For every positive integer $n$, we introduce a set $\mathcal{T}_n$ made of $(n+3)^2$ Wang tiles (unit squares with labeled edges). We represent a tiling by translates of these tiles as a configuration $\mathbb{Z}^2 \rightarrow \mathcal{T}_n$. A configuration is valid if the common edge of adjacent tiles has the same label. For every $n \geqslant 1$, we consider the Wang shift $\Omega_n$ defined as the set of valid configurations over the tiles $\mathcal{T}_n$. The family $\left\{\Omega_n\right\}_{n \geqslant 1}$ broadens the relation between quadratic integers and aperiodic tilings beyond the omnipresent golden ratio as the dynamics of $\Omega_n$ involves the positive root $\beta$ of the polynomial $x^2-n x-1$. This root is sometimes called the $n$-th metallic mean, and in particular, the golden mean when $n=1$ and the silver mean when $n=2$. The family gathers the hallmarks of other small aperiodic sets of Wang tiles. When $n=1$, the set of Wang tiles $\mathcal{T}_1$ is equivalent to the Ammann aperiodic set of 16 Wang tiles. The tiles in $\mathcal{T}_n$ satisfy additive versions of equations verified by the Kari-Culik aperiodic sets of 14 and 13 Wang tiles. Also configurations in $\Omega_n$ are the codings of a $\mathbb{Z}^2$-action on a 2-dimensional torus by a polygonal partition like the Jeandel-Rao aperiodic set of 11 Wang tiles. The tiles can be defined as the different instances of a square shape computer chip whose inputs and outputs are 3-dimensional integer vectors. There is an almost one-to-one factor map $\Omega_n \rightarrow \mathbb{T}^2$ which commutes the shift action on $\Omega_n$ with horizontal and vertical translations by $\beta$ on $\mathbb{T}^2$. The factor map can be explicitely defined by the average of the top labels from the same row of tiles as in Kari and Culik examples. We also show that $\Omega_n$ is self-similar, aperiodic and minimal for the shift action. Also, there exists a polygonal partition of $\mathbb{T}^2$ which we show is a Markov partition for the toral $\mathbb{Z}^2$-action. The partition and the sets of Wang tiles are symmetric which makes them, like Penrose tilings, worthy of investigation. Details can be found in the preprints available at https://arxiv.org/abs/ 2312.03652 (part I) and https://arxiv.org/abs/2403. 03197 (part II). The talk will present an overview of the main results.[-]

In the first part of the talk we will survey known results concerning Fibonacci numbers whose digital representations in base 10 display some interesting patterns. In the second part of the talk we will give the main steps of the proof of a recent result which states that $b = 4$ is the only integer ≥ 2 such that there are two Fibonacci numbers larger than 1 which are repunits in base b. In this case, $F_{5}=(4^{2}-1)/(4-1)$ and $F_{8}=(4^{3}-1)/(4-1)$. This is joint work with C. A. Gomez and J. C. Gomez from Cali, Colombia.[-]

Recently several papers appears on ArXiv, on various topics apparently unrelated such as: spin system observable (T. Helmuth, A. Shapira), Fibonacci polynomials (A. Garsia, G. Ganzberger), fully commutative elements in Coxeter groups (E. Bagno, R. Biagioli, F. Jouhet, Y. Roichman), reciprocity theorem for bounded Dyck paths (J. Cigler, C. Krattenthaler), uniform random spanning tree in graphs (L. Fredes, J.-F. Marckert). In each of these papers the theory of heaps of pieces plays a central role. We propose a walk relating these topics, starting from the well-known loop erased random walk model (LERW), going around the classical bijection between lattice paths and heaps of cycles, and a second less known bijection due to T. Helmuth between lattice paths and heaps of oriented loops, in relation with the Ising model in physics, totally non-backtracking paths and zeta function in graphs. Dyck paths, these two bijections involve heaps of dimers and heaps of segments. A duality between these two kinds of heaps appears in some of the above papers, in relation with orthogonal polynomials and fully commutative elements. If time allows we will finish this excursion with the correspondence between heaps of segments, staircase polygons and q-Bessel functions.[-]

The modular curve $Y^1(N)$ parametrises pairs $(E,P)$, where $E$ is an elliptic curve and $P$ is a point of order $N$ on $E$, up to isomorphism. A unit on the affine curve $Y^1(N)$ is a holomorphic function that is nowhere zero and I will mention some applications of the group of units in the talk.
The main result is a way of generating generators (sic) of this group using a recurrence relation. The generators are essentially the defining equations of $Y^1(N)$ for $n < (N + 3)/2$. This result proves a conjecture of Maarten Derickx and Mark van Hoeij.[-]