Day 1 (Freiburg)
lundi 21 mars 2022 -
09:30
lundi 21 mars 2022
09:30
Welcome coffee
Welcome coffee
09:30 - 10:30
10:30
Diophantine properties of Thue-Morse expansions
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Yann Bugeaud
Diophantine properties of Thue-Morse expansions
Yann Bugeaud
10:30 - 11:15
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).
11:30
Discussion of future plans
Discussion of future plans
11:30 - 12:00
12:00
Lunch
Lunch
12:00 - 14:00
14:00
On the distribution of algebraic numbers and its generalization in Arakelov geometry
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Robert Wilms
On the distribution of algebraic numbers and its generalization in Arakelov geometry
Robert Wilms
14:00 - 14:45
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.
14:45
Coffee break
Coffee break
14:45 - 15:30
15:30
$C^1$-triangulations of semi-algebraic sets
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Christoph Brackenhofer
$C^1$-triangulations of semi-algebraic sets
Christoph Brackenhofer
15:30 - 15:50
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.