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

Résumé

Riemann's non-differentiable function is a classic example of a continuous but almost nowheredifferentiable function, whose analytic regularity has been widely studied since it was proposedin the second half of the 19th century. But recently, strong evidence has been found that one ofits generalisation to the complex plane can be regarded as the trajectory of a particle in thecontext of the evolution of vortex filaments. It can, thus, be given a physical and geometricinterpretation, and many questions arise in these settings accordingly.It is the purpose of this dissertation to describe, study and prove geometrically and physicallymotivated properties of Riemann's non-differentiable function. In this direction, a geometricanalysis of concepts such as the Hausdorff dimension, geometric differentiability and tangentswill be carried out, and the relationship with physical phenomena such as the Talbot effect,turbulence, intermittency and multifractality will be explained.