London Mathematical Society -- EPSRC Durham Symposium

Permutation groups and transformation semigroups

2015-07-20 to 2015-07-30

Abstracts of Talks

Nate Ackerman: *Ergodic invariant measures as probabilistic structures*

In this series of three talks, we report on a program that examines ergodic measures invariant under the logic action. Consider the Borel space of all L-structures with underlying set the natural numbers, where L is a countable language. The symmetric group on the natural numbers acts on this space via the logic action, by permuting the elements of a given structure. We describe how the ergodic probability measures on this space that are invariant under the logic action provide a natural notion of "probabilistic structure". Associated to each such ergodic invariant measure is a complete consistent infinitary theory. It can be shown that these measures fall into two classes: (i) those that concentrate on a single isomorphism class of structures, i.e., a single orbit of the logic action, and (ii) those whose associated infinitary theory has no classical models. In the first case, we characterize those orbits admitting an invariant measure, and in particular describe when such a measure is unique; the latter involves the notion of a highly homogeneous permutation group. We provide sufficient conditions for the second case to occur, leading to new probabilistic constructions of Urysohn space and other structures. The study of such measures is closely tied to the theory of limits of dense graph sequences, or graphons, as in work of Lovász, Szegedy, and others. We describe this connection, and show how the model theory of invariant measures can shed light on the study of graphons. Joint work with Alex Kruckman, Aleksandra Kwiatkowska, Jaroslav Nešetřil, and Jan Reimann. Lecture 1: An overview of ergodic invariant measures as probabilistic structures Lecture 2: Graphons and ergodic invariant measures Lecture 3: Techniques for constructing ergodic invariant measures

Lovkush Agarwal: *Uncountably many maximal closed subgroups of Sym(w), via Henson digraphs*

In Ravello 2013, Macpherson asked whether there are uncountably many maximal closed subgroups of Sym(w), where G is maximal means that G is not equal to Sym(w) and there are no closed subgroups in between G or Sym(w). In this talk, I will present a positive answer to this question using Henson digraphs.

Will Anscombe: *Generalised measures on free homogeneous structures*

In this talk I will discuss how to view free homogeneous structures as `generalised measurable structures', which is a new definition (from my project with Macpherson-Steinhorn-Wolf) generalising the earlier definition of measurable structures (due to Macpherson-Steinhorn). Our main motivating example is the generic triangle-free graph.

João Araujo: *Groups and semigroups: from a duet to a chorus*

Semigroup theory naturally prompts questions on primitive groups that were totally hopeless 40 years ago, let alone 70 or 80 years ago when semigroup theory was in its infancy. The Classification of Finite Simple Groups, however, completely changed the situation. In this talk I will survey some of the recent results in semigroups that we now have courtesy of the Classification. I trust it will be clear that the two topics (semigroups and groups) that have been -to a large extent- singing different songs, are now ready to join in a beautiful chorus, leading to deep and elegant results, and to fruitful interconnections with other parts of mathematics. En passant, many problems will be proposed such as, for example, "Let G be a Suzuki group in its primitive action on a set X. Is it true that given any subset of X of size 3, say S, and any partition P of X into three parts, in the orbit of S under G there exists a transversal for P?"

Wolfram Bentz: *How synchronizing are primitive groups? *

Synchronization is a property of automata and can be understood as a method of error recovery. An automaton is synchronizing if there is an input sequence which always brings the automaton into a known state irrespectively of the original state of the automaton. Such an instruction set is called a \emph{reset} (or \emph{synchronizing}) word. We can translate this question into the realm of semigroup theory by asking if the transition semigroup associated to the automaton contains a constant map. An important case of this approach is the situation where this semigroup is generated by a permutation group $G$ together with a singular transformation $t$, both acting on the state set $X$. The primitivity of the group $G$ is a strong property that ``usually´´ forces synchronization in connection with a non-permutation. Our work examines how usual this situation is. We will present several new results about the synchronization properties of primitive groups, both negative and positive. We will also give an introduction to the graph-based methods used in our proofs. Further details on the computational issues involved in this project along with a variety of problems concerning primitive graphs, graph endomorphisms, and cores of vertex-transitive graphs will be discussed in upcoming talks by Gordon Royle. This is a joint work with {\sc Jo\~{a}o Ara\'{u}jo} (CEMAT Universidade de Lisboa), {\sc Peter J. Cameron} (Mathematical Institute, University of St Andrews), {\sc Gordon Royle} (Centre for the Mathematics of Symmetry and Computation, University of Western Australia), and {\sc Artur Schaefer} (Mathematical Institute, University of St Andrews).

