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.
