price one then rank gradient of sequences of lattices must vanish). This is joint work with Mikołaj Frączyk and Amanda Wilkens.

 The goal of this series of talks (part one of an undetermined number) is to explain the result in the title. I will give some historical context and motivation for the result, and then outline the structure of the argument. This outline is quite easy to follow, but requires a lot of black box type results. To truly understand the proof requires opening up a few of those black boxes (most of which are joint work of myself with Miklós Abért). The hope is that these talks will be informal, interactive, and have a choose-your-own-adventure quality. Which black boxes do you dare to open?

Cost is an important invariant of probability measure preserving and essentially free actions. A group has fixed price if every such action has the same cost. Whilst usually studied only for countable groups, it can be defined for arbitrary locally compact second countable unimodular groups, and has implications for lattices in such groups (for example, if the group has fixed price one then rank gradient of sequences of lattices must vanish). This is joint work with Mikołaj Frączyk and Amanda Wilkens.

 The goal of this series of talks (part one of an undetermined number) is to explain the result in the title. I will give some historical context and motivation for the result, and then outline the structure of the argument. This outline is quite easy to follow, but requires a lot of black box type results. To truly understand the proof requires opening up a few of those black boxes (most of which are joint work of myself with Miklós Abért). The hope is that these talks will be informal, interactive, and have a choose-your-own-adventure quality. Which black boxes do you dare to open?

Cost is an important invariant of probability measure preserving and essentially free actions. A group has fixed price if every such action has the same cost. Whilst usually studied only for countable groups, it can be defined for arbitrary locally compact second countable unimodular groups, and has implications for lattices in such groups (for example, if the group has fixed price one then rank gradient of sequences of lattices must vanish). This is joint work with Mikołaj Frączyk and Amanda Wilkens.

 The goal of this series of talks (part one of an undetermined number) is to explain the result in the title. I will give some historical context and motivation for the result, and then outline the structure of the argument. This outline is quite easy to follow, but requires a lot of black box type results. To truly understand the proof requires opening up a few of those black boxes (most of which are joint work of myself with Miklós Abért). The hope is that these talks will be informal, interactive, and have a choose-your-own-adventure quality. Which black boxes do you dare to open?

Cost is an important invariant of probability measure preserving and essentially free actions. A group has fixed price if every such action has the same cost. Whilst usually studied only for countable groups, it can be defined for arbitrary locally compact second countable unimodular groups, and has implications for lattices in such groups (for example, if the group has fixed price one then rank gradient of sequences of lattices must vanish). This is joint work with Mikołaj Frączyk and Amanda Wilkens.

 The goal of this series of talks (part one of an undetermined number) is to explain the result in the title. I will give some historical context and motivation for the result, and then outline the structure of the argument. This outline is quite easy to follow, but requires a lot of black box type results. To truly understand the proof requires opening up a few of those black boxes (most of which are joint work of myself with Miklós Abért). The hope is that these talks will be informal, interactive, and have a choose-your-own-adventure quality. Which black boxes do you dare to open?

The celebrated Ornstein-Weiss quasi-tiling machinery (or "generalized Rokhlin Lemma") has been an indispensable tool in the study of amenable group actions on probability spaces. While being tremendously successful for studying actions of amenable groups, this machinery appears at first glance to be of little use in the nonamenable setting due to the lack of Følner sets. In this talk I will discuss ongoing joint work with Damien Gaboriau and Tom Hutchcroft in which we are able to successfully apply the Ornstein-Weiss quasi-tiling machinery in the nonamenable context by taking advantage of amenable actions of nonamenable groups, and focusing specifically on generalized Bernoulli shifts associated to such amenable actions. This allows us to completely remove the normality assumption for a vast range of results about "normal coamenable subgroups."

Given a topological group G, a G-flow is a continuous action of G on a compact Hausdorff space X. This talk will discuss a new notion of ultraproduct for G-flows, and compare and contrast this notion to existing notions of ultraproduct for unitary representations of locally compact groups. In ongoing joint work with Gianluca Basso, we apply ultraproducts of G-flows to achieve a new characterization of those Polish groups G with the property that every minimal flow has a comeager orbit.