Catarina Carvalho: *On a subsemigroup of binary relations*

We introduce the monoid of surjective hyperoperations, as it appeared in connection with variation of the Constraint Satisfaction Problem. We relate it with the already studied subsemigroups of binary relations, give some of its structural properties and present an array of open questions about its structure.

Alonso Castillo-Ramirez: *Memoryless Computation and Universal Simulation*

Let A be a finite set and n an integer at least 2. Memoryless computation is the study of instructions of A^n (i.e. transformations of A^n with at most one nontrivial coordinate function) and the semigroups of transformations that they generate. This model of computation has been recently revitalised by several new results, such as the proof by Cameron, Fairbairn and Gadouleau that the full transformation semigroup on A^n may be generated by just n+1 instructions. In this talk we will introduce the concept of 'simulation' as a way of computing transformations of A^n using m instructions that may depend on m-n additional coordinates. A transformation of A^m is n-universal of size m if the instructions induced by its coordinate functions may simulate any transformation of A^n. We will establish that there is no n-universal transformation of size n, but there is one of size n+2. We will also introduce the notions of sequential, parallel and quasi-parallel simulation. This talk is based on joint work with Maximilien Gadouleau.

Gregory Cherlin: *Metrically Homogeneous Graphs of Generic Type*

I will discuss the conjectured classification of metrically homogeneous graphs. I will first define precisely what I mean by generic type, then discuss some or all of the following points. * The metrically homogeneous graphs of non-generic type are classified. * The conjectured classification in generic type is that the minimal forbidden structures are either triangles or clique-like constraints * Under a technical assumption called 4-triviality, the allowable patterns of forbidden triangles are known. * The infinite diameter case reduces to the finite diameter case - if the conjecture is correct in finite diameter.

Udayan Darji: *Generating infinite random graphs*

In this talk we discuss a probabilistic algorithm for generating infinite random graphs. It has a probabilistic flavor but at the same time some homogeneous structure seems to be lurking underneath. For example, we characterize when our algorithm generates the Rado graph. Under certain parameters, our process generates infinite random trees. Many interesting problems and connections with homogeneous structures arise. This is joint work with Csaba Bir\xc3\xb3.

Igor Dolinka: *Representing semigroups and groups by endomorphisms of Fraïssé limits (Part I)*

PART I (Semigroup embeddings): I will begin by giving a summary of the particular results concerning embedding abstract (countable) semigroups into End(M), where M is one of the 'most popular' Fraïssé limits. Then I will describe how this evolved into a general universality result for endomorphism monoids of countably infinite homogeneous structures, obtained in collaboration with D. Mašulović. This result will involve several category-theoretical properties of Fraïssé classes. PART II (Groups - overt & covert): In this part, my departing point will be the observation that, under some relatively mild conditions, a structure is isomorphic to a retract of a Fraïssé limit if and only if it is algebraically closed. Then I will lay out the main points of a recent joint work with R. Gray, J. McPhee, J. Mitchell, and M. Quick, which investigates automorphism groups of graph-like algebraically closed structures, thus representing them as a) maximal subgroups of endomorphism monoids of homogeneous graphs, and as b) Schützenberger groups of non-regular endomorphisms. Along the way, to provide a proper background, I will develop some basic semigroup-theoretical machinery specialised to endomorphism monoids.

Igor Dolinka: *Representing semigroups and groups by endomorphisms of Fraïssé limits (Part II)*

PART I (Semigroup embeddings): I will begin by giving a summary of the particular results concerning embedding abstract (countable) semigroups into End(M), where M is one of the 'most popular' Fraïssé limits. Then I will describe how this evolved into a general universality result for endomorphism monoids of countably infinite homogeneous structures, obtained in collaboration with D. Mašulović. This result will involve several category-theoretical properties of Fraïssé classes. PART II (Groups - overt & covert): In this part, my departing point will be the observation that, under some relatively mild conditions, a structure is isomorphic to a retract of a Fraïssé limit if and only if it is algebraically closed. Then I will lay out the main points of a recent joint work with R. Gray, J. McPhee, J. Mitchell, and M. Quick, which investigates automorphism groups of graph-like algebraically closed structures, thus representing them as a) maximal subgroups of endomorphism monoids of homogeneous graphs, and as b) Schützenberger groups of non-regular endomorphisms. Along the way, to provide a proper background, I will develop some basic semigroup-theoretical machinery specialised to endomorphism monoids.

