A review of differential geometry methods in classical and quantum mechanics through quantization

This thesis is a presentation of differential geometry methods in non-relativistic areas of mechanics. In the quantum level, we investigate the representation theory formulation through Heisenberg's Lie group. In the classical realm, we discuss a symplectic geometry methods in Hamiltonian mechanics to abstract manifolds. Additionally, we study symmetries of motion in both these scopes, paying special attention to generalized versions of Noether's Theorem. Finally, we unite both of the quantum and classical descriptions in a simplified analysis of geometric quantization, inspired by the failure of the canonical approach. For the reader unfamiliar with the methods of mathematical physics, we provide two lengthy but holistic discussions on abstract algebra and differential geometry, paying special attention to their physical applications.