Organizers:  Antoine Poulin, Marcin Sabok, & Anush Tserunyan 
Typical time (unless specified otherwise below):  Tuesday at 11:30 ET (50 minutes + ε) 
Online location:  Zoom meeting https://mcgill.zoom.us/j/87587975637, password: Bor▩▩ (the σalgebra generated by open sets) 
Physical location:  Burnside 1234 
2023 Dec 6  Speaker: Jing Yu (Georgia Tech) 
Title: Largescale geometry of graphs of polynomial growth  
AbstractKrauthgamer and Lee showed that every connected graph of polynomial growth admits an injective contraction mapping to (ℤ^{n}, ⋅_{∞}) 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 ℤ^{n}. Furthermore, we extend these results to Borel graphs. Namely, we show that graphs generated by free Borel actions of ℤ^{n} are in a certain sense universal for the class of Borel graphs of polynomial growth. This provides a general method for extending results about ℤ^{n}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 wellknown 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. 

2023 Nov 28  Speaker: Petr Naryshkin (University of Münster) 
Title: Free actions of ℤ_{2}≀ℤ^{2} have finite Borel asymptotic dimension  
AbstractWe recall the definition of (finite) Borel asymptotic dimension, due to Conley, Jackson, Marks, Seward, and TuckerDrob. 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. 

2023 Nov 21  Speaker: Assaf Shani (Concordia University) 
Title: Actions of nonCLI groups via metric structures  
AbstractThompson's theorem states that a nonCLI Polish group admits an action which is unpinned. One corollary of this is that a nonCLI 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. 

Typeset notes  
2023 Nov 14  Speaker: Elias Zimmermann (Leipzig University) 
Title: Strictly irreducible Markov operators and ergodicity properties of skew products  
Abstract
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.


Slides  
2023 Nov 7  Speaker: Alexander Kastner (UCLA) 
Title: Baire measurable perfect matchings  
AbstractMarks 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 nonbipartite 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 nonamenable groups. This is joint work with Clark Lyons. 

Slides  
2023 Oct 31  Speaker: Tamás Kátay (Eötvös Loránd University) 
Title: Generic properties of countably infinite groups  
Abstract
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 isomorphisminvariant subsets of 𝒢, and it makes sense to ask: what properties are generic (in the sense of Baire category)?


Notes  
2023 Oct 24  Speaker: Shaun Allison (University of Toronto) 
Title: Treeable CBERs are classifiable by an abelian Polish group  
Abstract
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 F_{2} is a treeable CBER, and not hyperfinite, this answers Hjorth’s question in the negative.


Slides  
2023 Oct 18  Speaker: Ran Tao (Carnegie Mellon University) 
Joint with the McGill Geometric Group Theory seminar  
Title: Quasitreeable CBERs are treeable via median graphs  
Abstract
A countable Borel equivalence relation (CBER) E on a Polish space X is said to be treeable if there is a Borel forest G ⊂ X^{2} whose trees are precisely the equivalence classes of said relation. E is quasitreeable if it has a Borel graphing, each of whose components is quasiisometric to a tree.


Slides  
2023 Oct 17  Speakers: Owen Rodgers & Sasha Bell (McGill University) 
Title: Behavior of Radon–Nikodym cocycles of oneended measure class preserving transformations  
AbstractAnswering a question of Tserunyan and TuckerDrob, we provide examples of behaviors of Radon–Nikodym cocycles for a countabletoone 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 nonsummable 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. 

Slides  
2023 Oct 3  Speaker: Gil Goffer (UC San Diego) 
Title: Probabilistic laws on infinite groups  
Abstract
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 rball 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.


Slides  
2023 Sep 19  Speaker: Sohail Farhangi (University of Adam Mickiewicz) 
Title: Van der Corput's Difference Theorem and the left regular representation  
AbstractVan 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 ℤactions, 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. 

Slides  
2023 Sep 12  Speaker: Joel Newman (McGill University) 
Title: Solving the Halting Problem with the Connes Embedding Problem  
Abstract
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.


Exposition  
2023 May 9  Speaker: Amanda Wilkens (UT Austin) 
Title: Higher rank groups have fixed price one, part 5  
AbstractIn 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 PoissonVoronoi 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. 

Annotated slides  
2023 May 3  Speaker: Sam Mellick (McGill University) 
Title: Higher rank groups have fixed price one, part 4  
AbstractCost 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 chooseyourownadventure quality. Which black boxes do you dare to open? 

Blackboard photos  
2023 May 2  Speaker: Sam Mellick (McGill University) 
Title: Higher rank groups have fixed price one, part 3  
AbstractCost 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 chooseyourownadventure quality. Which black boxes do you dare to open? 