Manfred Droste: *The normal subsemigroups of the monoid of injective maps*

We consider the monoid Inj(M) of injective self-maps of a set M and want to determine its normal subsemigroups by numerical invariants. This was established by Mesyan in 2012 if M is countable. Here we obtain an explicit description of all normal subsemigroups of Inj(M) for any uncountable set M. Joint work with Rüdiger Göbel (Essen).

James East: *Linear sandwich semigroups*

Let $M_{mn}$ denote the set of all $m\times n$ matrices over a field $F$, and fix some $n\times m$ matrix $A\in M_{nm}$. An associative operation $\star$ may be defined on $M_{mn}$ by $X\star Y=XAY$ for all $X,Y\in M_{mn}$, and the resulting ``sandwich semigroup'' is denoted $M_{mn}^A$. It seems these linear sandwich semigroups were introduced by Lyapin in his 1960 monograph, and they are related to the so-called generalized matrix algebras of Brown (1955), but they have not received a great deal of attention since some early papers by Magill and Subbiah in the 60s and 70s. In this talk, I will report on joint work with Igor Dolinka (Novi Sad) in which we investigate certain combinatorial questions regarding the linear sandwich semigroups, including: regularity, Green's relations, ideals, rank and idempotent rank. We also outline a general framework for studying more general sandwich semigroups: the context is a kind of partial semigroup related to Ehresmann-style arrows-only categories.

Attila Egri-Nagy: *Finite Diagram Semigroups: Extending the Computational Horizon *

Diagram semigroups are interesting algebraic and combinatorial objects, several types of them originating from questions in computer science and in physics. Here we describe diagram semigroups in a general framework and extend our computational knowledge of them. The generated data set is replete with surprising observations raising many open questions for further theoretical research. Joint work with James East, Andrew R. Francis and James D. Mitchell.

David Evans: *Reconstruction of endomorphism monoids of $\omega$-categorial structures*

Cameron Freer: *Ergodic invariant measures as probabilistic structures*

Title: Ergodic invariant measures as probabilistic structures Nathanael Ackerman, Cameron Freer, Rehana Patel Abstract: In this series of three talks, we report on a program that examines ergodic measures invariant under the logic action. Consider the Borel space of all L-structures with underlying set the natural numbers, where L is a countable language. The symmetric group on the natural numbers acts on this space via the logic action, by permuting the elements of a given structure. We describe how the ergodic probability measures on this space that are invariant under the logic action provide a natural notion of "probabilistic structure". Associated to each such ergodic invariant measure is a complete consistent infinitary theory. It can be shown that these measures fall into two classes: (i) those that concentrate on a single isomorphism class of structures, i.e., a single orbit of the logic action, and (ii) those whose associated infinitary theory has no classical models. In the first case, we characterize those orbits admitting an invariant measure, and in particular describe when such a measure is unique; the latter involves the notion of a highly homogeneous permutation group. We provide sufficient conditions for the second case to occur, leading to new probabilistic constructions of Urysohn space and other structures. The study of such measures is closely tied to the theory of limits of dense graph sequences, or graphons, as in work of Lovász, Szegedy, and others. We describe this connection, and show how the model theory of invariant measures can shed light on the study of graphons. Joint work with Alex Kruckman, Aleksandra Kwiatkowska, Jaroslav Nešetřil, and Jan Reimann. Lecture 1: An overview of ergodic invariant measures as probabilistic structures Lecture 2: Graphons and ergodic invariant measures Lecture 3: Techniques for constructing ergodic invariant measures

Max Gadouleau: *Transformation semigroups generated by idempotents of defect one*

It is well known that the semigroup of all singular transformations of a finite set is generated by idempotents of defect one. Such a transformation can be seen as an arc in a directed graph. We then study properties of semigroups generated by idempotents of defect one based on their digraph representation. We will first study some algebraic properties and then investigate the maximum length of a word using these generators.

Michael Giudici: *The classification of $\frac{3}{2}$-transitive permutation groups*

A permutation group on a set $\Omega$ is called $\frac{3}{2}$-transitive if it is transitive and for all $\alpha\in\Omega$ the orbits of the point stabiliser $G_\alpha$ on $\Omega\backslash\{\alpha\}$ all have the same length. Wielandt showed that a finite $\frac{3}{2}$-transitive group is either Frobenius or primitive. The talk will discuss the recent classification of all primitive $\frac{3}{2}$-transitive permutation groups that is the result of work by various subsets of Bamberg, Giudici, Liebeck, Saxl, Praeger and Tiep.

