Surprisingly, not much known about algebraic groups viewed as permutation groups (in their rational actions). The talk will survey some recent results in this area (joint work wth Gregory Cherlin).
A sample result: If an algebraic group G acts faithfully and primitively on an algebraic variety of dimension n, then dimension of G is bounded by a function of n.
(Easy examples show that the "primitiveness" assumption cannot be dropped.)
Our theory is developed in a wider context of the so-called omega-stable groups of finite Morley rank.
This class of groups includes algebraic groups over algebraically closed fields. Groups of finite Morley rank appear as "Galois groups" in model-theoretic algebra in approximately the same way as algebraic groups appear as "Galois groups" in differential algebra.