Differential Geometry is a branch of mathematics that studies spaces that can be described in a neighborhood of each point using a set of independent coordinates. This theory has its roots in the local theory of curves and surfaces in the Euclidean space, whose natural evolution is the notion of "manifold". Manifolds are ubiquitous both in mathematics and outside mathematics. In mathematics, they appear for example in the study of geometric aspects of PDEs (geometric analysis) and in differential topology. In physics, symplectic and Poisson manifolds are at the basis of Hamiltonian mechanics, pseudo-Riemannian manifolds are at the basis of Einstein's general theory of relativity, vector bundles and connections are at the basis of Yang-Mills theories (such as the ones describing the physics of elementary particles). Sub-Riemannian manifolds are used in control theory. Metric aspects of smooth manifolds are used in computer graphics and computer vision. This MOOC is a short introduction to smooth manifolds, where the student will learn the main definitions and the main theorems. Due to time and space constraints, some of the proofs are only sketched, but a meticulous student should be able to fill in the gaps in most of the proofs (and, in fact, this is a very good exercise). The lectures contain many examples that are essential to understand the theory. Every lecture begins with a not-too-technical video, that introduces the topic and gives some background and motivation.
- Docente: Francesco D'Andrea