Robert Gray: *Presentations for symmetric groups encoded by idempotents in the full transformation monoid*

In 1966, J. M. Howie proved that in the finite full transformation monoid, every non-invertible element can be written as a product of idempotents. Central to the study of idempotent generated semigroups are certain free objects, called free idempotent generated semigroups. In this talk I will speak about joint work with N. Ruskuc where we give a complete description of the maximal subgroups of the free idempotent generated semigroup over the finite full transformation monoid. It turns out that these groups are all either free, or are isomorphic to finite symmetric groups. I will explain how the proof comes down to uncovering an intricate encoding of the standard Coxeter presentations for finite symmetric groups hidden in the idempotent structure of the full transformation monoid, reflected in the combinatorics of kernels (partitions) and images (subsets) of transformations.

Peter Higgins: *Using transformations to embed finite semigroups of a certain type in 2-generated finite semigroups of the same type.*

We use transformation semigroups to show how finite semigroups in a particular class may be embedded in 2-generated finite semigroups of the same class. In particular, any finite orthodox semigroup (a regular semigroup in which idempotents form a subsemigroup) may be embedded in a 2-generated finite orthodox semigroup. Along the way we make use of the properties of a certain peculiar string of numbers known as the Mian-Chowla sequence.

Jan Hubicka: *Ramsey Classes by Partite Construction (Part I)*

The first lecture will focus on the Partite Construction - a technique of proving Ramsey property of a given class introduced by Nesetril and Rodl. I will show how to extend the technique to classes with algebraic closures and give examples of classes where the Ramsey Property can be shown by application of this method. The second lecture will follow by extending the Partite Construction to classes with forbidden homomorphic images. I will show how this leads to more systematic approach of finding new Ramsey lifts.

Jan Hubicka: *Ramsey Classes by Partite Construction (Part II)*

The first lecture will focus on the Partite Construction - a technique of proving Ramsey property of a given class introduced by Nesetril and Rodl. I will show how to extend the technique to classes with algebraic closures and give examples of classes where the Ramsey Property can be shown by application of this method. The second lecture will follow by extending the Partite Construction to classes with forbidden homomorphic images. I will show how this leads to more systematic approach of finding new Ramsey lifts.

James Hyde: *2-Generation In Groups Of Homeomorphisms Of The Cantor Set*

Let $S$ be a set of homeomorphisms of the Cantor set. We will call $S$ vigorous if and only if for any $A$ a non-empty clopen subset of the Cantor set and any $B$ and $C$ non-empty proper clopen subsets of $A$ there exists $g$ in $S$ with the support of $g$ contained in $A$ and with $Bg$ a subset of $C$. Our main theorem is that finitely generated simple vigorous groups of homeomorphisms of the Cantor set are $2$-generated. We will give sufficient conditions for a group of homeomorphism to be vigorous and give some examples.

Julius Jonusas: *Topological rank of automorphism groups of homogenous graphs*

Let $G$ be a Polish group. As usual define the rank of $G$ to be the cardinality of a minimal generating set for $G$. However, if $G$ happens to be uncountable there is little to be learned from the rank itself and thus various generalisations of the concept were introduced such as topological rank, Sierpinski rank and etc. Topological rank of $G$ is be the smallest rank of a dense subsemigroup of $G$. Mitchell and Darji showed that the symmetric group on natural number, automorphisms of the random graph, and order automorphisms of the rationals are all of topological rank 2. Moreover, they also classified how easy is to find these dense subgroups of rank 2. We will extend their results to the automorphism groups all homogenous undirected graphs.

Mark Kambites: *Amenability and geometry of semigroups*

I will discuss some recent joint research with Robert Gray on the connection between amenability, Følner conditions and the geometry of finitely generated semigroups.

Michael Kinyon: *Automorphic loops and their associated permutation groups*

An important permutation group associated with a loop $Q$ is its multiplication group $\mathrm{Mlt}(Q)$ generated by all left translations $L_x : y\mapsto xy$ and all right translations $R_x : y \mapsto yx$. The stabilizer of the identity element of $Q$ is the inner mapping group $\mathrm{Inn}(Q)$. A loop is \emph{automorphic} if every inner mapping is an automorphism of $Q$. Groups and commutative Moufang loops are examples of automorphic loops, but there are many others as well. The outstanding open problem in the theory of automorphic is to determine if there are any finite, nonassociative, simple automorphic loops. Simplicity of a loop $Q$ is characterized by $\mathrm{Mlt}(Q)$ acting primitively on $Q$, and thus one approach to searching for simple loops is to use an O'Nan-Scott attack. In this talk, I will give some background on automorphic loops and describe the current state of the art in the search for finite, nonassociative, simple automorphic loops.

