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Multifractality in the evolution of vortex filaments

Event Category:
Applied Mathematics and Computation Seminar
Speaker:
Daniel Eceizabarrena
Institution:
UMass Amherst
Webpage:

Vortex filaments that evolve according the binormal flow are expected to exhibit turbulent properties. Aiming to quantify this, I will discuss the multifractal properties of the family of functions
$$
R_{x_0}(t) = \sum_{n \neq 0} \frac{e^{2\pi i ( n^2 t + n x_0 ) } }{n^2},
\qquad
x_0 \in [0,1],
$$
that approximate the trajectories of regular polygonal vortex filaments. These functions are a generalization of the classical Riemann's non-differentiable function, which we recover when $x_0 = 0$. I will highlight how the analysis seems to critically depend on $x_0$, and I will discuss the important role played by Gauss sums, a restricted version of Diophantine approximation, the Duffin-Schaeffer theorem, and the mass transference principle.

This talk is based on the recent article
https://arxiv.org/abs/2309.08114 in collaboration with Valeria Banica (Sorbonne Universite), Andrea Nahmodinfo-icon (University of Massachusetts) and Luis Vega (BCAM, UPV/EHU).

Tuesday, February 13, 2024 - 4:00pm
LGRT 1681