The geodesics $\xi$ on the hyperbolic plane $\mathbf{HP}$ are determined by their endpoints $(\xi^-, \xi^+)$ in the real projective line. Consider a finite cover of the modular orbifold $S = \Gamma \ \mathbf{HP}$ and a hyperbolic geodesic $\xi$ whose projection to $S$ is simple (without double points). We then show that its endpoints $\xi^\pm$ are necessarily either rational or quadratic, or transcendental. This relies on the fact that simple geodesics define languages of sublinear complexity, and on a transcendence criterion for continued fractions having long repetitions due to Adamczewski and Bugeaud.

Matching is a phenomenon found in families of interval maps with specific number-theoretic properties. It can be the cause of piecewise monotone or even smooth dependence of the entropy, Lyapunov exponents and invariant densities on the parameter of the family. In this survey, we present some families where matching occurs and is even prevalent: open, dense and of full Lebesgue measure in parameter space. We also discuss the bifurcation theory of the matching phenomenon, hoping to illustrate some common period doubling and renormalization aspect that it appears to imply.

Matching is a phenomenon found in families of interval maps with specific number-theoretic properties. It can be the cause of piecewise monotone or even smooth dependence of the entropy, Lyapunov exponents and invariant densities on the parameter of the family. In this survey, we present some families where matching occurs and is even prevalent: open, dense and of full Lebesgue measure in parameter space. We also discuss the bifurcation theory of the matching phenomenon, hoping to illustrate some common period doubling and renormalization aspect that it appears to imply.

We propose a sufficient condition for the non-periodicity of a set of Wang tiles. It applies to sets of Wang tiles whose tiles have vertical or horizontal stripes. The proof is based on a geometric argument involving a quadrilateral circumscribed to a parabola from which we conclude the irrationality of the densities of the vertical and horizontal stripes. We apply the sufficient condition to propose new proofs of non-periodicity of known sets of Wang tiles, including an encoding of Penrose tilings into 24 Wang tiles and the family of metallic mean Wang tiles. Conversely, for every pair $(\alpha,\beta)\in[0,1]^2$ of irrational numbers in the same quadratic number field, we construct a finite aperiodic set of Wang tiles with stripes that admits a valid tiling whose density of vertical stripes is $\alpha$ and density of horizontal stripes is $\beta$. This is a joint work with Jarkko Kari and Pieter Mostert available at arXiv:2606.24693. In this board talk, we will present the first theorem and the method to find the quadrilateral from a set of Wang tiles with stripes.

We will consider multiplicative matrix equations of the knapsack problem type and special cases such as the mortality problem or the identity problem. In the first part an overview on classical results is given. In the second part connections to Skolem's problem are outlined. In particular, some important contributions of Florian Luca to this subject are presented and connections to analytic and diophantine number theory are discussed. The third part is devoted to special instances of the identity problem, which were published in a joint work by Heintze- Noubissie-Tichy (Combinatorics and NumberTheory, 2025). The main tools are effective diophantine methods.

Counting lattice points can be viewed as a geometric formulation of the Diophantine problem of finding integer solutions to a system of equations and/or inequalities. Several techniques have been developed to address this problem, most of them rooted in either Harmonic Analysis or Model Theory. This talk will primarily survey the latter approach, highlighting recent developments in the problem of counting solutions to systems of homogeneous forms. Particular emphasis will be placed on applications of these methods to a range of questions arising from the work of Athreya, Margulis, Sarnak, and others.

Counting lattice points can be viewed as a geometric formulation of the Diophantine problem of finding integer solutions to a system of equations and/or inequalities. Several techniques have been developed to address this problem, most of them rooted in either Harmonic Analysis or Model Theory. This talk will primarily survey the latter approach, highlighting recent developments in the problem of counting solutions to systems of homogeneous forms. Particular emphasis will be placed on applications of these methods to a range of questions arising from the work of Athreya, Margulis, Sarnak, and others.

