פעילויות השבוע

In this talk, we study the density properties in the real line of oscillating sequences of the form $( g(k) \cdot F(kα) )_{k \in \mathbb{N}}$, where $g$ is a positive increasing function and $F$ a real continuous $1$-periodic function. This extends work by Berend, Boshernitzan and Kolesnik who established differential properties on the function F ensuring that the oscillating sequence is dense modulo 1.

More precisely, when $F$ has finitely many roots in $[0,1)$, we provide necessary and sufficient conditions for the oscillating sequence under consideration to be dense in $\mathbb{R}$. All the related results are stated in terms of the Diophantine properties of $α$, with the help of the theory of continued fractions.

In 1959 Gerhard Ringel posed the following problem which remained open for over 60 years. Suppose we are given a finite family $\C$ of circles in the plane no three of which are pairwise tangent at the same point. Is it possible to always color the circles with five colors so that tangent circles get distinct colors.

When the circles are not allowed to overlap (i.e., the discs bounded by the circles are pairwise interiorly disjoint) then the number of colors that always suffice is four and this fact is equivalent to the Four-Color-Theorem for planar graphs.

We construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number. Moreover, no two circles are internally tangent and no two circles are concentric. This provides a strong negative answer to Ringel‘s 1959 open problem. The proof relies on a (multidimensional) version of Gallaiӳ theorem with polynomial constraints, which we derive using tools from Ramsey-Theory.

Joint work with James Davis, Chaya Keller, Linda Kleist and Bartosz Walczak

בכל קופסא של דגני בוקר ישנו קופון. יש $n$ סוגים של קופונים. הקופונים שווי שכיחות. כמה קופסאות יש לקנות בממוצע על מנת להשיג לפחות קופון אחד מכל סוג?

הבעייה ידועה כבעיית אוסף הקופונים. היא הוצגה כבר ע“י דה-מואבר לפני יותר מ-300 שנה.

נציג מספר תוצאות המתייחסות לבעייה ולואריאנטים שלה וכן מספר שימושים.

Ribet‘s lemma is an algebraic statement that Ribet used in his proof of the converse of Herbrand‘s theorem. Since then various generalisations of Ribet‘s lemma have been found, with arithmetic applications. In this talk I will discuss a joint work with Ariel Weiss in which we show that two measures of reducibility for two dimensional representations over a DVR are the same, thus answering a question of Bellaiche and Cheneveier, and deducing from it a particular generalisation of Ribet‘s lemma. An interesting feature of the proof is that it applies to both the residually multiplicity-free and the residually non-multiplicity-free cases. I will discuss an application to a local-global principle for isogenies of elliptic curves.