Coaches: Hanzhang Yun 员含章 (USTC)、 Muxi Li 李沐西 (USTC)

Location: A103

Course description:

In 1872, Felix Klein made a famous speech [1] on geometry, whose content is known as the Erlangen program in the history of mathematics. One significance of this program is that a geometry is defined to be the study of the invariant properties of geometric objects under a given transformation group. It had a great influence on the development of geometry and physics. In this course, we shall try to digest a big portion of this important document, that shall be learnt in a way accessible to modern language. Interesting enough, in §8.2 [1], entitled Analysis situs, Klein had foreseen the rising of a new branch of geometry, topology. Nowadays, topology has become a large topic of geometry, which has found spectacular applications in physics as well as other branches of science. Hence our course, centering around [1], has both old and new aspects of geometry. You may find some highlights of the course as below:

(i) Hilbert's axiomatic approach to geometry;
(ii) Motions of rigid bodies in E3, classification of quadratic surfaces;
(iii) Transformation groups and geometries: Klein's point of view;
(iv) Planar projective geometry and projective transformations;
(v) Topological spaces and topological transformations.

References

[1] F. Klein, A comparative review of recent researches in geometry, Bull. New York Math. Soc. 2 (1892-1893), 215-249. Available online.

Homeworks

Project 1
Project 2