A countable Borel equivalence relation (CBER) E on a standard Borel space X is an equivalence relation on X that is Borel (viewed as a subset of X²), and whose equivalence classes are countable. A compression of an equivalence relation E on a set X is an injective map f: X → X such that for every E-class C we have f(C) ⫋ C. We say a CBER E is compressible if it admits a Borel compression. In this talk we show that the notion of compressibility is effective, that is, if a Δ¹₁ (i.e. effectively Borel) CBER E is compressible, then it admits a Δ¹₁ compression. This follows from an effective version of Nadkarni's theorem, from which we also derive an effective ergodic decomposition theorem. Finally, we provide an example of a Δ¹₁ CBER admitting a Borel invariant probability measure but no Δ¹₁ invariant probability measure, and use this to construct a Δ¹₁ CBER which is Borel isomorphic to a Δ¹₁ compact subshift of (2^ℕ)^𝔽_∞ but admits no Δ¹₁ isomorphism with such a subshift. This is joint work with Alexander Kechris.

We will start with some motivation and background for the talk, and then discuss stable decompositions of a countable ergodic p.m.p. equivalence relation. We will explain the definition and show that the stabilization of any equivalence relation without central sequences in its full group (i.e. it is not "Schmidt") has a unique stable decomposition. This provides the first non-strongly ergodic such examples. We will discuss the main ideas behind the proof, which incorporates some techniques inspired by von Neumann algebras.

The Følner function is a fundamental invariant for finitely generated amenable groups, capturing "how well they are amenable" by providing for every n the smallest possible size of a subset whose boundary has relative size at most 1/n. For instance, the Følner function of ℤ is linear while that of ℤ² is quadratic. In this talk, we will explain why the Følner function is monotonous under regular maps between amenable groups, a result which was open even for coarse embeddings. We will have to make a detour through quantitative orbit equivalence and a more appropriate version of the Følner function called the isoperimetric profile. This is joint work with Thiebout Delabie, Juhani Koivisto and Romain Tessera.

Dye's theorem states that any two ergodic measure-preserving transformations on a standard probability space are orbit equivalent: up to conjugating one of the two, they share the same orbits. Belinskaya's theorem shows that the corresponding cocycles have to behave badly: if they are integrable then the two transformations are flip-conjugate. In a joint work with Carderi, Joseph and Tessera, we show that her result becomes false if one replaces integrability by being in L^p for all p<1. As I will explain, this relies crucially on a new family of Polish groups that we associate to every subadditive function and every measure-preserving transformation.

A partial automorphism of a graph is an isomorphism between induced subgraphs of the graph. In 1992, as a key ingredient for proving the small index property for the automorphism group of the random graph (Hodges, Hodkinson, Lascar and Shelah '93), Hrushovski proved the following purely combinatorial result: For every finite graph G there is a finite graph H containing G as an induced subgraph such that every partial automorphism of G extends to an automorphism of H. Since then, analogous results have been proved for various other classes of structures and connections with model-theory and topological dynamics have been well established. In this talk I will give an overview of the area.

We give an introduction to structural generalizations of the well known Ramsey theorem. We start by 1960's work of Laver and Devlin about coloring finite subsets of rational numbers and show some recent results in the area. In particular a new and relatively straighforward proof of Dobrinen's theorem stating that big Ramsey degrees of the triangle-free graphs are finite. We show generalizations of this proof to new Ramsey results and outline an emerging theory of big Ramsey structures.
This is a joint work with Balko, Chodounsky, Dobrinen, Konečný, Nešetřil, de Rancourt, Todorcevic and Zucker.

Motivated by the search for perfect matchings, we find explicit connected toasts for free Borel actions of polynomial growth groups. This proof relies on machinery built for Borel asymptotic dimension, as well as on geometric properties of Cayley graphs of finitely presented groups. This is joint work with Matt Bowen and Jenna Zomback.

I will discuss recent joint work with Haosui Duanmu and Daniel M. Roy, in which we give a precise characterization of admissibility in Bayesian terms, solving a long-standing problem in the field of statistical decision theory. This result uses so-called hyperpriors, which can give infinitesimal weight to events, to achieve this characterization. I will also discuss some classical, standard results (that is, results not mentioning hyperpriors or infinitesimals) that arise from this work.

