This is the title of a four-parts talk given at the CMU model theory seminar from February 6 to March 6, 2017.

Shelah's study of the first-order stability spectrum (the class of cardinals at which a theory is stable) led him to the definition of forking, one of the central notion of model theory. Shelah proved that for every stable first-order theory T, there is a cardinal κ(T) such that for every high-enough λ, T is stable in λ if and only if λ=λ^{<κ(T)}. Moreover κ(T) is the least cardinal κ such that every type does not fork over a set of size less than κ.

The simplest case is when κ(T)=ℵ_{0} (or equivalently when T is stable on a tail of cardinals). In this case T is called superstable. Such a T has a well-developed structure theory. For example, it is known that it has a saturated model in every high-enough cardinal. In fact, for every λ, the union of a chain of λ-saturated models is λ-saturated.

In this series of talk, I will discuss generalizing first-order superstability theory to the framework of abstract elementary classes (AECs). We will look at both the local context (where we study the behavior of the class at one specific cardinal) and the global context (where we study the class globally, assuming the existence of a monster model and usually also tameness: a locality property for Galois types introduced by Grossberg and VanDieren). The theory has recently seen several applications, including a proof of Shelah's eventual categoricity conjecture for universal classes. Our guiding thread will be the following result, which combines papers by Boney, Grossberg, Shelah, Vasey, and VanDieren.

*Theorem*: Let K be a tame AEC with a monster model. The following are equivalent:

- K is stable in all high-enough cardinals.
- K has no long splitting chains in all high-enough cardinals.
- K has a saturated model in all high-enough cardinals.
- For all high-enough λ, the union of a chain of λ-Galois-saturated models is λ-Galois-saturated.

John T. Baldwin,

*Categoricity*, University Lecture Series, vol. 50, American Mathematical Society, 2009.Saharon Shelah,

*Classification Theory for Abstract Elementary Classes*, Studies in Logic: Mathematical Logic and foundations, vols. 18 and 20, College Publications, 2009.Will Boney and Sebastien Vasey,

*A survey on tame abstract elementary classes*, Accepted, Beyond First Order Model Theory (José Iovino ed.), CRC Press. Preprint: pdf arXiv.Rami Grossberg and Sebastien Vasey,

*Equivalent definitions of superstability in tame abstract elementary classes*, Preprint: pdf arXiv.Sebastien Vasey,

*Toward a stability theory of tame abstract elementary classes*, Preprint: pdf arXiv.