The course delves further into general topology and introduces the basic notions of algebraic
and differential topology, focussing on the concept of differentiable manifold. Furthermore, the
elements of Riemannian geometry will be introduced as well.
The course, suitable to both curricula (didactic and applied) will be quite concrete and based
on examples also coming from other areas of mathematics.
General topology (continued). Separation. Quotients.
Fundamental group. Covering spaces.
De Rham's theory.
Curvature tensors (Riemann, sectional, Ricci, scalar).
Geodesics and their variational aspects.
Lie groups. Symmetric spaces.
Riemann surfaces and algebraic curves.
Vector bundles, Euler's class and number, Euler-Poincare' characteristic.
The Poincare'-Hopf theorem.
Written test, followed by an oral exam.