Blackboard photos  
2023 Apr 25  Speaker: Sam Mellick (McGill University) 
Title: Higher rank groups have fixed price one, part 2  
AbstractCost 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 chooseyourownadventure quality. Which black boxes do you dare to open? 

Blackboard photos  
2023 Apr 18  Speaker: Sam Mellick (McGill University) 
Title: Higher rank groups have fixed price one, part 1  
AbstractCost 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 chooseyourownadventure quality. Which black boxes do you dare to open? 

Notes  
2023 Apr 11  Speaker: Robin TuckerDrob (University of Florida) 
Title: Følner Tilings in the nonamenable setting  
AbstractThe celebrated OrnsteinWeiss quasitiling 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 OrnsteinWeiss quasitiling 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." 

2023 Apr 4  Speaker: Andy Zucker (University of Waterloo) 
Title: A notion of ultraproduct for flows of topological groups  
AbstractGiven a topological group G, a Gflow is a continuous action of G on a compact Hausdorff space X. This talk will discuss a new notion of ultraproduct for Gflows, 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 Gflows to achieve a new characterization of those Polish groups G with the property that every minimal flow has a comeager orbit. 

Notes  
2023 Mar 28  Speaker: Michael Wolman (Caltech) 
Title: An effective version of Nadkarni’s Theorem  
AbstractA 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 X^{2}), 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 Eclass 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 Δ^{1}_{1} (i.e. effectively Borel) CBER E is compressible, then it admits a Δ^{1}_{1} 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 Δ^{1}_{1} CBER admitting a Borel invariant probability measure but no Δ^{1}_{1} invariant probability measure, and use this to construct a Δ^{1}_{1} CBER which is Borel isomorphic to a Δ^{1}_{1} compact subshift of (2^{ℕ})^{𝔽∞} but admits no Δ^{1}_{1} isomorphism with such a subshift. This is joint work with Alexander Kechris. 

Live notes and prepared notes  
2023 Feb 20  Speaker: Pieter Spaas (University of Copenhagen) 
Title: Stable decompositions of countable equivalence relations  
AbstractWe 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 nonstrongly ergodic such examples. We will discuss the main ideas behind the proof, which incorporates some techniques inspired by von Neumann algebras. 

Notes  
2023 Feb 14  Speaker: François Le Maître (Université de Paris) 
Title: Monotonicity of the Følner function under coarse embeddings between amenable groups  
AbstractThe 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 ℤ^{2} 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. 

Notes  
2023 Jan 31  Speaker: François Le Maître (Université de Paris) 
Title: Belinskaya's theorem is optimal  
AbstractDye's theorem states that any two ergodic measurepreserving 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 flipconjugate. 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 measurepreserving transformation. 

Notes  
2023 Jan 24  Speaker: Matěj Konečný (Charles University) 
Title: Extending partial automorphisms  
AbstractA 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 modeltheory and topological dynamics have been well established. In this talk I will give an overview of the area. 

Slides  
2023 Jan 24  Joint with the McGill Discrete Mathematics and Optimization seminar 
Speaker: Jan Hubička (Charles University)  
Title: Introduction to big Ramsey degrees  
Abstract
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 trianglefree
graphs are finite. We show generalizations of this proof to new Ramsey
results and outline an emerging theory of big Ramsey structures.


Slides  
2022 Dec 6  Speaker: Antoine Poulin (McGill University) 
Title: Explicit connected toasts in one ended polynomial growth groups  
AbstractMotivated 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. 

Notes  
2022 Nov 22  Speaker: David Schrittesser (University of Toronto) 
Title: Nonstandard methods for statistics  
AbstractI 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 longstanding problem in the field of statistical decision theory. This result uses socalled 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. 

Notes  
2022 Nov 15  Speaker: Matthew Bowen (McGill University) 
Title: Definable matchings in oneended Borel graphs  
AbstractWe show that every degree regular oneended 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. 

Slides  
2022 Nov 8  Speaker: Andrei Alpeev (St. Petersburg State University) 
Title: Extensions of invariant random orders and amenability  
AbstractAn invariant random order on a group is a measure on the space of all orders on the group that is invariant under the natural shiftaction. Recently, Glasner, Lin and Meyerovitch proved that SL_{3}(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. 

Slides  
2022 Nov 1  Speaker: Nachi Avraham Re'em (Hebrew University of Jerusalem) 
Title: The orbit equivalence class of the nonsingular shift  
Abstract
Following Krieger's classification theorem of orbit equivalence of nonsingular 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.


