Grups de trenes, representació de Burau i categorificació de Khovanov-Seidel

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2023, Director: Ricardo García López[en] The braid group, together with its representations, is a fascinating mathematical structure, studied from different fields, such as group theory, topology... Moreover, it is a theory that extends beyond itself, with relations that go from the theory of knots and their invariants to concepts of theoretical physics. The main objective of the paper is the introduction of the notion of the braid group, the Burau representation and a categorification of it. We will begin by presenting braids as a mathematical structure and the different ways of interpreting the group they form. Then, we introduce the non-reduced and reduced Burau representations. This family of representations is faithful for $n4$, it is unknown if it is faithful for $n=4$. In this work, the case $n=5$ is not studied. Finally, the Seidel-Khovanov categorification of the Burau representation is presented, which, curiously, is faithful for all $n$

Network geometry and the SIS epidemic model

Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2022-2023, Tutor: Marián BoguñáWe give a short introduction to network geometry and its relation to the SIS epidemic model. Approximations and simulations are provided for four synthetic networks and two real-world networks. We compare the quality of the different mean-field approximations to the simulations. Finally, we study the effect of the nodes’s angular distribution in the hidden metric space on the prevalence of the epidemic