Nicholas Loughlin: *The Brauer Project*

The Brauer project started as a small undertaking to count and understand idempotents in certain semigroups of diagrams. I'll focus on recent developments in understanding the Jones monoids and some close relatives, including a newly-discovered family of semigroups, and there will be an emphasis on developing the theory into memory-efficient working algorithms for counting and indexing idempotents, which are of independent interest.

Stuart Margolis: *Translational hulls as the transformation monoid of continuous maps on combinatorial structures*

A finite aperiodic 0-simple semigroup is by Rees Theorem given by a {0,1} matrix C. By viewing C as an incidence structure, we gain a perspective that allows interactions between group theory, semigroup theory and combinatorics. We look at 35 year old results of Jeff Dinitz and the speaker in the case that C is the incidence matrix of a block design. The translational hull, is then the transformation monoid of all continuous partial maps on the design. This is the monoid of all partial functions on the points of the design such that the inverse image of any block is either empty or another block. We give interesting connections between the structure of the translational hull and the parameters of the design. In particular, we give a generalization of the fundamental theorem of projective geometry.

Dragan Masulovic: *Categorical Constructions and Ramsey Property*

It was obvious from the beginning that structural Ramsey property is a categorical property: it depends not only on the choice of objects, but also on the choice of morphisms involved. In this talk we explicitely put the Ramsey property and the dual Ramsey property in the context of categories of finite structures and investigate the invariance of these properties under some standard categorical constructions. We use elementary category theory to generalize some combinatorial results and using the machinery of very basic category theory provide new combinatorial statements (whose formulations do not refer to category-theoretic notions).

Chrystopher Nehaniv: *Coproducts for Permutation Groups, Transformation Semigroups, Automata and Related Categories*

The structure of coproducts of groups, monoids and semigroups is well-known: they are the so-called ``free products'' satisfying a universal mapping property and their elements can be written in a canonical form. Surprisingly, the structure of coproducts for faithful representations of groups by permutations, or for monoids and semigroups by transformations appears not to have been described in the literature. Indeed they may fail to exist (in some degenerate cases) for transformation semigroups. Moreover, the most obvious guesses of what the coproduct should be in these categories turn out to be wrong. Here we completely describe the structure of coproducts (including canonical forms for their state sets) in the categories of permutation groups, transformation monoids and transformation semigroups, with or without base point and also for partial transformation semigroups. It turns out this also allows us to completely describe the structure of coproducts of automata (whether in the categories of deterministic and complete, or partial automata), i.e. for discrete dynamical systems with inputs, leading to applications for computer science. This is joint work with Fariba Karimi and funded under the EC FP7 BIOMICS project.

Rehana Patel: *Ergodic invariant measures as probabilistic structures (Nathanael Ackerman, Cameron Freer, Rehana Patel)*

In this series of three talks, we report on a program that examines ergodic measures invariant under the logic action. Consider the Borel space of all L-structures with underlying set the natural numbers, where L is a countable language. The symmetric group on the natural numbers acts on this space via the logic action, by permuting the elements of a given structure. We describe how the ergodic probability measures on this space that are invariant under the logic action provide a natural notion of "probabilistic structure". Associated to each such ergodic invariant measure is a complete consistent infinitary theory. It can be shown that these measures fall into two classes: (i) those that concentrate on a single isomorphism class of structures, i.e., a single orbit of the logic action, and (ii) those whose associated infinitary theory has no classical models. In the first case, we characterize those orbits admitting an invariant measure, and in particular describe when such a measure is unique; the latter involves the notion of a highly homogeneous permutation group. We provide sufficient conditions for the second case to occur, leading to new probabilistic constructions of Urysohn space and other structures. The study of such measures is closely tied to the theory of limits of dense graph sequences, or graphons, as in work of Lovász, Szegedy, and others. We describe this connection, and show how the model theory of invariant measures can shed light on the study of graphons. Joint work with Alex Kruckman, Aleksandra Kwiatkowska, Jaroslav Nešetřil, and Jan Reimann. Lecture 1: An overview of ergodic invariant measures as probabilistic structures Lecture 2: Graphons and ergodic invariant measures Lecture 3: Techniques for constructing ergodic invariant measures

Christian Pech: *Reconstructing the topology of polymorphism clones*

Every clone of functions comes naturally equipped with a topology-the topology of pointwise convergence. A clone C is said to have automatic homeomorphicity with respect to a class K of clones, if every clone-isomorphism of C to a member of K is already a homeomorphism (with respect to the topology of pointwise convergence). I am going to talk about automatic homeomorphicity-properties for polymorphism clones of countable homogeneous relational structures. The results base on (and extend) previous results by Bodirsky, Pinsker, and Pongrácz.