Slides  
2022 Oct 25  Speaker: Nachi Avraham Re'em (Hebrew University of Jerusalem) 
Title: An introduction to the ergodic theory of orbit equivalence classification of group actions  
Abstract
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.


Slides  
2022 Oct 18  Speaker: Ruiyuan (Ronnie) Chen (University of Michigan) 
Title: Topology versus Borel structure for actions  
AbstractIt 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 nary relations; to groupoids; to nonHausdorff spaces; and even to pointfree "spaces". 

Annotated slides  
2022 Oct 4  Speaker: Sam Mellick (McGill University) 
Title: Kazhdan groups have cost one, after HutchcroftPete, part 2  
Abstract
I will discuss an alternative proof of the result in the title, part of ongoing joint work with Lukasz Grabowski and Hector JardonSanchez.


2022 Sep 27  Speaker: Sam Mellick (McGill University) 
Title: Kazhdan groups have cost one, after HutchcroftPete, part 1  
Abstract
I will discuss an alternative proof of the result in the title, part of ongoing joint work with Lukasz Grabowski and Hector JardonSanchez.


Slides  
2022 Sep 20  Speaker: Konrad Wrobel (McGill University) 
Title: Cost of inner amenable equivalence relations  
AbstractCost is a [1, ∞)valued measureisomorphism 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 TuckerDrob 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 TuckerDrob. 

Notes  
2022 Sep 13  Speaker: Konrad Wrobel (McGill University) 
Title: An introduction to inner amenable groups  
AbstractInner 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. 

Notes  
2022 Apr 26  Speaker: Jenna Zomback (University of Illinois UrbanaChampaign) 
Title: Gaboriau's fundamental theorem of cost  
AbstractIn 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. 

Notes and recording  
2022 Apr 19  Speaker: Antoine Poulin (McGill University) 
Title: Complexity of Archimedean orders  
AbstractOrderings 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. 

Slides  
2022 Apr 12  Speaker: Aristotelis Panagiotopoulos (Carnegie Mellon University) 
Title: Strong ergodicity phenomena for Bernoulli shifts of bounded algebraic dimension  
AbstractFor 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 ℵ_{1} 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 settheory 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. 

Slides  
2022 Apr 5  Speaker: Ádám Timár (University of Iceland and Rényi Institute) 
Title: Poisson matchings of optimal tail via matchings in graphings  
AbstractConsider 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. 

Slides  
2022 Mar 22  Speakers: Thomas Buffard, Gabriel Levrel, Sam Mayo (McGill University) 
Title: Borel determinacy in 50 (+ε) minutes  
AbstractBorel determinacy asserts that any twoplayer 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. 

Annotated slides  
2022 Mar 15  Speaker: Ruiyuan (Ronnie) Chen (CRM / McGill University) 
Title: Introduction to Poisson processes  
AbstractThe 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. 

Notes  
2022 Mar 8  Speaker: Julien Melleray (Université Lyon 1) 
Title: Generic properties of (chaintransitive) homeomorphisms preserving a given dynamical simplex  
AbstractGiven 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 Bairecategory 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 wellknown open problem). 

Annotated slides  
2022 Feb 22  Speaker: Jenna Zomback (University of Illinois UrbanaChampaign) 
Title: Pointwise ergodic theorems for semigroup actions  
AbstractWe discuss new pointwise ergodic theorems for free semigroup actions, where the averages are taken over trees. This is joint work with Anush Tserunyan. 

Annotated slides and recording  
2022 Feb 15  Speaker: Sam Mellick (ENS de Lyon) 
Title: Visualising actions, computing cost, and fixed price for G x Z  
Abstract
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.


Slides  
2022 Feb 8  Speaker: Konrad Wrobel (Steklov Math. Institute at St. Petersbourg) 
Title: Orbit equivalence, cofinitely equivariant maps, and wreath products  
AbstractWe 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 $A \wr F$ and $B \wr F$ 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 TuckerDrob. 

Notes  
2022 Feb 1  Speaker: Matthew Bowen (McGill University) 
Title: Oneended spanning trees and generic combinatorics  
Abstract
We show that every oneended bounded degree Borel graph admits a oneended 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 

Slides  
2021 Dec 13  Speaker: Colin Jahel (Carnegie Mellon University) 
Title: Invariant random subgroups of Polish groups (with a focus on S_{∞})  
Abstract(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_{∞}. 

Notes  
2021 Dec 6  Speaker: Forte Shinko (Caltech) 
Title: Lifts of Borel actions on quotient spaces  
AbstractGiven 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. 