We show that every degree regular one-ended bipartite Borel graph admits a Baire measurable perfect matching. If the graph is also hyperfinite and pmp then we prove the same result for measurable matchings. This talk is based on joint work with Kun and Sabok and with Poulin and Zomback.

An invariant random order on a group is a measure on the space of all orders on the group that is invariant under the natural shift-action. Recently, Glasner, Lin and Meyerovitch proved that SL₃(ℤ) has an order that could not be extended to an invariant random total order. Starting off of their result, I will show that amenability for groups is eqivalent to the property that any invariant random order could be extended to the invariant random total order.

Following Krieger's classification theorem of orbit equivalence of non-singular actions, many authors dealt with classifying actions on the symbolic space 2^G equipped with a product measure. In this context, there are fairly general results for actions that change finitely many coordinates, such as the finite permutations or the odometer. However, the shift action of G on 2^G is substantially harder to classify and it remained open for decades.
 In this talk I will introduce recent results on the orbit equivalence classification of the nonsingular shift, both for Bernoulli shift (product measure) and for Markov shifts (Markov measure). I will survey a new technique to compute the Kreiger's invariant of the "ratio set", using an interesting type of ergodic theorem due to Danilenko.

Two group actions are orbit equivalent if there is a Borel bijection between the underlying spaces that carries orbit to orbit. The classification of actions according to orbit equivalence is an old and important subject of study. In the framework of ergodic theory, we put a measure on the underlying spaces and study the orbit equivalence of an action up to zero measure sets. Since the orbit equivalence class of an action depends crucially on the measure, the ergodic theory of orbit equivalence is different than the theory of orbit equivalence without a measure.
 In this talk I will introduce the theory of orbit equivalence for measure preserving actions and for non-singular actions of amenable groups. I will survey the classical result by Dye on measure preserving actions, and the seminal work of Krieger on non-singular actions.

It is a classical result that any Borel set in a "nice" topological space can be made open in a finer "nice" topology. The Becker–Kechris theorem can be seen as characterizing the extent to which this remains true in the presence of a group action. We give a new proof of the Becker–Kechris theorem, and use it to extend the theorem in several directions: to n-ary relations; to groupoids; to non-Hausdorff spaces; and even to point-free "spaces".

I will discuss an alternative proof of the result in the title, part of ongoing joint work with Lukasz Grabowski and Hector Jardon-Sanchez.
The talks will be as self-contained as possible, with the only assumed background being knowing the definition of cost. If time permits, in the subsequent talk I will discuss further examples and applications of our work.

I will discuss an alternative proof of the result in the title, part of ongoing joint work with Lukasz Grabowski and Hector Jardon-Sanchez.
The talks will be as self-contained as possible, with the only assumed background being knowing the definition of cost. If time permits, in the subsequent talk I will discuss further examples and applications of our work.

Cost is a [1, ∞)-valued measure-isomorphism invariant of equivalence relations defined by Gilbert Levitt and heavily studied by Damien Gaboriau. For a large class of equivalence relations, including aperiodic amenable, the cost is 1. Yoshikata Kida and Robin Tucker-Drob 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.

Inner amenable groups were first introduced by Effros in connection to property Gamma of von Neumann algebras. This talk will introduce inner amenable groups and amenable actions and provide some examples. We'll then discuss some algebraic and ergodic theoretic consequences of inner amenability, as time permits.

In order to distinguish countable Borel equivalence relations (CBERs) up to measure isomorphism, Gilbert Levitt introduced an invariant called cost. I will present my understanding of Damien Gaboriau's "Mercuriale de groupes et de relations", which shows that any Borel treeing of a CBER achieves its cost. In particular, the cost of a free, ergodic, pmp action of the free group on n generators is n.

Orderings on groups have been studied from many angles. Such orders can be encoded into a Polish space, and the Borel complexity of isomorphisms of orders has been a recent subject of study. Motivated by a question of Calderoni, Marker, Motto Ros and Shani, we prove that the isomorphism relation on Archimedean orders of ℤ​² is hyperfinite, but not smooth.

