RésumésNathalie Aubrun
An application of Lovász local lemma in Symbolic Dynamics
Symbolic Dynamics is the study of subshifts, that are sets of colorings of Z^d (or another finitely generated group) by a finite alphabet, that respect some local constraints. To prove that a subshift is non empty, it is enough to exhibit a coloring that satisfies the constraints. In this talk, I will explain how Lovász local lemma can be helpful to prove nonemptiness of subshifts. As an application, we will show the existence of a strongly aperiodis subshift on any finitely generated group.
Laurent Bartholdi
Amenability
Amenability is one of the fundamental properties of infinite groups. In all cases, one learns something valuable about the group: if it is amenable, then its actions on various geometric objects must have fixed points; if it is nonamenable, then it admits paradoxical decompositions: finite partitions whose pieces, after lefttranslation, can be rearranged in two copies of the original group.
Remarkably, groups generated by automata served as a prominent source of examples of amenable groups, delineating quite richly the boundary between amenable and nonamenable. I will outline some constructions of groups generated by automata and some fundamental open questions.
Tara Brough
The power of symboltuples for automaton semigroup constructions
In 2014, Cain and I made the following conjectures [1], giving a proposed `small' counterexample in each case:
Conjecture 1 The class of automaton semigroups is not closed under taking free products.
Conjecture 2 The class of automaton monoids is not closed under taking wreath products with finite top monoid.
Both of our proposed counterexamples are in fact automaton semigroups, and Conjecture 2 is false [2]. What we were missing was that it turns out to be very powerful, in attempting to construct new automaton semigroups from known examples, to use tuples of symbols from the 'base' automata as the symbols of the new automaton.
In this talk I will illustrate several ways of using symboltuples to realise various standard semigroup constructions  such as free products, wreath products, strong semilattices and Rees matrix constructions (under certain conditions)  as automaton semigroups.
[1] T. Brough and A.J. Cain, `Automaton semigroup constructions', Semigroup Forum 90(3), 763774 (2014).
[2] T. Brough and A.J. Cain, `Automaton semigroups: new constructions results and examples of nonautomaton semigroups', Theoret. Comput. Sci. 674, 115 (2017).
Alan J. Cain
Daniele D'Angeli Automaton (semi)groups: on the undecidability of some problems In this talk I present some results obtained in collaboration with T. Godin I. Klimann, M. Picantin, E. Rodaro, and JP. Wächter.
Martin Delacourt Permutive oneway cellular automata and the finiteness problem for automaton groups
Francesca Fiorenzi Entropy of strongly irreducible subshifts
Pierre Gillibert Simulating Turing machines with invertible Mealy automata
Markus Lohrey Knapsack and subset sum for groups beyond the integers (joint work with Daniel König and Georg Zetzsche) Recently, Myasnikov, Nikolaev and Ushakov considered classical knapsack related decision problems for arbitrary finitely generated (f.g.) groups instead of the group of integers (the classical case).
Given elements g_1,..., g_k, g of G, decide whether there exist numbers e_1,..., e_k from {0,1} such that g = g_1^{e_1} … g_k^{e_k}.
Given elements g_1,...,g_k, g of G, decide whether there exist natural numbers e_1,..., e_k such that g = g_1^{e_1} … g_k^{e_k}. In the talk, I will speak about some recent results for knapsack and subset sum in various classes of groups: nilpotent groups, polycyclic groups, rightangled Artin groups (aka graph groups), cocontext free groups, certain wreath products. More precisely, we will discuss the following results and (if time permits) will prove some of them:
Jean Mairesse Uniform measures on braid monoids We are interested in the asymptotic properties of typical positive braids, respectively positive dual braids. Denoting by μk the uniform distribution on positive (dual) braids of length k , we prove that the sequence (μk)k converges to a unique probability measure μ on infinite positive (dual) braids. The key point is that the limiting measure μ has a Markovian structure which can be described explicitly using the combinatorial properties of braids encapsulated in the Möbius polynomial. As a byproduct, we settle a conjecture by Gebhardt and Tawn (J. Algebra, 2014) on the shape of the Garside normal form of large uniform braids. Joint work with S. Abbes, S. Gouëzel and Vincent Jugé
Irène Marcovici Ergodicity of Noisy Cellular Automata: The Coupling Method and Beyond When perturbating a cellular automaton by a random noise (positive probability of error, for each cell independently), the system is generally expected to be ergodic, meaning that during its evolution, it eventually forgets about its initial condition. For a high noise, this can be shown by coupling. However, for a small noise, ergodicity is often very difficult to prove. We present extensions of the coupling method to small noises when the cellular automaton has some specific properties (hardcore exclusion, nilpotency, permutivity).
Cyril Nicaud Random automata We’ll survey various result on the properties of random deterministic automata (including random generation and analysis of algorithms), with an emphasis on their typical algebraic properties.
Nicolas Ollinger The periodicity problem of cellular automata (joint work with J. Kari, A. Gajardo, and R. Torres)
Dmytro Savchuk Duality in reversible automata generating lamplighter type groups
We study a class of groups generated by reversible automata modeling multiplication in the ring of formal power series. The realization of the lamplighter type groups by similar types of automata has been studied by several authors since the beginning of 2000’s. We show that in many cases the group generated by the dual automaton is again of lamplighter type and that it also acts on series in certain rational function. This is a joint work with Ievgen Bondarenko.
Reem Yassawi
Linear cellular automata and pautomatic sequences
Let p be prime. We study the links between pautomatic sequences and asymptotic properties of linear cellular automata, emphasizing the constructive aspect. In particular, we outline a method that aims to identify nontrivial measures that are invariant under the action of both the shift and the cellular automaton under study. This is joint work with Benjamin Hellouin and Eric Rowland.
