## Eceizabarrena, Daniel

The project of my thesis is based on the interaction between the Talbot effect and Riemann's non-differentiable function, a rich and surprising relationship based on the Vortex Filament and Schrödinger equations.

The **Talbot effect** is a microscopic optical phenomenon that takes place when light goes through an equidistant diffraction grating. Interference makes the light create a copy of the grating at a certain distance, while at almost every intermediate distance, the grating is reproduced at different and smaller scales.

The Vortex Filament Equation (VFE) and therefore the Schrödinger equation are a good tool to describe the Talbot effect. If a planar regular polygon is taken as the initial datum for the VFE, the curvature of the solution shows the same behaviour as light in Talbot’s experiment. Also, **Riemann’s non-differentiable function** can be found if one follows the time trajectory of one of the corners.

In this context, I try to understand why the Talbot effect can be described by means of the Schrödinger equation instead of the wave equation. Also, in view of the geometric role of Riemann’s function above, I analyse the geometric regularity and the dimension of its image in the complex plane, as well as other geometric properties. Besides, I am interested in the mathematical description of intermittency in the context of turbulence, and in its application to Riemann’s function.

I carry out this work in BCAM under the supervision of professor Luis Vega, supported by a Spain's Ministry of Education FPU grant in the 2015 call (grant no. FPU15/03078). Previously, I obtained the Master's Degree in Mathematics and Applications at Universidad Autónoma de Madrid (UAM) in 2016 and the Bachelor's Degree in Mathematics at the University of the Basque Country (UPV/EHU) in 2015.