Slides  
2021 Nov 29  Speaker: Zoltán Vidnyánszky (Caltech) 
Title: Determinacy, measure, toasts, and the shift graph  
AbstractWe establish several new complexity results using the shift graph on [ℕ]^{ℕ}:


Annotated slides  
2021 Nov 15  Speaker: Filippo Calderoni (University of Illinois Chicago) 
Title: Rotation equivalence  
AbstractIn 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. 

2021 Nov 8  Speaker: Jan Grebik (University of Warwick) 
Title: Homomorphism graphs and Descriptive combinatorics  
Abstract
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.


Slides  
2021 Nov 1  Speaker: Matthew Bowen (McGill University) 
Title: Measurable integral flows and perfect matchings in hyperfinite graphings  
AbstractIn 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 dregular. These results will then be applied to give new proofs of measurable circle squaring. Based on joint work with Marcin Sabok and Gábor Kun. 

Annotated slides  
2021 Oct 25  Speaker: Ruiyuan (Ronnie) Chen (CRM / McGill University) 
Title: A representation theorem for cardinal algebras  
AbstractTarski'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 combinatoricsfree proof of Nadkarni's theorem on the existence of invariant measures. 

Annotated slides  
2021 Oct 18  Speaker: Dakota Ihli (University of Illinois UrbanaChampaign / McGill University) 
Title: Genericity of absolutely continuous interval homeomorphisms, part 2: proofs  
AbstractWe 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. 

Notes  
2021 Oct 4  Speaker: Dakota Ihli (University of Illinois UrbanaChampaign / McGill University) 
Title: Genericity of absolutely continuous interval homeomorphisms, part 1: introduction  
AbstractWe 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. 

Notes  
2021 Aug 6  Speaker: Dakota Ihli (University of Illinois UrbanaChampaign) 
Title: The group of absolutely continuous homeomorphisms of [0,1] is topologically 2generated  
AbstractAkhmedov and Cohen recently showed that the homeomorphism group of the interval is generically 2generated ¶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. 

Notes  
2021 Jul 20  Speaker: Marcin Sabok (McGill University) 
Title: On oneended spanning subforests and treeability of groups, part 8  
AbstractIn this part, we finish covering Section 3 of the paper of Conley, Gaboriau, Marks and TuckerDrob. We will discuss the proof of the fact that any Borel planar graph is measure treeable. 

Notes  
2021 Jul 9  Speaker: Marcin Sabok (McGill University) 
Title: On oneended spanning subforests and treeability of groups, part 7  
AbstractIn this part, we cover the first part of Section 3 of the paper of Conley, Gaboriau, Marks and TuckerDrob. We will discuss the proof of the fact that any Borel planar graph is measure treeable. 

Notes  
2021 Jul 2  Speaker: Anush Tserunyan (McGill University) 
Title: On oneended spanning subforests and treeability of groups, part 6: localglobal bridges, 2ended graphs, and the 1% lemma 

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

Notes  
2021 Jun 25  Speaker: Anush Tserunyan (McGill University) 
Title: On oneended spanning subforests and treeability of groups, part 5: the 99% lemma and the Kaimanovich–Elek theorem 

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

Notes  
2021 Jun 18  Speaker: Anush Tserunyan (McGill University) 
Title: On oneended spanning subforests and treeability of groups, part 4: combining the hyperfinite and superquadratic growth cases 

AbstractAssuming a characterization of µhyperfiniteness in terms of the isoperimetric constant (Kaimanovich–Elek theorem), we explain how nowhere µhyperfiniteness implies the existence of a Borel a.e. oneended spanning subforest. 

Notes  
2021 Jun 11  Speaker: Jenna Zomback (University of Illinois UrbanaChampaign) 
Title: On oneended spanning subforests and treeability of groups, part 3: oneended subforests in pmp graphs of superquadratic growth 

AbstractIn 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. 

Notes  
2021 Jun 4  Speaker: Matthew Bowen (McGill University) 
Title: On oneended spanning subforests and treeability of groups, part 2: oneended subforests in hyperfinite graphs  
AbstractWe give a brief introduction to the use of oneended spanning trees and forests in descriptive graph combinatorics and characterize which hyperfinite locally finite Borel graphs admit a.e. oneended spanning subforests. 

Annotated slides  
2021 May 28  Speaker: Ruiyuan (Ronnie) Chen (University of Illinois UrbanaChampaign) 
Title: On oneended spanning subforests and treeability of groups, part 1: introduction  
Annotated slides  
2021 May 20  Speaker: Nishant Chandgotia (TIFR Bangalore) 
Title: About Borel and almost Borel embeddings for ℤ^{d} actions  
AbstractKrieger'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. 

