Welcome coffee

Diophantine properties of Thue-Morse expansions

• Yann Bugeaud

The Thue-Morse word over the alphabet {a, b} is the fixed point starting with a of the morphism sending a to ab and b to ba, that is, it is the limit of the sequence of finite words a, ab, abba, abbabaab, abbabaabbaababba, … We survey the Diophantine properties of real numbers whose expansion in some integer base or whose continued fraction expansion is given by a Thue-Morse word (here, a and b are distinct positive integers). We also discuss the Diophantine properties of p-adic numbers whose Hensel expansion is a Thue-Morse word and of power series over a finite field F whose continued fraction expansion is a Thue-Morse word (here, a and b are distinct nonconstant polynomials with coefficients in F).

Discussion of future plans

Lunch

On the distribution of algebraic numbers and its generalization in Arakelov geometry

• Robert Wilms

I will talk about a new equidistribution result for the roots of polynomials with bounded Bombieri norm. This can be formulated and proven in a much broader setting for the zero sets of small sections of hermitian line bundles on arithmetic varieties in any dimension. I will discuss the main ingredients of the proof coming from geometry of numbers and from complex analysis. This is work in progress.

Coffee break

$C^1$-triangulations of semi-algebraic sets

• Christoph Brackenhofer

If one wants to treat integration of differential forms over semi-algebraic sets analogous to the case of smooth manifolds, it is desirable to have triangulations of semi-algebraic sets that are globally of class $C^1$. We will present a proof of the existence of such triangulations using the 'panel beating' method introduced by Ohmoto-Shiota (2017) and discuss possible generalizations.