# Abstracts

### Chronologically

All talks are in **MS.01**.

Monday 8 July

11:00 Joel Moreira: *A sumset conjecture of Erdös*

12:00 Ben Green: *Monochromatic sums and products*

14:30 Andreas Thom: *Obstructions to stability*

16:00 Vitaly Bergelson: *Polynomial Ramsey theory*

Tuesday 9 July

9:30 Clinton Conley: *Equitable colorings*

11:00 Andrew Marks: *Measurable realizations of abstract systems of congruences*

12:00 Robin Tucker-Drob: *Dynamical alternating groups, stability, property Gamma, and inner amenability
14:30 Anush Tserunyan: Hyperfinite subequivalence relations of treed quasi-pmp equivalence relations
16:00 Damien Gaboriau: On IRS of surface groups and higher dimensional manifolds
17:00 Łukasz Grabowski: Almost finite actions of non-amenable groups*

Wednesday 10 July

9:30 Nikolay Nikolov: *Gradients in discrete and profinite groups*

11:00 Tamar Ziegler: *Sections of high rank varieties and application*

Thursday 11 July

9:30 Gábor Elek: *Continuous graph limits and the spaces of Cantor actions*

11:00 Gábor Kun: *Nonamenable groups that do not admit a sofic approximation by expanders*

12:00 Ágnes Backhausz: *Action convergence of operators and graphs*

14:30 Tom Sanders: *Bootstrapping partition regularity of linear systems*

16:00 Tianyi Zheng: *The double commutator lemma for IRSs*

17:00 Endre Csóka: *Phase transitions in the independent sets of random graphs*

Friday 12 July

9:30 Brandon Seward: *Borel asymptotic dimension and hyperfinite equivalence relations*

11:00 Lewis Bowen: *A topological dynamical system with two different positive sofic entropies*

12:00 Hanfeng Li: *Entropy and combinatorial independence*

### Alphabetically

Title: *Action convergence of operators and graphs*

Abstract: In this talk we present the notion of action convergence, whose main motivation was to extend certain concepts of graph limit theory such that they can be used for dense random matrices. Graph limit theory, as a powerful combination of tools of analysis, probability theory and combinatorics, already has many applications about random graphs or in extremal graph theory, for example. In order to widen the family of models for which these techniques may be used, we proposed action convergence. We proved that normalized random sign matrices (with i.i.d. zero-mean entries) have non-trivial subsequental limits; before, such a convergence notion had not been known. We also showed that action convergence unifies dense graph limits and local-global convergence of bounded degree graphs. Furthermore, it provides non-trivial limit objects for various graph sequences of intermediate density as well (e.g. for hypercubes). The concept of action convergence and these results will be overviewed in the talk. Joint work with Balazs Szegedy.

Title: *Polynomial Ramsey theory*

Abstract: Polynomial Szemeredi Theorem and Polynomial Hales-Jewett Theorem (obtained jointly with A. Leibman) provide examples of Ramsey-theoretical results of polynomial nature. We will discuss some natural open problems and conjectures related to these results as well as the more general ergodic-theoretical and combinatorial phenomena of "non-linear" nature.

Title: *A topological dynamical system with two different positive sofic entropies*

Abstract: A sofic approximation to a countable group is a sequence of finite graphs that converges locally on average to a Cayley graph of the group. A group is sofic if it admits a sofic approximation. Sofic entropy theory is a generalization of classical entropy theory in dynamics to actions by sofic groups. However, the sofic entropy of an action may depend on a choice of sofic approximation. Until now, it was unknown whether there exists an action with two different positive sofic entropies (depending on sofic approximation). I’ll explain an explicit example of this, inspired by work of Coja-Oghlan and Zdeberova on phase transitions in random hypergraph 2-coloring. This is joint work with Dylan Airey and Frank Lin.

Title: *Equitable colorings*

Abstract: A proper coloring of a graph is called equitable if every color class has (approximately) the same number of vertices. In the finite setting, the celebrated Hajnal–Szemerédi theorem establishes the existence of equitable (d+1)-colorings, where d is a bound on the vertex degrees. We discuss the existence of such colorings in the measure-theoretic and purely Borel contexts. This is joint work with Anton Bernshteyn.

Title: *Phase transitions in the independent sets of random graphs*

Abstract: We analyze the asymptotic relative size of the largest independent set of a random $d$-regular graph on $n \to \infty$ vertices. For $d \ge 20$, we have a formula for this ratio, but for $3 \le d \le 19$, the ratio is unknown, and $d = 3$ seems to be the most difficult case. We are trying to explore the highly non-trivial phase diagram of the structures of the independent sets depending on the relative size of them and the degree $d$. These phase transitions are related to algorithmic thresholds, mixing properties, counting, graph reconstruction, graph limits and other questions. Our tools are partially coming from statistical physics.

Title: *Continuous graph limits and the spaces of Cantor actions*

Abstract: I will introduce the notion of weak equivalence for continuous, free actions of a countable group on the Cantor set. This is the continuous analogue of the weak equivalence of measure preserving, essentially free actions of countable groups introduced by Kechris. Then, I will explain how continuous weak equivalence is related to a very natural analogue of the Global-Local Graph Convergence (Hatami-Lovasz-Szegedy).