Yann Peresse: *Long chains of subsemigroups*

The length of a semigroup S is defined to be the largest size of a chain of subsemigroups of S. An exact formula for the length of the symmetric group on n points was found by Cameron, Solomon and Turull; the length is roughly 3n/2. In general, it follows by Lagrange's Theorem that the length of a group is at most the logarithm of the group order. Semigroups refuse to be as well-behaved. The only valid upper bound for the length of an arbitrary semigroup is its size. For example, any zero-semigroup has length equal to its size. Even for less degenerate and more natural examples of semigroups, the contrast to groups is noticable. We will see that the length of the full transformation semigroup on n points, the semigroup analogue to the symmetric group, is asymptotically at least a constant multiple of its size.

Michael Pinsker: *Topological clones (Part I)*

Every algebra carries, in addition to its algebraic structure, a natural topological structure: this structure is given by the topology of pointwise convergence on its term functions. Topological clones are the abstract algebraic and topological objects which capture both the algebraic and topological structure of algebras, similarly to topological groups which appear as the algebraic and topological abstraction of permutation groups. In my two lectures I am going to explain what we can tell about an algebra from its topological clone. This will lead me in particular into complexity theory, where certain computational problems, called Constraint Satisfaction Problems, are investigated systematically via their polymorphism algebras, and subsequently via topological clones. I will moreover address the often non-trivial interference between the algebraic and the topological structure of algebras. Finally, I will show how the algebraic modelling of function clones can be altered in order to be more suitable for constraint satisfaction.

Michael Pinsker: *Topological clones (Part II)*

Every algebra carries, in addition to its algebraic structure, a natural topological structure: this structure is given by the topology of pointwise convergence on its term functions. Topological clones are the abstract algebraic and topological objects which capture both the algebraic and topological structure of algebras, similarly to topological groups which appear as the algebraic and topological abstraction of permutation groups. In my two lectures I am going to explain what we can tell about an algebra from its topological clone. This will lead me in particular into complexity theory, where certain computational problems, called Constraint Satisfaction Problems, are investigated systematically via their polymorphism algebras, and subsequently via topological clones. I will moreover address the often non-trivial interference between the algebraic and the topological structure of algebras. Finally, I will show how the algebraic modelling of function clones can be altered in order to be more suitable for constraint satisfaction.

Christian Rosendal: *Large scale geometry of automorphism groups (Part III)*

The large scale geometry of finitely or compactly generated groups has long been a central part of geometry, topology and group theory. An abstract approach to certain aspects of large scale geometry is given by the coarse spaces due to J. Roe of which finitely or compactly generated groups, metric spaces and Banach spaces are particular examples. By analogy with a classical description of the left-invariant uniformity, we define a canonical left-invariant coarse structure on every topological group. This coincides with the previously mentioned cases, but also identifies natural structure in more general topological groups such as homeomorphism groups and other topological transformation groups. We develop this theory in the particular setting of automorphism groups of first order structures, which will allow us to identify a canonical geometry of the model theoretical structures themselves.

Christian Rosendal: *Large scale geometry of automorphism groups (Part I)*

The large scale geometry of finitely or compactly generated groups has long been a central part of geometry, topology and group theory. An abstract approach to certain aspects of large scale geometry is given by the coarse spaces due to J. Roe of which finitely or compactly generated groups, metric spaces and Banach spaces are particular examples. By analogy with a classical description of the left-invariant uniformity, we define a canonical left-invariant coarse structure on every topological group. This coincides with the previously mentioned cases, but also identifies natural structure in more general topological groups such as homeomorphism groups and other topological transformation groups. We develop this theory in the particular setting of automorphism groups of first order structures, which will allow us to identify a canonical geometry of the model theoretical structures themselves.

