There are compelling and long-established connections between automata theory and tame geometry, including Büchi automata and the additive group of real numbers.
We say a subset X of the reals is "k-regular" if there is a Büchi automaton that accepts (one of) the base-k representations of every element of X, and rejects the base-k representations of each element in its complement.
We say an expansion of the real additive group is k-regular if all of its definable sets are k-regular. In this talk we will see a complete characterization of the so-called tame geometries that may arise in a k-regular expansion of the reals—how many different geometries can such an expansion have and what exactly do they look like?
We will unpack the connection between this question and some of the staples of tame topology, such as open cores and pairs of structures. | 