The construction of normal numbers by concatenating function values has a long history in number theory. The references range from the first construction of Champernowne using the identity to show that \[0.1\,2\,3\,4\,5\,6\,7\,8\,9\,10\,11\,12\,\ldots\] is normal to base $10$; to the construction of Nakai and Shikawa using polynomials with real coefficients. In recent years functions that grow polynomially became of interest and they are also used for the construction of normal numbers. Considering subsequences like the primes those constructions should also yield normal numbers. For example the construction of Copeleand and Erd\H{o}s showing that \[0.2\,3\,5\,7\,11\,13\,17\,19\,23\,29\,\ldots\] is normal to base $10$. However, there are not so many proved examples involving polynomials in primes and in the present talk we want to shed some light on recent results and their connection with symbolic dynamical systems.

The classical technique of Hensel lifting describes the zeros of a polynomial map module $p^e$ using its zeros modulo $p$. The same technique fails in many aspects for non-polynomial maps. In this talk, we introduce a version of Hensel lifting for linear-exponential maps over a local ring of positive characteristic. This allows us to characterize zeros of linear-exponential maps over rings by reducing to the case of fields. Based by joint work with Doron Shafrir.

Bilu et al recently introduced the concept of twisted rational zeros to explain some phenomena regarding formulae for the p-adic valuation of linear recurrence sequences (LRS). They left some questions open; namely, given a twisted rational zero of an LRS, are there infinitely many primes p for which it is identified as (or avoids being) a p-adic zero? I will talk about recent work that answers these questions and gives connections to local-global principles for integer zeros of LRS. In particular, the answer to the questions above on twisted rational zeros and p-adic zeros may be applied to prove a local-global principle for simultaneous zeros of two coprime LRS, subject to the p-adic Schanuel conjecture.

I describe algorithms to linearise the dynamics of Frobenius acting on varieties over finite fields and discuss applications to point counting and factoring polynomials.

Suppose we have a deterministic finite-state transducer $\mathcal{A}$ and an infinite word $x$, and run $\mathcal{A}$ on $x$ to obtain an infinite word $\mathcal{A}(x)$. Which properties of $x$ are guaranteed to also hold for $\mathcal{A}(x)$? In this talk, we consider this preservation question for various well-known classes of words having desirable combinatorial properties, e.g., recurrent words, primitive morphic words, and words that admit factor frequencies. The celebrated Krohn-Rhodes theorem provides the framework for proving our preservation results, and our techniques are based on the ergodic theory of symbolic dynamical systems, i.e., shift spaces. This talk is based on joint work with Valérie Berthé, Herman Goulet-Ouellet, Toghrul Karimov, and Dominique Perrin.

In 1997, Vulakh characterized the Markoff spectra of Hecke groups of index $q\ge 3$ up to their first accumulation points. In this talk, we develop an expansion of real numbers using the Hecke groups and completely characterize the real numbers corresponding to the initial discrete parts of the Markoff spectra of the Hecke groups. This is joint work with Byungchul Cha.

We consider an inhomogeneous multiplicative analogue of the Lagrange spectrum, focusing on its combinatorial and symbolic structure. More precisely, we study the set \[ L(10;\eta) = \left\{ \limsup_{n\to\infty}\|\xi 10^n-\eta\| \mid \xi \in {\bf R} \right\}. \] As the parameter \(\eta\) varies, the relevant extremal order on infinite words changes, and this causes a bifurcation in the discrete part of the spectrum. The main role is played by explicit extremal words generated from the substitution \(\tau(1)=2\) and \(\tau(2)=211\). These words describe the isolated values below the first accumulation point. In particular, for a sequence of parameters tending to \(0\), the number of such isolated values is exactly \(k+2\), giving a staircase-type bifurcation in the discrete spectrum. In this way, the talk highlights a word-combinatorial mechanism behind a staircase-type bifurcation in an arithmetic spectrum.

Automatic sets are sets of natural numbers whose expansions in a fixed base are recognized by finite automata. Cobham's theorem describes the limitations of recognizability in two multiplicatively independent bases: a set recognizable in both must be ultimately periodic. This raises a related question: what can be said about the intersection of two sets, each recognizable in one of the bases? Sparse automatic sets provide a setting in which this question admits a finiteness theorem, together with quantitative bounds in terms of the bases and the accepting automata. This result connects the structure of sparse regular languages with Diophantine equations and also extends to higher dimensions.