Title: *On IRS of surface groups and higher dimensional manifolds*

Abstract: The groups of the title have plenty of IRS's. More precisely: The fundamental group *G* of an aspherical compact surface of genus g is known to have *cost(G)* = 2*g*-1 (resp. *cost(G)* = *g-*1 when non-orientable). We show when *G* is non-amenable that the cost of any pmp **non-free** action of *G* is strictly less than *cost(G).* Let ${\cal R}$ is an ergodic pmp standard equivalence relation (on the atomless standard probability space). We show: If *cost*(${\cal R}$) < *cost(G)* , then there is a highly faithful action of *G* which defines ${\cal R}$, such that *G* is dense in the full group [${\cal R}$] and thus is totally non-free and highly transitive on the orbits. The natural map (given by conjugation action) from ergodic atomless IRS's of *G* to the ergodic pmp equivalence relations of cost strictly less than *cost(G)* is onto and has uncountable fibers.

This is joint work with A. Carderi, P. Fima and F. le Maître. Concerning higher dimensional manifolds, I use an invariant of ${\ell}$^{2} type allowing to distinguish uncountably many IRS's of any residually finite group *G* which is the fundamental group of a compact orientable *d*-manifold decomposable as a connected sum M#N of infinite fundamental group manifolds.

Title: *Almost finite actions of non-amenable groups*

Abstract: Recently C. T. Conley, S. Jackson, D. Kerr, A. Marks, B. Seward and R. Tucker-Drob proved that the graphing given by any probability-measure-preserving action of an amenable group can be measurably tiled with Foelner sets. This is often shortened to saying that actions of amenable groups are "almost finite". This talk is a progress report on a joint work with Gabor Elek, where we investigate the analogous question for amenable actions of non-amenable groups. In particular we show that the actions of the free groups on their boundaries are almost finite. If time permits I'll also discuss some other "amenability-like" conditions for graphings.

Title: *Monochromatic sums and products*

Abstract: I will talk about joint work with Tom Sanders from a couple of years ago in which we show that if p is large enough then any r-colouring of Z/pZ contains many x, y, x+y and xy of the same colour.

Title: *Nonamenable groups that do not admit a sofic approximation by expanders*

Abstract: We show that the direct product of an infinite Kazhdan Property T group and a not LEF (for example finitely presented, but not residually finite) amenable group admits no sofic approximation by finite expander graphs. (Joint work with Andreas Thom.)

Title: *Entropy and combinatorial independence*

Abstract: When a group G acts on a compact space X, a subset H of G is called an independence set for a finite family W of subsets of X if for any finite subset M of H and any map f from M to W, there is a point x of X with sx in f(s) for all s in M. I will discuss how positivity of entropy can be described in terms of density of independence sets, and give a few applications including the relation between positive entropy and Li-Yorke chaos. This is joint work with Zheng Rong.

Title: *Measurable realizations of abstract systems of congruence*

Abstract: An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and n-divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the 2-sphere. This answers a question of Wagon. We also construct Borel realizations of abstract systems of congruences for the action of PSL_2(Z) on P^1(R).

Title: *A sumset conjecture of Erdös*

Abstract: Erdös conjectured that any set of natural numbers with positive density contains the arithmetic sum A+B of two infinite sets A and B of natural numbers. In joint work with Richter and Robertson we recently solved this problem using techniques from ergodic theory. In the talk I will present the main ideas that go into the proof.

Title: *Gradients in discrete and profinite groups*

Abstract: Let $G$ be a finitely generated residually finite group. The growth of invariants (e.g. rank or torsion in abelianization, minimal number of generators, deficiency, etc.) of subgroups of finite index in $G$ has received a lot of attention recently in connection with measurable group actions, L2 invariants and topology. There are still many open questions. Now all of the above gradients can be defined in the world of profinite groups, where the analogous questions are usually much easier to solve, but new intriguing questions arise. In this talk I will survey the different flavour of results in the discrete and the profinite world.

Title:* Bootstrapping partition regularity of linear systems*

Abstract: Suppose that $A$ is a $k \times d$ matrix of integers such that fir any $r$ there is some $N$ such that any $r$-colouring of $\{1,\dots,N\}$ contains a monochromatic solution to $A$, meaning there is a colour class $C$ and $x \in C^d$ such that $Ax=0$. Not all matrices $A$ have this property (consider, for example, when all the entries of $A$ are positive), but when they do they are called partition regular. In this talk we consider what bounds can be given on $N$ in terms of $r$ (and $A$) when $A$ is partition regular.

Title: *Borel asymptotic dimension and hyperfinite equivalence relations*

Abstract: A well known and long-standing open problem in the theory of Borel equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. Previous progress on this problem has been confined to groups possessing coarse euclidean geometry and polynomial volume growth (ultimately leading to a positive answer for groups that are either virtually nilpotent or locally nilpotent). In this talk I will discuss the coarse geometric notion of asymptotic dimension and discuss its applications to this problem. Relying upon the framework of asymptotic dimension, it is possible to both significantly simplify the proofs of prior results and uncover the first examples of solvable groups of exponential volume growth all of whose Borel actions generate hyperfinite equivalence relations. This is joint work with Clinton Conley, Steve Jackson, Andrew Marks, and Robin Tucker-Drob.

Title: *Obstructions to stability*

Abstract: I will review recent work on obstructions to stability of almost homomorphisms in various setups. In joint work with Marcus de Chiffre, Lev Glebsky and Alex Lubotzky, we used cohomological methods to study questions of stability of almost homomorphisms in the Frobenius metric. In joint work with Oren Becker and Alex Lubotzky, we studied stability of sofic approximations for amenable groups.

Title: Dynamical alternating groups, stability, property Gamma, and inner amenability

Title: *The double commutator lemma for IRSs*

Abstract: The so called double commutator lemma for normal subgroups, dating back to the work of Higman in the 50s, is a useful tool in showing simplicity of certain classes of groups. We show an extension this lemma to IRSs of countable groups. A quantitative version of it can be applied towards showing Neretin’s groups admit no non-trivial IRSs.

Title: *Sections of high rank varieties and application*

Abstract: TBD