Évènements

In this talk, we will study a class of metric spaces that satisfy a strengthened version of the triangle inequality, known as metric spaces with small rough angles (SRA), and that capture the idea that all metric angles determined by triples of points are, in a certain sense, small. We will explore different scenarios in which an arbitrary metric space may fall into one of two possibilities: either it does not contain sufficiently large subsets satisfying the SRA condition, or it admits arbitrarily large subsets that do satisfy this property. We will also see that these conditions are particularly relevant for investigating whether self-contracted curves, a class of curves that provides a natural framework for the study of convex gradient-type dynamical systems, have finite length.

We will discuss recent progress on several conjectures of Erdős and collaborators concerning the arithmetic function ω(n), including a conjecture of Erdős and Straus on long strings of integers with few prime factors, Erdős’s irrationality conjecture for a series involving ω(n), and the Erdős–Pomerance–Sárközy conjecture on the frequency of solutions to ω(n)=ω(n+1). A common theme is the interplay between probabilistic methods, sieves, and quantitative correlation estimates for multiplicative functions. I will outline how these tools allow us to resolve the first two conjectures and to verify the third for almost all values of x. This is based on joint work with Terence Tao.