The $k$-automatic sets, those whose base-$k$ representations form a regular language, are a well-studied class of subsets of the natural numbers. In this talk, we discuss our proof of the following dichotomy: given a $k$-automatic subset $X$ of the natural numbers, either $X$ is definable from Presburger arithmetic expanded by a predicate for the powers of $k$, or else the Presburger arithmetic expanded by $X$ itself defines any other $k$-automatic set.

For multiplicatively independent natural numbers $\alpha$ and $\beta$, Villemaire showed in 1992 that the first-order theory of Presburger arithmetic expanded with both Büchi predicates $V_\alpha$ and $V_\beta$ is undecidable, as it encodes multiplication. In recent years, Hieronymi and Schulz showed that Presburger arithmetic expanded with the weaker power predicates $\alpha^\mathbb{N} = \{\alpha^n: n \in \mathbb{N}\}$ and $\beta^\mathbb{N}$ is also undecidable, while Karimov et al. showed that the existential fragment of this theory is decidable. These results left open the natural problem of determining the decidability of the existential fragment of Villemaire's original expansion. We settle this question for coprime $\alpha$ and $\beta$. Specifically, we give a quantifier-elimination argument that proves the decidability of the existential fragment of $\mathsf{FO}(\mathbb{Z};<,+, V_\alpha, V_\beta)$.

Recent advances in artificial intelligence — from large language models capable of engaging with formal proofs to automated theorem-proving systems and AI-assisted conjecture generation — are beginning to reshape how mathematicians think about their discipline.
This session offers an open forum for mathematicians at all career stages to discuss the practical, intellectual, and professional implications of these tools. This session is structured as a facilitated conversation, drawing on participants' own experiences and perspectives to surface open questions, shared concerns, and emerging opportunities.

A linear dynamical system (LDS) is governed by a map $x \mapsto Mx$, where $x$ is a $d$-dimensional vector and $M$ is a $d \times d$ matrix. Various classical reachability problems of LDS are intimately connected to open problems in number theory, e.g. the Skolem Problem and the Positivity Problem. Recently, however, o-minimality has emerged as a complementary perspective on LDS. I will discuss the following two results which illustrate the basic techniques for analysing orbits of LDS.
(A) Given an LDS over $\mathbb{R}^d$, an initial point $s$, and a semialgebraic partition $T$ of $\mathbb{R}^d$, the coding of the orbit of $s$ with respect to $T$ is (ultimately) equal to a coding of a toral translation.
(B) For any o-minimal function $f$, the Birkhoff averages of $(f(M^n s))_n$ converge to a limit that can be expressed as an integral.

A special case of the Sarnak conjecture states that all substitutive words $a$ with zero entropy satisfy that \[ \sum_{n\leq N} a(n)\mu(n) = o(N), \] where $a(n)$ is the $n$-th letter of the word and $\mu(n)$ is the Möbius function. In 2017, Müllner showed that this holds in the case where $a$ is an automatic sequence and later Drmota, Müllner and Spiegelhofer proved it for the Fibonacci word. In this talk, I extend these results to certain morphic words and other classes of substitutive words.

Which real vectors have bounded Brun, Jacobi–Perron, or Selmer expansions, and how large is the set of such vectors?
For regular continued fractions, this question is classical. In higher dimensions, the corresponding sets are non-conformal iterated function systems, and much less is known about their size. As far as I know, the only result in this direction is the recent work of Berthé and Lee, who obtained bounds on the Hausdorff dimension using the Ostrowski skew-product.
I will present a tool for studying these sets: win-lose inductions, elementary inductions associated with labelled finite graphs that can describe all of these algorithms. A combinatorial condition on a subgraph yields both ergodicity and an explicit upper bound for the Hausdorff dimension of the fractal it defines. I will apply this framework to sets of vectors with bounded digits and discuss further directions.