The quantum Hall effect (QHE) is a central topic in condensed matter physics, highlighting the role of topology in quantum mechanics. The fractional quantum Hall effect, driven by electron-electron interactions, leads to the formation of exotic quasi-particles with fractional charge and non-standard statistics. Recent experiments have revealed numerous fractional Hall states, particularly in bilayer systems like graphene or double quantum wells, where strong interlayer interactions and spin-orbit coupling introduce new complexities to the QHE landscape. This internship will focus on understanding the impact of spin-orbit coupling on integer QHE systems. The first phase of the project will explore how spin-orbit coupling modifies the wavefunctions and edge states of non-interacting electrons within a single Landau level, including the effects of trapping potentials. The second phase will examine Landau level crossings in bilayer systems and incorporate electron interactions using a mean-field Hartree-Fock approach. This project combines analytical and numerical methods to enhance our understanding of topological phases in quantum Hall systems. Possible PhD extensions of this work include studying transport and disorder effects, investigating light-matter coupling in QED cavities, and developing more precise numerical treatments to explore fractional phases.