Deciding whether two one-dimensional subshifts are conjugate remains one of the most important question in symbolic dynamics. In this talk, we will highlight a new approach, using the diagrammatic calculus approach popular in category theory and especially in categorical quantum mechanics. We will explain how matrices (and subshifts of finite type) can be represented graphically and how this representation may help us find new conjugacy invariants.[-]

Recall that normality is a elementary form of randomness: an infinite word is normal to a given alphabet if all blocks of symbols of the same length occur in the word with the same asymptotic frequency. We consider a notion of independence on pairs of infinite words formalising that two words are independent if no one helps to compress the other using one-to-one finite transducers with two inputs. As expected, the set of independent pairs has Lebesgue measure 1. We prove that not only the join of two normal words is normal, but, more generally, the shuffling with a finite transducer of two normal independent words is also a normal word. The converse of this theorem fails: we construct a normal word as the join of two normal words that are not independent. We construct a word x such that the symbol at position n is equal to the symbol at position 2n. Thus, x is the join of x itself and the subsequence of odd positions of x. We also show that selection by finite automata acting on pairs of independent words preserves normality. This is a counterpart version of Agafonov's theorem for finite automata with two input tapes.
This is joint work with Olivier Carton (Universitéé Paris Diderot) and Pablo Ariel Heiber (Universidad de Buenos Aires).[-]

Carleson's Theorem states that for $1 < p < \infty$, the Fourier series of a function $f$ in $L^p[-\pi,\pi]$ converges to $f$ almost everywhere. We consider this theorem in the context of computable analysis and show the following two results.
(1) For a computable $p > 1$, if $f$ is a computable vector in $L^p[?\pi,\pi]$ and $t_0 \in [-\pi,\pi]$ is Schnorr random, then the Fourier series for $f$ converges at $t_0$.
(2) If $t_0 \in [-\pi,\pi]$ is not Schnorr random, then there is a computable function $f : [-\pi,\pi] \rightarrow \mathbb{C}$ whose Fourier series diverges at $t_0$.
This is joint work with Timothy H. McNicholl, and Jason Rute.[-]

We discuss subshifts of finite type (tilings) that combine virtually opposite properties, being at once very simple and very complex. On the one hand, the combinatorial structure of these subshifts is rather simple: we require that all their configurations are quasiperiodic, or even that all configurations contain exactly the same finite patterns (in the last case a subshift is transitive, i.e., irreducible as a dynamical system). On the other hand, these subshifts are complex in the sense of computability theory: all their configurations are non periodic or even non-computable, or all their finite patterns have high Kolmogorov complexity, the Turing degree spectrum is rather sophisticated, etc.
We start with the simplest example of such centaurisme with an SFT that is minimal and contains only aperiodic (and quasiperiodic) configurations. Then we discuss how far these heterogeneous properties can be strengthened without getting mutually exclusive.
This is a joint work with Bruno Durand (Univ. de Montpellier).[-]

Barmpalias and Lewis-Pye recently proved that if $\alpha$ and $\beta$ are (Martin-Löf) random left-c.e. reals with left-c.e. approximations $\{\alpha_s \}_{s \in\ omega}$ and $\{\beta_s \}_{s \in\ omega}$, then
\[
\begin{equation}
\frac{\partial\alpha}{\partial\beta} = \lim_{s\to\infty} \frac{\alpha-\alpha_s}{\beta-\beta_s}.
\end{equation}
\]
converges and is independent of the choice of approximations. Furthermore, they showed that $\partial\alpha/\partial\beta = 1$ if and only if $\alpha-\beta$ is nonrandom; $\partial\alpha/\partial\beta>1$ if and only if $\alpha-\beta$ is a random left-c.e. real; and $\partial\alpha/\partial\beta<1$ if and only if $\alpha-\beta$ is a random right-c.e. real.

