Regularization of bi-interpretations (with examples related to algebraic groups)

Roughly speaking, an interpretation is a way to define one algebraic structure inside another using first order logic, and a bi-interpretation of two structures is a pair of mutual interpretations that are inverse to each other in a certain sense. In general, a bi-interpretation may use parameters from the corresponding structures, which can be a problem for certain applications. Such applications often require a regular bi-interpretation. In the first part of the talk we discuss a general approach to turning a bi-interpretation with parameters into a regular one. It turns out that a certain connection between automorphism groups of the structures is required. In the second part of the talk we discuss how recent results on abstract automorphisms of certain algebraic groups may be applied to establish a regular bi-interpretation between a commutative ring and an algebraic group over that ring.