A class of finite semigroups V is said to be decidable if the membership problem for V has a solution, that is, if we can construct an algorithm to test whether a given semigroup lies in V. Decidability of pseudovarieties is not preserved by some of the most common pseudovariety operators, such as semidirect product, Mal’cev product and join [1, 17]. In particular Rhodes [17] has exhibited a de...

We combine two notions in AECs, tameness and good λ-frames, and show that they together give a very well-behaved nonforking notion in all cardinalities. This helps to fill a longstanding gap in classification theory of tame AECs and increases the applicability of frames. Along the way, we prove a complete stability transfer theorem and uniqueness of limit models in these AECs.

Many animal species have been domesticated over the course of human history and became tame as a result of domestication. Tameness is a behavioral characteristic with 2 potential components: (1) reluctance to avoid humans and (2) motivation to approach humans. However, the specific behavioral characteristics selected during domestication processes remain to be clarified for many species. To qua...

The concept of tameness of a pseudovariety was introduced by Almeida and Steinberg [1] as a tool for proving decidability of the membership problem for semidirect products of pseudovarieties. Recall that the join V ∨ W of two pseudovarieties V and W is the least pseudovariety containing both V and W. This talk is concerned with the problem of proving tameness of joins. This problem was consider...

We study the problem of extending an abstract independence notion for types of singletons (what Shelah calls a good frame) to longer types. Working in the framework of tame abstract elementary classes, we show that good frames can always be extended to types of independent sequences. As an application, we show that tameness and a good frame imply Shelah’s notion of dimension is well-behaved, co...

We contribute to the fully formal verification of Hales’ proof of the Kepler Conjecure by analyzing the enumeration of all tame plane graphs. We sketch a formalization of plane graphs, tameness and Hales’ enumeration procedure in Higher Order Logic. The correctness of the enumeration is partially verified (which uncovered a small mismatch between Hales’ definition of tameness and his enumeratio...