A Pisot number is a real algebraic integer whose Galois conjugates are less than 1 in absolute value. In this survey lecture, we present various results inspired by the (substitutive) Pisot conjecture. This conjecture states that Pisot morphisms (i.e., morphisms acting on words whose expansion factor is a Pisot number) are conjectured to generate symbolic codings of the simplest dynamical systems, namely group translations on compact abelian groups, such as e.g. circle translations.

Suppose we have a deterministic finite-state transducer $A$ and an infinite word $x$, and run $A$ on $x$ to obtain an infinite word $A(x)$. Which properties of $x$ are guaranteed to also hold for $A(x)$? 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.

The Grothendieck period conjecture, a version of which was made famous by Kontsevich and Zagier, is something of a fantasy, suggesting that much of transcendental number theory must be subsumed by polynomial algebra over the rationals. Even in the limited cases where this conjecture is known to hold, translating the problem across these two mathematical fields and then solving the underlying algebra problem is highly elaborate. In 2025, with J. Ouaknine (MPI-SWS) and J. Worrell (Oxford), we made this translation algorithmic for $\overline{\mathbb{Q}}$-linear identities between $1$-periods. These are the period integrals of single-variable algebraic functions, of which $\pi$, $\log 2$, and elliptic integrals are the first examples.
For $2$-periods and beyond, what is known indicates that we are not even close to having the theoretical foundations for a period conjecture. There is, however, a well-known source of integer-linear relations between certain $2$-periods, due to Lefschetz in the 1920s, which was itself the inspiration for the period conjecture. There is a curious asymmetry here: integral relations are easy to guess by numerical approximation, but cannot be proven in this way. Meanwhile, although Lefschetz's geometric insight indicates a purely mechanical route to a proof, carrying out this "routine" calculation is practically impossible. In joint work with Edgar Costa (MIT), we are able to go back and forth repeatedly between these two sides, improving the situation at each pass, until a proof is found. The process is systematic and rigorous, and we have showcased it in hundreds of examples, though the total CPU time has long exceeded the lifespan of any mortal.

One topic of study in mathematical logic concerns the first-order theories of the natural numbers with various algebraic structures thereupon. Questions answerable in this topic include which such theories are decidable (i.e. recursive) and what the definable sets and relations are in each theory. We will present an overview of the existing knowledge in this direction as it applies to the specific context of k-automatic structure, which is the structure given by regular properties of the base-k representations of numbers, including previously known results as well as results proven by the speaker and their coauthors.

A 1965 problem due to Danzer asks whether there exists a set with finite density in Euclidean space (i.e. « not containing too many points ») intersecting any convex body of volume one. A suitable weakening of the volume constraint leads one to the (much more recent) problem of constructing dense forests. Progress towards these problems have so far involved a very wide range of areas in mathematics (including number theory, computer science, ergodic theory, geometry and harmonic analysis). After surveying some of the known results related to the Danzer Problem and to the construction of dense forests, the talk will present some new constructions.

I will provide an introduction to a collection of results known as "arithmetic equidistribution theorems" that describe the geometry of points (in an algebraic variety) with small arithmetic complexity (defined by a notion of height with good properties). I will give simple examples and mention a few fancy versions that have been useful in different settings, with applications to the study of algebraic dynamical systems. Quantitative versions also exist.

In this talk we'll discuss some sequences of rational numbers associated to iteration of algebraic functions. These include well-known sequences such as Fibonacci and other linear recurrence sequences, more rapidly growing sequences arising from orbits of points under iteration of morphisms, and rather mysterious sequences arising as algebraic degrees of iterates of birational maps. I'll introduce the techniques that arithmetic dynamicists use to tackle these problems, and focus on the many still-open questions about these sequences.

We show that for non-degenerate integer linear recurrence sequences $(u_n)_n$ with exactly two simple dominant roots, the first-order theories of $\langle \mathbb{N}; <,\, n \mapsto \max\{0, u_n\} \rangle$ and $\langle \mathbb{N}; +,\, \{u_n \ge 0\} \rangle$ are undecidable. Our approach is to use Diophantine approximation to show that such $(u_n)_n$ contain, in a specific sense, all finite sequences over $\mathbb{N}$, an idea that we borrow from the proof of Hieronymi and Schulz that the first-order theory of $\langle \mathbb{N}; <, +, \{2^n\}, \{3^n\} \rangle$ is undecidable. In a similar way, we harness a contemporary result about quasi-randomness in the values of the Ramanujan tau function to show that the first-order theory of $\langle \mathbb{N}; <,\, n \mapsto |\tau(n)| \rangle$ is undecidable.

