Octahedrally and Icosahedrally Symmetric Solutions of the N-Vortex Problem on the Sphere

We investigate the octahedrally and icosahedrally symmetric solutions of the N-vortex problem on the sphere, for N = 24 and N = 60 respectively, in the case of equal vorticities. Such solutions are characterized by the property that the motion of the entire system can be recovered from that of a single vortex through the action of the octahedral and icosahedral symmetry groups, respectively. Using the construction of C. García-Azpeitia and L.C. García-Naranjo (J. Nonlinear Sci. 32 (2022)), which exploits the symmetries arising from rotations and permutations of the vortices, the problem is reduced to the analysis of a one-degree-of-freedom Hamiltonian system on the sphere. The original contributions of this thesis concern the explicit derivation and analysis of this reduced system yielding in particular a classification of the equilibrium and collision configurations with the prescribed symmetries. The computational complexity of the analysis is overcome by exploiting the associated discrete symmetries through the invariant theory of reflection groups and Palais' principle of symmetric criticality, and, in some cases, by resorting to numerical approximations.