Zusammenfassung für den Vortrag am 11.02.2020 (12:00 Uhr)

Arbeitsgemeinschaft ANGEWANDTE ANALYSIS

Daniel Eceizabarrena (Basque Center for Applied Mathematics, Bilbao)
Riemann’s non-differentiable function: a physical and geometric description

It is believed that in the 1860s Riemann proposed the function

\[R(x) = \sum_{n=1}^{\infty}\frac{\sin(n^2 x)}{n^2}, \qquad x \in \mathbb{R},\] as an example of a continuous but nowhere differentiable function, but no proof was given at the time. Since then, many authors have worked on its analytic properties, first trying to solve Riemann’s guess (proven a century later, in 1970) and studying further regularity properties afterwards.

However, recent results suggest that the generalisation of Riemann’s function

\[\phi(x) = \sum_{k \in \mathbb{Z}}\frac{e^{-4\pi^2ik^2x}-1}{-4\pi^2k^2}, \qquad x \in \mathbb{R}\] is a surprisingly precise approximation to a temporal trajectory in an experiment concerning vortex filaments. It can be argued then that it has also an intrinsic physical and geometric structure.

In this talk, I will present my work concerning geometric properties of \(\phi(\mathbb{R})\). We will see that its Hausdorff dimension is upper bounded by 4/3.