Christian Rosendal: *Large scale geometry of automorphism groups (Part II)*

The large scale geometry of finitely or compactly generated groups has long been a central part of geometry, topology and group theory. An abstract approach to certain aspects of large scale geometry is given by the coarse spaces due to J. Roe of which finitely or compactly generated groups, metric spaces and Banach spaces are particular examples. By analogy with a classical description of the left-invariant uniformity, we define a canonical left-invariant coarse structure on every topological group. This coincides with the previously mentioned cases, but also identifies natural structure in more general topological groups such as homeomorphism groups and other topological transformation groups. We develop this theory in the particular setting of automorphism groups of first order structures, which will allow us to identify a canonical geometry of the model theoretical structures themselves.

Gordon Royle: *Endomorphisms and Synchronisation *

In an earlier talk, Wolfram Bentz discussed the question of whether a finite primitive permutation group is necessarily *almost-synchronising* (that is, synchronises every non-uniform transformation) and explained how this reduces to finding endomorphisms of vertex-primitive graphs. In this talk, I will discuss some computational approaches to resolving this question for small orders along with some infinite families of examples, whose discovery relied on examining the computational data.

Artur Schaefer: *Synchronization Theory and Links to Combinatorics*

If we are given a transformation semigroup S on n points, the graph Gr(S) has vertex set 1,...,n where two vertices v,w are adjacent, if there is no f in S with vf=wf. Moreover, a graph X is called a hull, if X=Gr(S), where S is its endomorphism monoid. This construction and, in particular, hulls were introduced by Cameron and Kazanidis and have been used to establish the important connection between synchronization theory and graph endomorphisms. Their theorem says that a group G is synchronizing, if and only if there is no G-invariant graph which is a hull, admitting singular endomorphisms. In this talk, we will discuss the major role of hulls for synchronization theory, interpret graph endomorphisms as well known combinatorial objects, and discuss some structural properties of some non-synchronizing semigroups. Firstly, we will talk about hulls, provide examples of hulls and non-hulls and discuss (minimal) generating sets for Gr(S). Secondly, we will consider Hamming graphs, which are hulls, and related graphs. In particularly, we will link their endomorphisms to the existence of Latin hypercuboids of class R, to minimum distance separable mixed codes and to tilings. Ultimately, we introduce a new set of examples of non-synchronizing semigroups induced by tilings and provide general properties of these semigroups. Consequently, such a semigroup can be decomposed into a disjoint union of subsemigroups.

Slawomir Solecki: *Homogeneity of the pseudo-arc through permutation groups*

The pseudo-arc is the generic compact connected subset of the plane (or the Hilbert cube). By a fundamental result of Bing, it is homogeneous as a topological space. By work of Irwin and myself, the pseudo-arc is represented as a quotient of a dual Fraisse limit, which allows for a discretization of a continuous situation and makes it possible to apply combinatorial/permutation group methods. In this joint work with Tsankov, we determine the correct partial homogeneity of the dual Fraisse limit. Further, we prove a transfer theorem, through which we recover Bing's homogeneity of the pseudo-arc from our partial homogeneity of the dual Fraisse limit.

Benjamin Steinberg: *The representation theory of finite monoids (Part I)*

I will provide an introduction to the representation theory of finite monoids focusing on irreducible representations, character theory and applications to combinatorics and Markov chains. Important examples such as the full transformation monoid and the monoid of nxn matrices over a finite field will be used as examples.

Benjamin Steinberg: *The representation theory of finite monoids (Part II)*

I will provide an introduction to the representation theory of finite monoids focusing on irreducible representations, character theory and applications to combinatorics and Markov chains. Important examples such as the full transformation monoid and the monoid of nxn matrices over a finite field will be used as examples.

Benjamin Steinberg: *The representation theory of finite monoids (Part III)*

I will provide an introduction to the representation theory of finite monoids focusing on irreducible representations, character theory and applications to combinatorics and Markov chains. Important examples such as the full transformation monoid and the monoid of nxn matrices over a finite field will be used as examples.

Simon Thomas: *Invariant random subgroups of locally finite groups*

Let $G$ be a countable discrete group and let $\text{Sub}_{G}$ be the compact space of subgroups $H \leqslant G$. Then a probability measure $\nu$ on $\text{Sub}_{G}$ which is invariant under the conjugation action of $G$ on $\text{Sub}_{G}$ is called an invariant random subgroup. In this talk, I will explain how the pointwise ergodic theorem reduces questions concerning the invariant random subgroups of locally finite groups to problems in the asymptotic theory of finite permutation groups.

