PROGRAM, TITLES & ABSTRACTS

Program

[All the presentations will be held in "Sala Consiliare" of the Comune di Cogne] 

Wednesday 26th June



Thursday 27th June



TITLES AND ABSTRACTS


Marzio Mula 

Transcendence in ♠️ words, part 1: families of non-periodic words.

Abstract: 

A word on a set A is any string of letters – i.e. elements – in A. When A is a set of integers, any word w can be associated with the continued fraction whose partial quotients are the letters of w. It is conjectured that whenever A is a finite set of positive integers, all infinite non-periodic words on A correspond to transcendental numbers.

In the first part of this talk, we will survey some results that address special cases of this conjecture – namely, Thue-Morse words and quasi-periodic words. Later on, we will explore a more recent result by Bugeaud, which encompasses all these special cases and more, covering the set of so-called "♠️-words". This set is, in fact, transversal to the set of words on finite alphabets, and it also contains all words that can be produced by a finite automaton. Finally, we will see how similar questions arise in the context of p-adic numbers, i.e. when A is a suitable set of {p}-integers.




LAura Capuano

Transcendence in ♠️ words, part II: an analogue of a theorem of Bugeaud in the p-adic setting


Abstract: 

A result of Bugeaud asserts that, whenever a real number $\alpha$ has a continued fraction expansion whose partial quotients form a “♠️ word” (a notion introduced in the previous talk, which generalizes for example automatic sequences or Sturmian words), then $\alpha$ is either quadratic or transcendental. Bugeaud’s proof strongly relies on Schmidt Subspace Theorem, which is a celebrated result in Diophantine approximation that can be seen as a higher-dimensional analogue of Thue-Siegel-Roth theorem.  After an introduction to this result, we will see how this is applied in the proof of Bugeaud’s result, and how similar ideas can be used to apply the p-adic version of Schmidt Subspace Theorem due to Evertse and Schlickewei to deduce an analogue of Bugeaud’s result in the p-adic setting.  This is a joint work in progress with S. Checcoli, M. Mula and L. Terracini.


Sachi Hashimoto

Rational points on X_0(N)* when N is non-squarefree

Abstract: 

Let N be a non-squarefree number. The curve X_0(N)* is the quotient of X_0(N) by the full group of Atkin--Lehner involutions. In ongoing work with Samuel Le Fourn and Timo Keller, we aim to show that the rational points on X_0(N)* are CM points or cusps for N >> 0. Our strategy follows the work of Mazur, Momose, and Bilu--Parent for the families X_0(p) and X_0(p^r)+. We first show that the non-cuspidal rational points have bounded denominators, and then seek to obtain a bound on the integer points. In this brief overview, we discuss our current progress and the challenges we have encountered.



Denis Simon

An estimation of the degree of a subtorus

Abstract: 

A subtorus $T$ of $G_m^n$ is characterized by a sublattice $\Lambda$ of $Z^n$. The aim of this presentation is to compare the degree of $T$ as a subvariety of the projective space $P^n$ and the covolume of $\Lambda$.