Torsors

After recalling the notion of algebraic groups, I will introduce their torsors (also known as principal bundles). While the torsors can be viewed as vector bundles with additional structures, their scope is much wider. For example, the theory of quadratic forms is essentially equivalent to the theory of torsors under the orthogonal groups O(n).

After giving examples, I will discuss basic properties of the torsors. Then I will briefly discuss the role of torsors in the geometric Langlands program and formulate a particular case of an old conjecture of Grothendieck and Serre that was proved by I. Panin and the speaker 11
years ago.

This talk will be paced much slower than the previous talks, and some results will actually be proved. For the most of the talk I will assume only basic knowledge of algebraic geometry (e.g., complex algebraic varieties and the Zariski topology)