**Prelectura de Tesis **

**Raquel Sánchez Cauce**

**Fecha y hora: 24 de septiembre de 2018 a las 11:00h**

**Lugar: Aula 520, modulo 17 **

** Título: Differential Galois Theory for some Spectral Problems **

** Conferenciante: Raquel Sánchez Cauce**

**Directores de tesis: Juan José Morales Ruiz (Dpto. de Matemática Aplicada. Universidad Politécnica de Madrid) y María Ángeles Zurro Moro (Dpto. de Matemáticas. Universidad Autónoma de Madrid)**

**Resumen: In this talk we will introduce the Picard-Vessiot Theory for integrable systems and the Darboux transformations. First, we will present our results on the differential Galois groups for Ablowitz-Kaup-Newell-Segur systems, which are an important kind of integrable systems depending on a spectral parameter $lambda$. **

** Next we will focus on the Schrödinger equation $(-partial ^{^2}+u)psi=-lambda^{^2} psi$ associated to the Korteweg de Vries hierarchy (KdV hierarchy for short). We will show the algebraic structure of the fundamental matrices for the Schrödinger equation with potential $u$ in a fixed family of KdV rational potentials. As a by product, we will obtain the differential Galois groups associated with the mentioned spectral problem. We will also compute non trivial examples in the $1+1$ dimensional case using SAGE.**

Moreover, we will establish the deep relationship between the singularities of the spectral curves, the Darboux transformations and the fundamental matrices for the KdV hierarchy.

Secondly, we will present a family of rational complex potentials $u$ depending on a parameter. We will show that these functions are KdV potentials and compute fundamental matrices for the corresponding Schrödinger equation.

Finally, we will use Darboux transformations for studying orthogonal differential systems from a galoisian point of view. Here the techniques of tensor products of differential systems are essential tools. Explicit formulas for these matrix Darboux transformations are computed using Maple.