Plenary Speakers

Confirmed Invited Speakers

Laurent Bartholdi
(Saarland University, Germany)

Automatic Actions

Abstract: I will present a general notion of automatic action, based on Büchi automata, and show how it unifies a large number of subclasses, in particular the automatic groups by Cannon, Thurston et al.; the transducer groups by Aleshin, Grigorchuk, Sushchansky, Sidki et al.; substitutional subshifts; and complex dynamics.

I will present some algorithms for these groups, and in particular show under an extra condition (boundedness) that their orbit relation is computable and regular. This will have strong decidability consequences, such as that the order problem, aperiodicity, minimality, etc. for automatic transformations is decidable.

I will detail applications to symbolic dynamics: in particular, it is decidable whether a substitutional subshift is aperiodic, minimal, topologically transitive, etc.; and to complex dynamics: it is decidable e.g. if a Julia set is a Sierpinski carpet.

Persi Diaconis
(Stanford University, USA)

What makes “Wild” wild?

Abstract: How do you show a problem is 'hard'? I'll discuss the problem of describing the conjugacy classes(or characters) of the group of n x n uni-upper triangular matrices over F_p. Of course, when n=3, it's the Heisenberg group and everything is fine. When n and p grow, we can't do it and, maybe, nobody can ever do it. How does one prove such a thing? This problem is 'wild' and perhaps there is a word problem that would be solved if you had a reasonable description of the classes. I'll make the connection to Carlos Andre's 'superclasses' but the bottom line is 'the community can't do it'.

James East
(Western Sydney University, Australia)

Projection algebras and free regular *-semigroups

Abstract:
It has long been known that the set of projections of a regular *-semigroup can be given the structure of a unary algebra. This gives rise to a forgetful functor from the category of regular *-semigroups to the category of projection algebras. It turns out that this functor has a right adjoint, which maps a projection algebra to an associated free (projection-generated) regular *-semigroup. The free semigroup is built from a natural `chain groupoid', which can be understood topologically as the fundamental groupoid of a certain simplicial 2-complex. In this talk I will outline the above theory, and discuss some results and consequences. For example, it turns out that Temperley-Lieb monoids are themselves free regular *-semigroups (over their own projection algebras), and are unique among diagram monoids in this regard. For another example, one can start with the projection algebra of an `adjacency semigroup' (in the sense of Jackson and Volkov), and the free functor produces an apparently new kind of graph semigroup. This is all joint work with Bob Gray, Nik Ruškuc and Azeef Muhammed PA.


Victoria Gould
(York University, UK)

Coherency for monoids and purity for their acts

Abstract: A monoid $S$ may be represented via mappings of sets or, equivalently and more concretely, by $S$-acts. A right $S$-act is a set $A$ together with a map $A\times S\rightarrow A$ where $(a,s)\mapsto as$, such that for all $a\in A$ and $s,t\in S$ we have $a1=a$ and $(as)t=a(st)$. The monoid $S$ itself, and any right ideal of $S$, are right $S$-acts under the monoid multiplication. Cyclic right $S$-acts are of the form $S/\rho$ where $\rho$ is a right congruence. We say that $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented and weakly right coherent if every finitely generated right ideal of $S$ has a finite presentation. 

Coherency properties are fascinating and somewhat elusive. They arise naturally from several directions, including model theory and ring theory. The corresponding notions of right coherency and weak right coherency for a ring $R$ (where, of course, $S$-acts are replaced by $R$-modules) coincide, essentially because every cyclic $R$-module is determined by a right ideal. This is not the case for monoids and we must develop new strategies. 

This talk will examine coherency notions for monoids from several angles, including the connection with other finitary conditions such as being right noetherian, axiomatisability of classes of algebraically closed right $S$-acts, and so-called purity properties, which are related to injectivity of right $S$-acts. 


This work is drawn from a number of sources, the most recent being joint with Matthew Brookes, Yang Dandan and Nik Ru\v{s}kuc.

Olga  Kharlampovich
(CUNY Graduate Center, USA)

Subgroup Separability in RAAGs via representations

Abstract: We answer the question asked by Louder, McReinolds and

Patel  and prove the following statement. Let L be a RAAG, H a

cubically convex-cocompact subgroup of L, then there is a finite

dimensional representation of L that separates the subgroup H

in the induced Zariski topology. As a corollary, we establish a

polynomial upper bound on the size of the quotients used to separate H

in L. This implies the same statement for a virtually special group L and, in

particular, a fundamental group of a hyperbolic 3-manifold.


 For any finitely generated subgroup H of a limit group L we prove the

same results and, in addition, show that  there exists a finite-index

subgroup K containing H, such that K is a subgroup of a group obtained

from H  by a series of extensions of centralizers and free products

with infinite cyclic group. If H is non-abelian, the K  is fully

residually H.  A corollary is that a hyperbolic limit group satisfies

the Geometric Hanna Neumann conjecture.


These are  joint results with K. Brown and A. Vdovina.

Dimitri Leemans
(Université Libre de Bruxelles, Belgium)

Highly symmetric polytopes: recent results and future challenges

Abstract:

Highly symmetric polytopes consist mainly of abstract regular and abstract chiral polytopes.

The regular polytopes are those with highest level of symmetry. Their automorphism group acts regularly on the flags. The chiral ones have maximal amount of rotational symmetries but no reflections. Their automorphismes group has two orbits on the flags such that every two adjacent flags lie in distinct orbits.

In this talk, we will show some nice results that have been obtained recently on highly symmetric polytopes whose automorphism groups are almost simple groups and we will suggest some open problems.


Martin Liebeck
(Imperial College, UK)

On a problem of Araújo and Cameron

Abstract: Araújo and Cameron posed the following problem, which is a linear version of a result of theirs on regular transformation semigroups: characterize the subgroups G of GL(n,q) with the property that for all singular matrices A, there exists g in G such that rank(A) = rank(AgA). I will show how this leads to a natural problem in representation theory, the solution of which gives rise to some rather nice families of examples of such groups G.

Stuart Margolis
(Bar-Ilan University, Israel)

Complexity 1 is Decidable For Finite Semigroups

The complexity of a finite semigroup S is the least number of groups needed in any decomposition of S as a divisor of a wreath product of groups and aperiodic (group-free) semigroups. Such a number is guaranteed by the Krohn-Rhodes Theorem.


We prove that it is decidable if a finite semigroup has complexity 1. This  settles a problem first posed in 1965. The talk will give background in complexity theory and discuss the tools needed to prove the theorem. Time permitting, we will outline the proof.


This is joint work with John Rhodes and Anne Schilling.


Anne Schilling
 (University of California, USA)

Pascal Weil
 (CNRS, ReLaX, Chennai)

Average-case complexity of certain algorithms on subgroups of free groups

Joint work with Mallika Roy (UPV, Bilbao) and Enric Ventura (UPC Barcelona)


We will discuss the average-case complexity of the Uniform Membership Problem in free groups (given elements h, g_1, …, g_k of a free group, decide whether h is in the subgroup H generated by the g_i?), and show that it is orders of magnitude smaller than the worst-case complexity of the same problem. We will also discuss the Relative Primitivity Problem (is h primitive in H, that is, is h an element of some basis of H?)