Model Theoretic Classification and its Applications to Algebra

Classification theory is a program initiated by Shelah in the early 70s with the aim of classifying complete first order theories into "tame" and "wild" theories by some relatively simple combinatorial invariants. Examples of such theories are stable, dependent and simple theories. 

From the beginning, the applications stemming from this work moved in two intertwining directions. 

The first, a detailed study of the geometry arising in such theories and applying these notions to algebra and geometry (e.g. Hrushovski's proof of Mordell-Lang). 

The second, studying and classifying algebraic objects whose theories enjoy these properties (e.g. Shelah and Cherlin proved that every superstable field is algebraically closed). 

In this talk I will introduce the main model theoretic notions and mainly follow the second direction of applications. I will review some old results and some newer results by myself and others especially in the dependent setting, where an open conjecture of Shelah's leads the way.

No prior knowledge in model theory or logic is required.