Solvency games, introduced in 2008 by Berger, Kapur, Schulman, and Vazirani, are a class of very simple one-player gambling problems in which the player's sole objective is to remain solvent, i.e., not go bankrupt. Although memoryless deterministic optimal strategies are known always to exist, in general we do not know how to compute them, nor whether they are unique and/or ultimately periodic. For an important subclass of solvency games, such questions are tightly connected to a particular class of recurrence sequences which we introduce and discuss.

In this talk, I will provide a general context where discrepancy for sequences appears, its relations with word combinatorics and symbolic dynamics. We will go through different tools such as Ostrowski numeration, bounded remainder sets, balancedness, Rauzy fractals.

Consider the triangular grid; put a lamp at each vertex, and a switch at each upwards-pointing triangle. You begin with a lone lamp at the origin, and wish to push switches (which toggle the three adjoining lamps) in such a manner that a ball of radius n around the origin is cleared. Show that you will, at one moment, have at least $\log_2(n)$ lamps on.
This simple puzzle in in fact related to three topics:
(1) models in statistical physics for glass (which do not undergo any phase transition as the temperature T goes to 0, but rather whose viscosity increases continuously as $\exp\!\left(\frac{1}{T^2}\right)$;
(2) ergodic theory, and the almost-mixing properties of symbolic dynamical systems
(3) combinatorial group theory, and more precisely the complexity of the word problem in certain groups.
I will try to explain the connections between these areas, and the main ideas behind the combinatorial statements and their application to statistical physics. This is joint work with Ivailo Hartarsky and Ivan Mitrofanov

A sequence $(u_n)$ is holonomic (or P-finite) if it satisfies a recurrence $$ a_d(n)\,u_{n+d} = a_{d-1}(n)\,u_{n+d-1} + \cdots + a_0(n)\,u_n, $$ with polynomial coefficients $a_i(n)$. In the case where the coefficients are constants, we call $(u_n)$ a C-finite sequence, and in this setting, the celebrated Skolem–Mahler–Lech theorem states that the set of zeros is a union of finitely many arithmetic progressions and a finite set. This has since been refined to give effective upper bounds on the number of arithmetic progressions and the size of the finite set *based only on the order $d$*. In the world of P-finite sequences, an analogue of the Skolem–Mahler–Lech theorem is not known in general. However, a special case is known (since 2012) for P-finite sequences where the first and last coefficients are non-zero modulo a prime $p$. In this talk, I will discuss recent/ongoing work in which we refine this result to give effective upper bounds on the number of arithmetic progressions and the size of the finite set based only on the order $d$, the degrees of the coefficients $a_i(n)$, and the prime $p$*.

A short introduction to Presburger arithmetic is given, along with a short history of results about undecidability of arithmetic theories. We mention results of Gödel, Turing, Matiyasevich, Buchi and Xiao. Building on the result of Xiao we delve into finding lower bounds for undecidability of Presburger arithmetic extended with a square predicate in a fixed number of variables. We then show decidability of Presburger arithmetic extended with polynomial predicates in a single variable, introducing simultaneous Pell equations and relative density along the way.We discuss related hard problems standing in the way of generalization.

TBA.

After the Hartmanis–Stearns conjecture, the transcendence of numbers with $\beta$-expansions in algebraic bases $\beta$ related to morphic sequences has been studied. In 2024, Pavol Kebis, Florian Luca, Joël Ouaknine, Andrew Scoones, and James Worrell proved that such expansions yield transcendental numbers or lie in $\mathbb{Q}(\beta)$ when given by so-called echoing sequences. In 2025, they proved such expansions yield transcendental numbers when the echoing sequences hold strong echoing condition. They also showed that Arnoux–Rauzy sequences are echoing and $k$-bonacci sequences are strong echoing. We extend these results to $S$-adic sequences such that the associated $S$-adic shift has purely discrete spectrum and the condition PRICE of Berthé, Steiner and Thuswaldner (2019) holds. Furthermore, we show that when such $S$-adic shift is dendric, the sequence in the $S$-adic shift is strong echoing. Our approach builds upon the geometrical interpretation of these sequences through their Rauzy fractals. This is joint work with Wolfgang Steiner.

TBA.

A Carmichael number is a composite positive integer $N$ which behaves like a prime number with respect to Fermat’s Little theorem; that is $a^N - a$ is a multiple of $N$ for all integers $a$. It is known that there are infinitely many such numbers. In this talk, we will explore such numbers which have the form $2^n k + 1$ for some odd integer $k$. The questions we can ask are. Fix $k$. What can we say about the numbers $n$ such that $2^n k + 1$ is Carmichael? Or, what can we say about the odd positive integers $k$ such that $N = 2^n k + 1$ is a Carmichael number for some positive integer $n$? We will give some partial answers to these questions.