Given $n$ multiplicatively independent rational functions $f_1, \ldots, f_n$ with rational coefficients, there are at most finitely many complex numbers $\alpha$ such that $f_1(\alpha), \ldots, f_n(\alpha)$ satisfy two independent multiplicative relations. This was proved independently by Maurin and by Bombieri, Habegger, Masser and Zannier, and it is an instance of more general conjectures of unlikely intersections over tori made by Bombieri, Masser and Zannier and independently by Zilber. We consider a positive characteristic variant of this problem, proving that, for sufficiently large primes, the cardinality of the set of $\alpha \in \mathbb F_p$ such that $f_1(\alpha), \ldots, f_n(\alpha)$ satisfy two independent multiplicative relations with exponents bounded by a constant $K$ is bounded independently of $K$ and $p$. We prove analogous results for products of elliptic curves and for split semiabelian varieties $E^n \times \mathbb G_m^n$. This is a joint work with F. Barroero, L. Mérai, A. Ostafe and M. Sha.

A field of algebraic numbers has the Northcott property (N) if it contains only finitely many elements of bounded absolute logarithmic Weil height. While for number fields property (N) follows immediately by Northcott's theorem, to decide whether it holds for infinite extensions of the rationals is, in general, a difficult problem. This property was introduced in 2001 by Bombieri and Zannier, who, among other results, investigated its validity for fields satisfying certain local conditions. In this talk I will present some results obtained in this context with Arno Fehm.

We discuss progress towards conjectures of Bertrand and Villegas on arithmetic properties of subgroups of units of a number field, suggesting that one can expect lower bounds interpolating from the question of Lehmer (one dimensional subgroups) all the way up to lower bounds for the regulator (full unit group). This is a joint work with F. Amoroso.

The method that gave the Schinzel--Zassenhaus conjecture did not yield anything of note on the deeper problem: whether or not the Salem numbers may get arbitrarily near to 1. Here the bound obtained is rather poorer than Dobrowolski's bound, which remains the state of the art. Instead, I will explain how effectively the SZ argument does also prove something new and sharp about the Salem case. Namely, sharpening over a theorem of Mahler, I will detail a proof of the qualitatively best-possible (that is: exponential in minus the sum of the degrees) separation bound between two Salem numbers lying in a fixed interval. The talk will then be concluded by results and questions about counting the Salem ensemble in a fixed interval.

The starting point of this talk is Mahler's upper bound for the discriminant of a complex polynomial in terms of its degree and Mahler measure. In the same 1964 paper he bound from below the distance between two roots of a squarefree polynomial. Later, Mignotte found a lower bound for the product of distances between pairs of roots and Schönhage considered a single triple of roots. I will present an inequality that involves general tuples of roots. The inequality can be used to bound from above the transfinite diameter of sets that arise in Dimitrov's recent proof of the Schinzel-Zassenhaus Conjecture.

Maurin's Theorem of 2008 is the definitive result about multiplicative relations on algebraic curves defined over $\overline{\bf Q}$. It has been generalized to curves over $\bf C$. Here we consider analogues in characteristic $p$. In 2014 I proved a version for ${\bf G}_{\rm m}^3$, and in 2017 with Brownawell a version for additive ${\bf G}_{\rm a}^3$ where the two relations have the form $\alpha_1 x_1+\alpha_2 x_2+\alpha_3 x_3=0$ for $\alpha_1,\alpha_2,\alpha_3$ in the Frobenius ring ${\bf F}_p[{\cal F}]$ with ${\cal F}x=x^p$. I describe our recent work (also for ${\bf G}_{\rm a}^3$) on the Carlitz ring ${\bf F}_p[{\cal C}]$ now with ${\cal C}x=tx+x^p$, also with an arbitrary field of definition. We also have results for varying $p$; for example there are exactly 23 Carlitz roots of unity whose reciprocals are roots of unity.

The Mahler measure of an algebraic number is again an algebraic number. Hence, taking the Mahler measure defines a dynamical system on the set of algebraic numbers. We will study the possible size of the (forward) orbit of elements in this dynamical system. In particular, we will discuss the problem which number fields contain algebraic numbers with an infinite orbit. We can completely solve this problem for abelian number fields and number fields with a symmetric Galois group over the rationals. This is joint work with P. Fili and M. Zhang.

In this talk, I will present recent results obtained with St\'ephane Fischler concerning the values taken at algebraic points by families of $G$-functions. These results are dual in some sense to a classical result of Chudnovsky on the subject. Our results hold not only for points inside the disk of convergence of a given $G$-function (the usual situation), but also for points in a suitable star-shaped domain at the origin to which the $G$-function can be extended.

Bernstein's theorem allows to predict the number of solutions of a system of Laurent polynomial equations in terms of combinatorial invariants. When the coefficients of the system are algebraic numbers, one can ask about the height of these solutions. Based on an on-going project with Roberto Gualdi (Regensburg), I will explain how one can approach this question using tools from the Arakelov geometry of toric varieties.

I will give an overview of results on the essential minimum of a subvarieties V of a torus and shortly explain how Francesco and me were bounding the degrees of the closure of the small points in V.
I will then present some applications and related results for the closure of the torsion.

After Apéry's pioneering result on the irrationality of $\zeta(3)$, Beukers found a different and simpler construction of Ap\'ery's sequences of rational approximations to $\zeta(2)$ and $\zeta(3)$ based upon the expressions of these constants as periods. I will give a survey on several qualitative and quantitative results of irrationality of periods proved since.

The additive complexity of an $m\times n$ matrix $A$ with coefficients in a field $F$ is defined as the minimum number of additions sufficient to compute some nonzero multiples of the $m$ linear forms in $n$ variables defined by $A$, from the variables and constants in $F$. A generic matrix $A$ has maximal additive complexity $m(n-1)$. We produce explicit matrices with integer entries having maximal complexity.

We shall consider families of tori $\mathbb G_m^n$, considered over a base curve $B$, and a subvariety $V$ of the total space. Given a finitely generated subgroup $\Gamma$, defined over $k(B)$, we shall consider the problem of bounding the height of the $b\in B$ when some element of $ \Gamma$ lies on $V$. This has several applications. The results are object of joint work with Amoroso and Masser.