ACFA is the model companion of the theory of a field endowed with a distinguished endomorphism. This theory has been extensively studied by Chatzidakis and Hrushovski. Notably, it was shown that any non-principal ultraproduct of algebraically closed fields with powers of the Frobenius map gives rise to a model of ACFA.

In this talk, I will discuss the model theory of pairs of ACFA. In particular, we will give an axiomatization of those pairs in which the smaller one is transformally algebraically closed in the larger one. These are precisely the ultraproducts of pairs of algebraically closed fields equipped with powers of the Frobenius map. This theory also provides an example of beautiful pairs in the sense of Cubides Kovacsics, Hils, and Ye.

This is joint work with Martin Hils, Udi Hrushovski, and Jinhe Ye.

In this talk from Alexander Ravnanger, he provides a dynamical description of the largest AF-ideal in certain crossed products by the integers. In the case of the uniform Roe algebra of the integers, this reveals an interesting connection to a well-studied object in topological semigroup theory. On the way, he gives an overview of what is known about the abundance of projections in such crossed products, the structure of the simple quotients, and concepts of low-dimensionality for uniform Roe algebras.