Differentiable Manifolds

This master's course explores several facets of Differential Geometry and Mathematical Physics, focusing on the role of symmetries motivated by physical systems. It is aimed at students of a Master's course in Mathematics or Physics.

The course commences with an exploration of Differential Topology, which relates the shape of space with the underlying geometrical structure. Motivated by physical systems, instead of plunging into Riemannian Geometry, we will direct our attention to Lie theory and Symplectic Geometry. The study of Lie groups is motivated by their actions on sets and manifolds. Many of them are naturally attached to physical systems, such as the Lorentz group classically associated with General Relativity, or the Heisenberg group associated with quantum mechanics. Symplectic geometry provides a generalization of the classical (position, momentum) models of a physical system on a cotangent bundle. With the language of differential forms we can formulate Hamiltonian systems in an elegant geometrical language.

Throughout the course, attention will be given to some of the open problems within these fields and to motivating examples from physics.

Course guide