Talks

Here is a (definitely incomplete) list of talks that I gave over the years. Since I do not know how to organize this list, it may appear chaotic (though, technically, the list is ordered: the talks are grouped together depending on their topic, and then each group is sorted by reversed chronological order on the latest occurrence of the talks; but I agree this is completely arbitrary 😅).

I try to provide the slides that come with most of them (assuming there are any), although – since I often repeat my talks on different occasions – the slides attached here may not coincide with what you might have seen if you attended the talk in person.

Like anyone who does research in symbolic dynamics, I now possess a large collection of TikZ figures that you can admire in those slides. I’d happily share their code with anyone asking!

• Computable topology
  • Descriptive complexity on represented spaces.
    • ASL 2021 (Invited talk) – Online, 23 June 2021.
    • CCC 2023 (Informal talk) – Online, 03 September 2020.
    • CIE 2020 (Informal talk) – Online, 01 July 2020.
    • STACS 2020 (Contributed talk) – Montpellier (France), 11 March 2020.
    Abstract:

    (Joint work with Mathieu Hoyrup).
    In the framework of Type-2 Theory of Effectivity (TTE), represented spaces (topological spaces equipped with a representation) form a general context in which computability has been extended.
    Descriptive Set Theory (DST) provides two competing measures of complexity for sets in such spaces. The first one is topological, and stratifies sets according to the number of Boolean operations required to obtain them from open sets. The second one measures the complexity of effectively testing membership in the set. As it measures the complexity of the symbolic representation of the set, we call it the symbolic complexity.
    In this talk, we investigate these two measures of complexity. While they coincide on countably-based spaces (as proved in [De Brecht, 2012]), topological and symbolic complexity may differ on more general spaces. We suggest that this difference is related to the mismatch between topological and sequential aspects of the topology of these spaces.

• Symbolic dynamics and tilings
  • Pattern complexity, aperiodic domino problem and group geometry.
    • Journées SDA2 (Seminar) – IMT, Toulouse (France), 29 March 2023.
    Abstract:

    (Joint work with Benjamin Hellouin de Menibus and Ville Salo).
    The Domino problem asks whether there exists a tiling in which none of the forbidden patterns given as input appear. We consider the aperiodic version of the Domino problem: given as input a family of forbidden patterns, does it allow an aperiodic tiling?
    In this talk, we relate the pattern complexity of subshifts with the aperiodic Domino problem by proving the following: if a subshift on $\Z^d$ has infinite "surface entropy" (in particular, if it has positive entropy), then it contains an aperiodic configuration. We then generalize this result from $\Z^d$ to some other groups by introducing a new geometrical notion of "confluence" for ordered group.

  • Distortion in the automorphism group of a full-shift.
    • Séminaire SymPA (Seminar) – LAMFA, Amiens (France), 28 November 2023.
    • Séminaire Pythéas Fogg (Seminar) – FRUMAM, Marseille (France), 03 March 2023.
    • Séminaire Rauzy (Seminar) – FRUMAM, Marseille (France), 03 March 2023.
    • Math seminar (Seminar) – UTU, Turku (Finland), 17 June 2022.
    Abstract:

    (Joint work with Ville Salo).
    Let $X$ be a full shift, i.e. the set of bi-infinite words on a given alphabet. Once equipped with the prodiscrete topology, $X$ is a Cantor space. The set $\Aut(X)$ of its automorphisms (shift-equivariant homeomorphisms) is a group for composition.
    The structure of $\Aut(X)$ can be quite complex, and many things are not known. One direction of study consists in studying groups that can be embedded into $\Aut(X)$ (e.g. every finite group, finitely generated abelian groups, and their products, direct sums, etc…, or the lamplighter group), or conversely to study restrictions that prevent groups from being embedded into $\Aut(X)$.
    It is still open whether the Bausmlag-Solitar groups $BS(m,n)$ or the Heisenberg group embed in $\Aut(X)$. To study this question, this talks focuses on the existence of distortion elements in $\Aut(X)$, i.e. automorphisms such that the word norm of their powers grows sublinearly (relatively to a finte set of generators).
    We show that every full shift contains a distortion element (this element being, morally, the SMART machine introduced by J. Cassaigne, N. Ollinger and R. Torres-Avilés). As a consequence, the multi-dimensional Brin-Thompson group $2V$ contains a distortion element.

  • The aperiodic Domino problem.
    • Journées SDA2 (Seminar) – ULiège, Liège (Belgium), 13 June 2022.
    • Séminaire AMACC (Seminar) – GREYC, Caen (France), 07 June 2022.
    • STACS 2022 (Contributed talk) – Online, 15 March 2022.
    Abstract:

    (Joint work with Benjamin Hellouin de Menibus).
    The classical Domino problem asks whether there exists a tiling in which none of the forbidden patterns given as input appear. In this talk, we consider the aperiodic version of the Domino problem: given as input a family of forbidden patterns, does it allow an aperiodic tiling? The input may correspond to a subshift of finite type, a sofic subshift or an effective subshift.
    The Aperiodic Domino is undecidable, because it reduces to the (classical) Domino problem. In this talk, we study the computational complexity of the Aperiodic Domino problem: $\Pi^0_1$-complete for $\Z^2$ subshifts (by a result from A. Grandjean, B. Hellouin and P. Vanier), it becomes $\Sigma^1_1$-complete (ie. much harder, namely analytic) in higher dimension: $d \geq 4$ in the finite type case, $d \geq 3$ for sofic and effective subshifts.
    These results are surprising for two reasons: first, the Aperiodic Domino separates 2- and 3-dimensional subshifts, wheareas most subshift properties are the same in dimension 2 and higher; second, this gap unexpectedly large.

  • Surface entropies of $\Z^2$ subshifts of finite type.
    • ICALP 2021 (Contributed talk) – Online, 13 July 2021.
    • Séminaire AMACC (Seminar) – GREYC, Caen (France), 30 March 2021.
    • Journées SDA2 (Seminar) – Online, 03 December 2020.
    Abstract:

    (Joint work with Pascal Vanier).
    Consider subshifts of finite type (SFTs), i.e. sets of colorings over $\Z^2$ defined by finite families of forbidden patterns. These objects enjoy a deep link with computability theory. This link was uncovered by the characterization of entropies of SFTs as the right recursively enumerable numbers by Hochman and Meyerovitch.
    Here, we focus on the related notion of surface entropy introduced in Dennis Pace's thesis. If the number of $n \times n$ patterns of the subshift grows as $2^{hn^2 + 2Cn}$, then $h$ is the classical topological entropy while $C$ is the surface entropy. In this talk, we prove that the surface entropies of $\Z^2$ SFTs are exactly the $\Pi^0_3$ real numbers of the arithmetical hierarchy of real numbers.
    To obtain this characterization, we introduce the sparse squares, a construction which realizes linear terms (instead of the usual quadratic terms) in the complexity function.