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

Geometric differentiability of Riemann's non-differentiable function

Riemann’s non-differentiable function is a classic example of a continuous function which is almost nowhere differentiable, and many results concerning its analytic regularity have been shown so far. However, it can also be given a geometric interpretation, so questions on its geometric regularity arise. This point of view is developed in the context of the evolution of vortex filaments, modelled by the Vortex Filament Equation or the binormal flow, in which a generalisation of Riemann’s function to the complex plane can be regarded as the trajectory of a particle. The objective of this document is to show that the trajectory represented by its image does not have a tangent anywhere. For that, we discuss several concepts of tangent vectors in view of the set’s irregularity.Ministerio de Educación, Cultura y Deporte - FPU15/0307

On the Hausdorff dimension of Riemann's non-differentiable function

Recent findings show that the classical Riemann's non-differentiable function has a physical and geometric nature as the irregular trajectory of a polygonal vortex filament driven by the binormal flow. In this article, we give an upper estimate of its Hausdorff dimension. We also adapt this result to the multifractal setting. To prove these results, we recalculate the asymptotic behavior of Riemann's function around rationals from a novel perspective, underlining its connections with the Talbot effect and Gauss sums, with the hope that it is useful to give a lower bound of its dimension and to answer further geometric questions

Some geometric properties of Riemann’s non-differentiable function

Riemann’s non-differentiable function is a celebrated example of a continuous but almost nowhere differentiable function. There is strong numeric evidence that one of its complex versions represents a geometric trajectory in experiments related to the binormal flow or the vortex filament equation. In this setting, we analyse certain geometric properties of its image in C. The objective of this note is to assert that the Hausdorff dimension of its image is no larger than $4/3$ and that it has nowhere a tangent.FPU15/03078 - Ministerio de Educación, Cultura y Deporte

Pointwise Convergence over Fractals for Dispersive Equations with Homogeneous Symbol

We study the problem of pointwise convergence for equations of the type $i\hbar\partial_tu + P(D)u = 0$, where the symbol $P$ is real, homogeneous and non-singular. We prove that for initial data $f\in H^s(\mathbb{R}^n)$ with $s>(n-\alpha+1)/2$ the solution $u$ converges to $f$ $\mathcal{H}^\alpha$-a.e, where $\mathcal{H}^\alpha$ is the $\alpha$-dimensional Hausdorff measure. We improve upon this result depending on the dispersive strength of the symbol. On the other hand, we prove negative results for a wide family of polynomial symbols $P$. Given $\alpha$, we exploit a Talbot-like effect to construct regular initial data whose solutions $u$ diverge in sets of Hausdorff dimension $\alpha$. However, for quadratic symbols like the saddle, other kind of examples show that our positive results are sometimes best possible. To compute the dimension of the sets of divergence we use a Mass Transference Principle from Diophantine approximation theory

Counterexamples for the fractal Schrödinger convergence problem with an Intermediate Space Trick

We construct counterexamples for the fractal Schrödinger convergence problem by combining a fractal extension of Bourgain's counterexample and the intermediate space trick of Du--Kim--Wang--Zhang. We confirm that the same regularity as Du's counterexamples for weighted $L^2$ restriction estimates is achieved for the convergence problem. To do so, we need to construct the set of divergence explicitly and compute its Hausdorff dimension, for which we use the Mass Transference Principle, a technique originated from Diophantine approximation.FJC2019-039804-

El efecto de Talbot: de la óptica a la ecuación de Schrödinger

El objetivo de este artículo es dar a conocer un bello efecto óptico que se denomina efecto de Talbot. Primero, describiremos el fenómeno y comentaremos su descubrimiento a mediados del siglo XIX. A continuación, analizaremos las razones del fenómeno y lo justificaremos mediante algunos cálculos apoyados en las leyes físicas que gobiernan la propagación de ondas, así como por medios analíticos. Por último, veremos que este efecto se reproduce en algunas soluciones de la ecuación de Schrödinger, que muestran el mismo comportamiento que la luz.Formación de profesorado universitario FPU15/03078, Ministerio de Educación, Cultura y Deporte