Slides  
2021 Apr 28  Speaker: Sohail Farhangi (Ohio State University) 
Title: Connections between van der Corput's Difference Theorem and the Ergodic Hierarchy of Mixing  
AbstractWe 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. 

Slides  
2021 Apr 21  Speaker: Antoine Poulin (McGill University) 
Title: On metrizability of universal minimal flows of homeomorphism groups of manifolds, part 2  
AbstractIn these two talks, we will discuss a result by Gutman, Tsankov, and Zucker, where nonmetrizability 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. 

Notes  
2021 Apr 14  Speaker: Antoine Poulin (McGill University) 
Title: On metrizability of universal minimal flows of homeomorphism groups of manifolds, part 1  
AbstractIn these two talks, we will discuss a result by Gutman, Tsankov, and Zucker, where nonmetrizability 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. 

Notes  
2021 Apr 7  Joint with the McGill Geometric Group Theory seminar 
Speaker: Joshua Frisch (Caltech)  
Title: The ICC property in Random Walks and Dynamics  
Abstract
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”.


2021 Mar 31  Speaker: Sławomir Solecki (Cornell University) 
Title: Random continuum and iterated Brownian motion  
AbstractWe 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 Wienertype measure on the space of all chainable continua. This is joint work with Viktor Kiss. 

Slides  
2021 Mar 24  Speaker: Anton Bernshteyn (Georgia Tech) 
Title: Probabilistic tools in continuous combinatorics  
AbstractIn this talk I will describe probabilistic tools that can be used to construct continuous solutions to combinatorial problems on zerodimensional 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. 

Slides  
2021 Mar 17  Speaker: Prakash Panangaden (McGill University) 
Title: The Logical Characterization of Probabilistic Bisimulation  
AbstractProbabilistic 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. 

2021 Mar 10  Speaker: Shrey Sanadhya (University of Iowa) 
Title: Generalized Bratteli Vershik model for substitution on infinite alphabets  
AbstractWe consider substitutions on countably infinite alphabets as Borel dynamical system and build their BratteliVershik models. We prove two versions of Rokhlin’s lemma for such substitution dynamical systems. Using the BratteliVershik model we give an explicit formula for a shiftinvariant measure (finite and infinite) and provide a criterion for this measure to be ergodic. This is joint work with Sergii Bezuglyi and Palle Jorgensen. 

Annotated slides  
2021 Mar 3  Speaker: Matthew Bowen (McGill University) 
Title: Descriptive graph combinatorics and the KechrisSoleckiTodorcevic dichotomy, part 2  
AbstractIn 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 KechrisSoleckiTodorcevic (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. 

2021 Feb 24  Speaker: Matthew Bowen (McGill University) 
Title: Descriptive graph combinatorics and the KechrisSoleckiTodorcevic dichotomy, part 1  
AbstractIn 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 KechrisSoleckiTodorcevic (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. 

2021 Feb 17  Speaker: Michael Wolman (Caltech) 
Title: Probabilistic Programming Semantics for Name Generation, part 3: the proof  
Abstract
In this series of talks we present a probabilistic model for name generation. Specifically, we interpret the nucalculus, a simplytyped lambdacalculus with name generation, in the category of quasiBorel spaces, an extension of the category of standard Borel spaces supporting both measure theory and higherorder programming. We prove that this model is fully abstract at firstorder types. This is joint work with Marcin Sabok, Sam Staton and Dario Stein.


2021 Feb 3  Speaker: Michael Wolman (Caltech) 
Title: Probabilistic Programming Semantics for Name Generation, part 2  
AbstractIn this series of talks we present a probabilistic model for name generation. Specifically, we interpret the nucalculus, a simplytyped lambdacalculus with name generation, in the category of quasiBorel spaces, an extension of the category of standard Borel spaces supporting both measure theory and higherorder programming. We prove that this model is fully abstract at firstorder types. This is joint work with Marcin Sabok, Sam Staton and Dario Stein. 

2021 Jan 26  Speaker: Michael Wolman (Caltech) 
Title: Probabilistic Programming Semantics for Name Generation, part 1  
AbstractIn this series of talks we present a probabilistic model for name generation. Specifically, we interpret the nucalculus, a simplytyped lambdacalculus with name generation, in the category of quasiBorel spaces, an extension of the category of standard Borel spaces supporting both measure theory and higherorder programming. We prove that this model is fully abstract at firstorder types. This is joint work with Marcin Sabok, Sam Staton and Dario Stein. 