For every Polish permutation group P≤Sym(ℕ) let A↦[A]_P be the assignment which maps every A⊆ℕ to the set of all k ∈ ℕ whose orbit under the action of the stabilizer P_A of A is finite. Then A↦[A]_P is a closure operator and hence it endows P with a natural notion of dimension dim(P). This notion of dimension has been extensively studied in model theory when A↦[A]_P satisfies additionally the exchange principle; that is, when A↦[A]_P forms a pregeometry. However, under the exchange principle every Polish permutation group P with dim(P)<∞ is locally compact and therefore unable to generate any "wild" dynamics. In this talk we will discuss the relationship between dim(P) and certain strong ergodicity phenomena in the absence of the exchange principle. In particular, for every n ∈ ℕ  we will provide a Polish permutation group P with dim(P)=n whose Bernoulli shift P↷ℝ^ℕ is generically ergodic relative to the injective part of the Bernoulli shift of any permutation group with dim(Q)<n. We will use this to exhibit an equivalence relation of pinned cardinal ℵ₁ which strongly resembles Zapletal’s counterexample to a question of Kechris, but which does not Borel reduce to the latter. Our proofs rely on the theory of symmetric models of choiceless set-theory and in the process we establish that a vast collection of symmetric models admit a theory of supports similar to the basic Cohen model. This is joint work with Assaf Shani.

Consider the following purely probabilistic problem. Take two infinite random discrete sets of points in the Euclidean space whose distributions are invariant under isometries. Find a "factor" perfect matching between the two, where factor means, intuitively, that every point can determine its pair using local information and using the same method. We want to make the probability that some fixed point is at distance at least r from its pair decay as fast as possible. A recent result of Bowen, Kun, and Sabok has become an important tool in settling this question for Poisson point processes, where we found a construction with optimal tail, significantly improving on previous ones.

Borel determinacy asserts that any two-player game of perfect information with a Borel payoff set is determined. The theorem was proved by Donald Martin in 1975, and while it holds much importance across descriptive set theory, its proof is technically difficult. In this talk we will outline the main ideas of the proof, including background on infinite games and determinacy. If time permits we will discuss the details of Martin’s most recent simplification of the proof using taboos, which we streamlined.

The Poisson process provides a canonical way to build a probability space from a possibly infinite measure space. We give an introduction to Poisson processes from a descriptive set theorist's perspective, with some applications to constructing free pmp actions of Polish groups.

Given a set K of Borel probability measures on the Cantor space X, consider the set G_K of all homeomorphisms which preserve every measure in K and which do not fix any nontrivial clopen set. For some Bernoulli measures µ, A. Yingst proved that the set of invariant measures of a generic element of G_µ is as small as possible (equal to {µ} in certain cases; this gives many interesting examples of totally ergodic homeomorphisms). I will explain why Yingst's result is a particular case of a theorem about dynamical simplices, i.e. sets of invariant measures of minimal homeomorphisms. I will recall the characterization of a dynamical simplex, try to motivate their study and the problem at hand; if time permits I will describe another Baire-category fact (describing when their exist nonmeager conjugacy classes in G_K) and why it rules out a potential approach towards computing the Borel complexity of conjugacy of minimal homeomorphisms (a well-known open problem).

We discuss new pointwise ergodic theorems for free semigroup actions, where the averages are taken over trees. This is joint work with Anush Tserunyan.

Actions of locally compact groups can be profitably studied by looking at their lacunary sections. In particular, lacunary sections can be used to define the cost of essentially free pmp actions of such groups.

In this talk, I will explain how you can visualise a lacunary section as an "invariant point process" on the group, and I will show how this picture can be exploited to give the first new technique for computing the cost of actions of nondiscrete groups.

No knowledge of probability theory will be required.

Joint work with Miklós Abért.

We prove various antirigidity and rigidity results around the orbit equivalence of wreath product actions. Let F be a nonabelian free group. In particular, we show that the wreath products A ≀ F and B ≀ F are orbit equivalent for any pair of nontrivial amenable (possibly finite) groups A, B. This is most interesting when the group A is finite. In order to accomplish this, we introduce the notion of a cofinitely equivariant map between shift spaces. This is joint work with Robin Tucker-Drob.

We show that every one-ended bounded degree Borel graph admits a one-ended component spanning tree on a comeagre Borel set. As an application, we show that such graphs admit Borel perfect matchings generically if they are bipartite and d-regular, and admit balanced orientations generically if they are 2d-regular. This talk is based on upcoming work with Poulin and Zomback.

