Topology and Geometry

Syllabus

The goal of the lecture is to give good notions of "curved spaces", as well as tools to study and work with these spaces.

Elements of general topology We will study topological spaces obtained as quotients, and by gluing together spaces. We will also study the notion of homotopy (continuous deformation), retract. The fundamental group will be introduced as a tool to distinguish some spaces. Basics of Morse theory.

Elements of differential geometry. Notions of differential (sub)manifolds. Local charts. Tangent and cotangent spaces. Vector fields (differential equations on submanifolds). Notions of differential forms and integration on submanifolds.

Pré-requis

Basis of topology of metric spaces and differential calculus, as generally seen in a bachelor degree.