Seminar in Mathematics MAT 402,
Geometry and Algebra of homogeneous spaces

The classical Euclidean Geometry has lots of relatives. The closest of them are barely mentioned in the standard undergraduate curriculum, although they belong to the core Mathematics. The most known of them study spaces of constant curvature such as spheres, hyperbolic spaces, projective spaces. There are also geometries, in which the role of points are played by simple figures of other geometries (such as lines or circles). All these classical geometries are based on geometric transformations, which form large Lie groups acting transitively on the underlying spaces. The underlying spaces of these geometries are called homogeneous. Lie groups and homogeneous spaces are objects of fundamental importance in modern mathematics and physics.

In the seminar, we are going to study geometries of homogeneous spaces and Lie groups. In Lie groups, algebraic properties of elements define stratifications which are not yet well understood. This is, hopefully, where we will come to the front edge of the subject. The research component of the seminar is based on the reseach texts listed in the links.

The seminar will start with several introductory lectures. They will be followed by students' talks based on fragments of textbooks and original research papers. The theory that we will study provides various techniques for solving problems of different levels, and we will work on the problems.