(This is a work in progress with Matthieu Joseph.) The notion of invariant random subgroups (IRS) classically describes the conjugacy invariant measures on the (compact) space of closed subgroups of a given locally compact group. Our idea is to explore this notion when working with a Polish group instead of a locally compact one. In particular, the permutation group of the integers, S_∞, is a very rich example of a Polish group that yields interesting results when it comes to IRSs. I will define all the notions mentioned in this abstract, spending in particular some time to describe subgroups of S_∞.

Given a countable Borel equivalence relation E, we consider the problem of lifting a Borel action of a countable group Gamma on X/E to a Borel action of Gamma on X. This is always possible when E is compressible, but it can happen that there are Borel bijections of X/E which do not lift to Borel automorphisms of E. This leads us to consider the problem of lifting outer actions, that is, actions on X/E induced by Borel automorphisms of E. We show that for many classes of groups Gamma, such as amenable groups and amalgamated products of finite groups, it is possible to lift any outer action on any X/E, and we show that any such group must be treeable. This is joint work with Joshua Frisch and Alexander Kechris.

We establish several new complexity results using the shift graph on [ℕ]^ℕ​​:

1. Using measure theory and graphs with large expansion, we show that there it is hard to decide the Borel chromatic number of locally finite, acyclic, bounded degree graphs.
2. Using determinacy and Marks' method, we prove the optimal result in the Borel context.
3. Using toasts, we show a Borel analogue of the Hell-Nesetril theorem.

In this talk we will present some results about the Euclidean spheres in higher dimensions and the corresponding orbit equivalence relations induced by the group of rational rotations. We will show that such equivalence relations are not treeable in dimension greater than 2. Also we will discuss progress on the conjecture that such equivalence relation are not universal CBERs.

I will introduce a new type of Borel graphs, homomorphism graphs, and show how to extend the celebrated determinacy method of Marks to these graphs. The main idea, rather surprisingly, comes from the adaptation of Marks' technique to the LOCAL model of distributed computing. In the talk, I will discuss this adaptation as well as some applications of this approach in descriptive combinatorics.
This is a joint work with Brandt, Chang, Grunau, Rozhoň and Vidnyánszky.

In this talk I will discuss problems related to finding measurable integral flows and perfect matching in hyperfinite graphings (probability measure preserving Borel graphs), as well as applications to measurable equideompositions. In particular, I will show that for one ended hyperfinite graphings, admitting a measurable integral flow is equivalent to admitting a (not necessarily measurable) flow, and that such graphings also admit measurable perfect matchings if they are bipartite and d-regular. These results will then be applied to give new proofs of measurable circle squaring. Based on joint work with Marcin Sabok and Gábor Kun.

Tarski's 1949 theory of cardinal algebras seeks to axiomatize key features of cardinal arithmetic without assuming the axiom of choice. The theory is remarkable in its efficiency: from a few simple axioms, Tarski (and later authors) derive seemingly all conceivable "natural" properties of countable addition in familiar algebras such as [0,∞]. In this talk, I will present a result that partly explains this phenomenon: every cardinal algebra A embeds into an algebra of Borel [0,∞]-valued functions (on a standard Borel space when A is countably presented, and more generally on a locale). As an application, I will sketch an abstract, nearly combinatorics-free proof of Nadkarni's theorem on the existence of invariant measures.

We present a new result, namely that the group of absolutely continuous homeomorphisms of the interval admits a comeagre conjugacy class (i.e. it admits generic elements). Furthermore, we give a characterization of the generic elements.

We present a new result, namely that the group of absolutely continuous homeomorphisms of the interval admits a comeagre conjugacy class (i.e. it admits generic elements). Furthermore, we give a characterization of the generic elements.

Akhmedov and Cohen recently showed that the homeomorphism group of the interval is generically 2-generated ¶mdash; that is, the generic pair of elements generate a dense subgroup. In this talk we outline the proof of this result, and we show how it may be altered to show the same result for the group of absolutely continuous homeomorphisms of the interval.

In this part, we finish covering Section 3 of the paper of Conley, Gaboriau, Marks and Tucker-Drob. We will discuss the proof of the fact that any Borel planar graph is measure treeable.

In this part, we cover the first part of Section 3 of the paper of Conley, Gaboriau, Marks and Tucker-Drob. We will discuss the proof of the fact that any Borel planar graph is measure treeable.

