Low dimensional classical groups and their geometries

The finite simple classical groups form a major part of finite simple groups, and their geometries play a crucial role in understanding their maximal subgroup structures. They play an important role in the applications of O’Nan-Scott Theorem and the classification of finite simple groups to various group theoretical problems and combinatorial problems. There are also important substructures in their associated polar spaces that have close connections to other branches of mathematics. This introductory minicourse focuses on the low dimensional classical groups and their geometries. We shall define those classical groups, examine their subgroup structures and study important substructures of finite classical polar spaces.

Outline:
Lecture 1. Projective lines and conics
Lecture 2. Klein correspondence and Plucker coordinates
Lecture 3. Polarities and classical polar spaces
Lecture 4. Rank 2 polar spaces and generalized quadrangles
Lecture 5. Intriguing sets, ovoids and spreads
Lecture 6. Tensor algebras and related modules
Lecture 7. Clifford algebras and spin modules
Lecture 8. The maximal subgroups of classical groups