Course Content (Syllabus)
Affine spaces and the affine group. Projective spaces: charts, their topology. Group actions and examples. Projective maps (homographies) and the projective group. Projective bases, they determine unique homography, examples. Projective subspaces, independence. Theorems of Pappus and Desargues, proofs. Perspectives. Cross ratio. Duality for vector spaces, annihilator, pencils. Projective quadrics.
Spherical and elliptic geometry: area, angle, Girard’s formula. Intrinsic metric.
Introduction to hyperbolic geometry: the three geometries, need for models of hyperbolic space. The hyperboloid model, elementary geometry and trigonometry, the projective model, other models (Klein, Poincaré, half-space).