We will begin the talk by explicitly stating and explaining the local-global bridge lemmas in the pmp setting that were used in various constructions, e.g. to go from nowhere µ-hyperfiniteness to exponential growth. We then discuss properties of 2-ended graphs, maximal hyperfinite connected subrelations, and prove the 1% lemma — a conservation property for µ-nonhyperfiniteness.

We begin by presenting µ-hyperfiniteness of locally finite graphs as almost finiteness (the 99% lemma) and use this to prove a characterization of µ-hyperfiniteness in terms of the isoperimetric constant (the Kaimanovich–Elek theorem).

Assuming a characterization of µ-hyperfiniteness in terms of the isoperimetric constant (Kaimanovich–Elek theorem), we explain how nowhere µ-hyperfiniteness implies the existence of a Borel a.e. one-ended spanning subforest.

In the third talk on this paper, we continue to investigate which graphs have a.e. spanning subforests. We prove that any pmp graph of superquadratic growth has an a.e. spanning subforest by demonstrating a sufficient condition for having such a subforest.

We give a brief introduction to the use of one-ended spanning trees and forests in descriptive graph combinatorics and characterize which hyperfinite locally finite Borel graphs admit a.e. one-ended spanning subforests.

Krieger's generator theorem says that all free ergodic measure preserving ℤ actions (under natural entropy constraints) can be modelled by a full shift. Following results by Anush Tserunyan and answering a question by Benjamin Weiss, in a sequence of two papers Mike Hochman noticed that this theorem can be strengthened: He showed that all free homeomorphisms of a Polish space (under entropy constraints) can be Borel embedded into the full shift. In this talk we will discuss some results along this line from a recent paper with Tom Meyerovitch and ongoing work with Spencer Unger.

We will begin with an overview of the classical van der Corput Difference Theorem and some of its Hilbertian variants that are useful in Ergodic Theory, including the variant that is used in the proof of Szemeredi's Theorem. We will then briefly review the ergodic hierarchy of mixing and point out the similarities to the existing variants of van der Corput's Theorem. Afterwards, we will state generalizations of the existing variants of van der Corput's Difference Theorem in Hilbert spaces that demonstrate connections to weak mixing, mild mixing, strong mixing, and Bernoulli (this last connection is more delicate than the rest). We will also be able to state a new Hilbertian variant of van der Corput's Difference Theorem corresponding to ergodicity. If time permits, we will state mixing van der Corput Difference Theorems in the context of uniform distribution.

In these two talks, we will discuss a result by Gutman, Tsankov, and Zucker, where non-metrizability of the universal minimal flow of the homeomorphism groups of high dimensional manifolds is established. The proof uses ingenious technology in the form of the space of maximal connected chains, as well as geometric property inherited from the charts.

In these two talks, we will discuss a result by Gutman, Tsankov, and Zucker, where non-metrizability of the universal minimal flow of the homeomorphism groups of high dimensional manifolds is established. The proof uses ingenious technology in the form of the space of maximal connected chains, as well as geometric property inherited from the charts.

A topological dynamical system (i..e a group acting by homeomorphisms on a compact Hausdorff space) is said to be proximal if for any two points p and q we can simultaneously "push them together" (rigorously, there is a net g_n such that limg_n(p) = limg_n(q)). In his paper introducing the concept of proximality, Glasner noted that whenever ℤ acts proximally, that action will have a fixed point. He termed groups with this fixed point property “strongly amenable”.
  The Poisson Boundary of a random walk on a group is a measure space that corresponds to the space of different asymptotic trajectories that the random walk might take. Given a group G and a probability measure μ on G the Poisson boundary is trivial (i.e. has no non-trivial events) if and only if G supports a bounded mu-harmonic function. A group is Called Choquet Deny if its Poisson Boundary is trivial for every μ.
  In this talk I will discuss work giving an explicit classification of which groups are Choquet Deny, which groups are strongly amenable, and what these mysteriously equivalent classes of groups have to do with the ICC property. I will also discuss why strongly amenable groups can be viewed as strengthening amenability in at least three distinct ways thus proving the name is extremely well deserved.
  This is joint work with Yair Hartman, Omer Tamuz, and Pooya Vahidi Ferdowsi.

