The main purpose of the course is to develop a unified approach to the study of different branches of modern geometry, such as Euclidean, spherical, hyperbolic, elliptic, and projective geometry. This approach is based on the fundamental idea going back to F. Klein, who regarded geometry as a set endowed with a group action satisfying certain additional conditions. This point of view will enable us to study uniformly not only the geometries listed above but also related objects such as tilings and kaleidoscopes, as well as the Platonic solids and their higher‑dimensional analogues. No prerequisites.
Syllabus
- The Erlangen program: Klein geometry, elements of group theory, examples of Klein geometries; toy geometries, discrete geometries, tilings, and kaleidoscopes.
- Finite geometries: the Platonic solids and their classification; regular polytopes in higher dimensions.
- Non‑Euclidean geometry I: hyperbolic plane and hyperbolic space, Poincaré and Cayley–Klein models, hyperbolic trigonometry.
- Non‑Euclidean geometry II: spherical geometry, Riemann’s elliptic geometry, models of elliptic geometry.
- Projective geometry: projective duality in the plane and in space; theorems of Desargues, Pappus, and Pascal; the viewpoint that projective geometry underlies many other geometries.
- Geometries over finite fields: projective and affine geometries over finite fields.
Textbooks
- A. B. Sossinsky, Geometries.
- M. Berger, Geometry.
- H. S. M. Coxeter, Introduction to Geometry.