Syllabus
- Euclid's Elements (their axiomatic structure and implicit presuppositions),
- the discovery of space in Renaissance painting,
- the birth of projective geometry in the 17th century,
- the geometry of the projective plane,
- the geometry of Lobachevski,
- Beltrami's model and its metric,
- Klein's Erlanger program and the classification of geometries,
- basic notions of algebraic topology.
Annotation
The aim of the seminar is to outline the main changes of the notion of space in the development of geometry from Euclid to the birth of algebraic topology at the dawn of the twentieth century. It attempts to offer a broader perspective on the various areas of classical geometry which the students are been taught in their courses during their study of mathematics.
Special seminars are devoted to Euclid's Elements (both their axiomatic construction as well as their implicit presuppositions), to the discovery of space in the Renaissance, to the birth of projective geometry in the 17th century, to the geometry of the projective plane (its non-orientability, one sidedness, as well as the duality of points and straight lines) and to projective coordinates. Then follow seminars devoted to non-Euclidean geometry, to Beltrami model and its metric, to Klein's Erlanger program, and to the classification of geometries. The final third series of seminars is devoted to fundamental notions of algebraic topology in Riemann and Poincare (homotopy, homology, and the fundamental group). The exposition is based on classical texts and it is rather informal and intuitive.