John Truss: *Definability properties of the monoid of endomorphisms of the rational numbers*

I write M for the monoid of self-embeddings of the rationals, G for its automorphism group, and E for the monoid of all its endomorphisms. Thus G is a subset of M which in turn is a subset of E. I study various definable subsets of these monoids, and discuss the relationship between them. One difference for instance is that in E it is easy to represent the individual rational numbers (via constant maps), but in M this is less obvious. This work is directed at reconstruction results for the polymorphism clones on the rationals under either the strict or reflexive relation (joint work with Edith Vargas Garcia).

Todor Tsankov: *Banach representations of dynamical systems and model theory*

It is well-known that the automorphism group of an omega-categorical structure encodes all model-theoretic information about the structure. Recently, an interesting correspondence has been discovered between properties of the theory (stability, omega-stability, NIP) and classes of Banach spaces on which certain dynamical systems (the automorphism group acting on type spaces over the model) can be represented. In the stable case, those dynamical systems also carry the structure of a semigroup that can be exploited. I will discuss what is known about this correspondence as well as some open questions. This is joint work with Itaï Ben Yaacov and Tomás Ibarlucía.

Lionel Van The: *Ramsey-type phenomena from fixed points in compactifications*

Ramsey theory (which is, roughly, the study of the necessary appearance of very organized substructures inside of any sufficiently large structure) has lately largely benefited from its connection to various other fields, especially dynamics and functional analysis. In this talk, I will illustrate this further by showing how the existence of fixed points in certain group compactifications allows to derive new Ramsey-type results.

Edith Vargas-Garcia: *Introduction to reconstructing the topological monoid of endomorphisms of the rationals.*

Transformation Monoids carry a natural topology, provided by the topology of point-wise convergence, with respect to which the composition is continuous. A Topological Monoid is an abstract monoid equipped with a topology under which the composition is continuous. The endomorphism monoids $End \left(\mathcal{A}\right)$ of a relational structure $\mathcal{A}$ are viewed abstractly as a topological monoids whose topology is the natural topology. Moreover, they are the closed submonoids of the space of all unary functions on $\mathcal{A}$. We study when $End \left(\mathbb{Q},\leq \right)$ has the property that every monoid isomorphism to the endomorphism monoid of other relational structure $\mathcal{B}$ is automatically a homeomorphism.

Mikhail Volkov: *Representations of finite J-trivial monoids and finite block-groups by transformations and relations *

Finite J-trivial monoids and finite block-groups play a distinguished role in the algebraic theory of regular languages. In turn, several purely algebraic representations of these monoids by transformations and binary relations have been established by language-theoretic tools, and moreover, mastering direct proofs for these representation results appears to constitute quite a serious challenge. We overview the area, focusing on recent advances and questions that still remain open. (2) Cerny's type problem for transformation semigroups The famous \v{C}ern\'{y} conjecture about synchronizing automata can be restated in terms of transformation semigroups as follows: for each subset $\Sigma$ of the full transformation monoid on the $n$-element set, if the subsemigroup $S$ generated by $\Sigma$ contains a constant transformation, then already a product of some $(n-1)^2$ transformations in $\Sigma$ is a constant. Since the problem has proved to be hard, it seems to be reasonable to approach it by imposing some natural constraints on the transformations in $\Sigma$ or on the subsemigroup $S$ as a whole. Several partial results in the area fit into this scheme; we survey them and comment on possible next steps in this line of research.

Mikhail Volkov: *Cerny's type problem for transformation semigroups*

Finite J-trivial monoids and finite block-groups play a distinguished role in the algebraic theory of regular languages. In turn, several purely algebraic representations of these monoids by transformations and binary relations have been established by language-theoretic tools, and moreover, mastering direct proofs for these representation results appears to constitute quite a serious challenge. We overview the area, focusing on recent advances and questions that still remain open. (2) Cerny's type problem for transformation semigroups The famous \v{C}ern\'{y} conjecture about synchronizing automata can be restated in terms of transformation semigroups as follows: for each subset $\Sigma$ of the full transformation monoid on the $n$-element set, if the subsemigroup $S$ generated by $\Sigma$ contains a constant transformation, then already a product of some $(n-1)^2$ transformations in $\Sigma$ is a constant. Since the problem has proved to be hard, it seems to be reasonable to approach it by imposing some natural constraints on the transformations in $\Sigma$ or on the subsemigroup $S$ as a whole. Several partial results in the area fit into this scheme; we survey them and comment on possible next steps in this line of research.