We extend their results to the d.c.e. reals, which clarifies what is happening. The extension is straightforward. Fix a random left-c.e. real $\Omega$ with approximation $\{\Omega_s\}_{s\in\omega}$. If $\alpha$ is a d.c.e. real with d.c.e. approximation $\{\alpha_s\}_{s\in\omega}$, let
\[
\partial\alpha = \frac{\partial\alpha}{\partial\Omega} = \lim_{s\to\infty} \frac{\alpha-\alpha_s}{\Omega-\Omega_s}.
\]
As above, the limit exists and is independent of the choice of approximations. Now $\partial\alpha=0$ if and only if $\alpha$ is nonrandom; $\partial\alpha>0$ if and only if $\alpha$ is a random left-c.e. real; and $\partial\alpha<0$ if and only if $\alpha$ is a random right-c.e. real.

As we have telegraphed by our choice of notation, $\partial$ is a derivation on the field of d.c.e. reals. In other words, $\partial$ preserves addition and satisfies the Leibniz law:
\[
\partial(\alpha\beta) = \alpha\,\partial\beta + \beta\,\partial\alpha.
\]
(However, $\partial$ maps outside of the d.c.e. reals, so it does not make them a differential field.) We will see how the properties of $\partial$ encapsulate much of what we know about randomness in the left-c.e. and d.c.e. reals. We also show that if $f\colon\mathbb{R}\rightarrow\mathbb{R}$ is a computable function that is differentiable at $\alpha$, then $\partial f(\alpha) = f'(\alpha)\,\partial\alpha$. This allows us to apply basic identities from calculus, so for example, $\partial\alpha^n = n\alpha^{n-1}\,\partial\alpha$ and $\partial e^\alpha = e^\alpha\,\partial\alpha$. Since $\partial\Omega=1$, we have $\partial e^\Omega = e^\Omega$.

Given a derivation on a field, the elements that it maps to zero also form a field: the $ \textit {field of constants}$. In our case, these are the nonrandom d.c.e. reals. We show that, in fact, the nonrandom d.c.e. reals form a $ \textit {real closed field}$. Note that it was not even known that the nonrandom d.c.e. reals are closed under addition, and indeed, it is easy to prove the convergence of [1] from this fact. In contrast, it has long been known that the nonrandom left-c.e. reals are closed under addition (Demuth [2] and Downey, Hirschfeldt, and Nies [3]). While also nontrivial, this fact seems to be easier to prove. Towards understanding this difference, we show that the real closure of the nonrandom left-c.e. reals is strictly smaller than the field of nonrandom d.c.e. reals. In particular, there are nonrandom d.c.e. reals that cannot be written as the difference of nonrandom left-c.e. reals; despite being nonrandom, they carry some kind of intrinsic randomness.[-]

I will discuss two recent interactions of the field called randomness via algorithmic tests. With Yokoyama and Triplett, I study the reverse mathematical strength of two results of analysis. (1) The Jordan decomposition theorem says that every function of bounded variation is the difference of two nondecreasing functions. This is equivalent to ACA or to WKL, depending on the formalisation. (2) A theorem of Lebesgue states that each function of bounded variation is differentiable almost everywhere. This turns out to be equivalent WWKL (with some fine work left to be done on the amount of induction needed). The Gamma operator maps Turing degrees to real numbers; a smaller value means a higher complexity. This operator has an analog in the field of cardinal characteristics along the lines of the Rupprecht correspondence [4]; also see [1]. Given a real p between 0 and 1/2, d(p) is the least size of a set G so that for each set x of natural numbers, there is a set y in G such that x and y agree on asymptotically more than p of the bits. Clearly, d is monotonic. Based on Monin's recent solution to the Gamma question (see [3] for background, and the post in [2] for a sketch), I will discuss the result with J. Brendle that the cardinal d(p) doesn't depend on p. Remaining open questions in computability (is weakly Schnorr engulfing equivalent to "Gamma = 0"?) nicely match open questions about these cardinal characteristics.[-]

One of the most fundamental problem in tiling theory is to decide, given a surface, a set of tiles and a tiling rule, whether there exist a way to tile the surface using the set of tiles and following the rules. As proven by Berger in the 60's, this problem is undecidable in general.
When formulated in terms of tilings of the discrete plane by unit tiles with colored constraints, this is called the Domino Problem and was introduced by Wang in an effort to solve satisfaction problems for ??? formulas by translating the problem into a geometric problem.
In this course, we will give a brief description of the problem and to the meaning of the word “undecidable”, and then give two different proofs of the result.[-]