We describe a probabilistic model involving iterated Brownian motion for constructing a random chainable continuum. We show that this random continuum is indecomposable. We use our probabilistic model to define a Wiener-type measure on the space of all chainable continua. This is joint work with Viktor Kiss.

In this talk I will describe probabilistic tools that can be used to construct continuous solutions to combinatorial problems on zero-dimensional spaces. I will also discuss some applications of these tools. In particular, I will outline an equivalence between certain problems in two seemingly disparate subjects: continuous combinatorics and distributed computing.

Probabilistic bisimulation is an equivalence relation on the states of a Labelled Markov Process that captures behavioural equivalence. It was introduced by Larsen and Skou in the late 1980s following the definition of bisimulation for nondeterministic transition systems in the 1970s by Park and Milner. I and my coworkers extended the theory to systems with continuous state spaces. In particular we showed that one can characterize bisimulation by a modal logic, which, surprisingly, was much simpler than the logic previously used to characterize probabilistic bisimulation on discrete state spaces. We were able to do this by using ideas from descriptive set theory specifically the concept of smooth equivalence relation and the unique structure theorem for analytic spaces. Later we extended these results to cover simulation as well. Still later this work was extended to MDPs and to metric analogues of bisimulation. I will give an expository talk assuming the audience knows all the relevant measure theory and descriptive set theory but not the computer science concepts like bisimulation. I will use a tablet to give a “chalkboard” talk rather than slides. This is joint work with Josée Desharnais, Abbas Edalat and then later with Josée Desharnais, Radha Jagadeesan and Vineet Gupta and finally with Florence Clerc, Nathanael Fijalkow and Bartek Klin.

We consider substitutions on countably infinite alphabets as Borel dynamical system and build their Bratteli-Vershik models. We prove two versions of Rokhlin’s lemma for such substitution dynamical systems. Using the Bratteli-Vershik model we give an explicit formula for a shift-invariant measure (finite and infinite) and provide a criterion for this measure to be ergodic. This is joint work with Sergii Bezuglyi and Palle Jorgensen.

In this series of two talks, we will give a brief introduction to the field of descriptive graph combinatorics and present a new proof of the Kechris-Solecki-Todorcevic (KST) dichotomy discovered independently by Anton Bernshteyn and Ben Miller. During the first talk we will discuss some key examples and results from this field, including the KST dichotomy and its applications. In the second talk we will go over Anton and Ben's proof in detail.

In this series of two talks, we will give a brief introduction to the field of descriptive graph combinatorics and present a new proof of the Kechris-Solecki-Todorcevic (KST) dichotomy discovered independently by Anton Bernshteyn and Ben Miller. During the first talk we will discuss some key examples and results from this field, including the KST dichotomy and its applications. In the second talk we will go over Anton and Ben's proof in detail.

In this series of talks we present a probabilistic model for name generation. Specifically, we interpret the nu-calculus, a simply-typed lambda-calculus with name generation, in the category of quasi-Borel spaces, an extension of the category of standard Borel spaces supporting both measure theory and higher-order programming. We prove that this model is fully abstract at first-order types. This is joint work with Marcin Sabok, Sam Staton and Dario Stein.
  In part 3 of this series, we present the proof of full abstraction of the nu-calculus in the category of quasi-Borel spaces. It will be a nice mix of programming language theory and descriptive set theory. On the programming language side, we will use logical relations to construct a normal form for the nu-calculus eliminating private names. On the descriptive set theory side, we will use a pmp action and a pair of Borel-inseparable sets to prove that passing to the normal form is valid in QBS.

In this series of talks we present a probabilistic model for name generation. Specifically, we interpret the nu-calculus, a simply-typed lambda-calculus with name generation, in the category of quasi-Borel spaces, an extension of the category of standard Borel spaces supporting both measure theory and higher-order programming. We prove that this model is fully abstract at first-order types. This is joint work with Marcin Sabok, Sam Staton and Dario Stein.

In this series of talks we present a probabilistic model for name generation. Specifically, we interpret the nu-calculus, a simply-typed lambda-calculus with name generation, in the category of quasi-Borel spaces, an extension of the category of standard Borel spaces supporting both measure theory and higher-order programming. We prove that this model is fully abstract at first-order types. This is joint work with Marcin Sabok, Sam Staton and Dario Stein.