A geometric and physical study of Riemann's non-differentiable function

Riemann's non-differentiable function is a classic example of a continuous but almost nowhere differentiable function, whose analytic regularity has been widely studied since it was proposed in the second half of the 19th century. But recently, strong evidence has been found that one of its generalisation to the complex plane can be regarded as the trajectory of a particle in the context of the evolution of vortex filaments. It can, thus, be given a physical and geometric interpretation, and many questions arise in these settings accordingly. It is the purpose of this dissertation to describe, study and prove geometrically and physically motivated properties of Riemann's non-differentiable function. In this direction, a geometric analysis of concepts such as the Hausdorff dimension, geometric differentiability and tangents will be carried out, and the relationship with physical phenomena such as the Talbot effect, turbulence, intermittency and multifractality will be explained.Ministerio de Educación, Cultura y Deporte - FPU15/0307