In Part 1 of the book, we provide a concrete example of the applicability of

the abstract methods to a mathematical problem. That is, we describe suﬃcient

conditions for categoricity in all powers and elaborate a specific algebraic example.

In Parts 2 and 3 we study more general contexts where only partial categoricity

transfer methods are known. We close the book in Part 4 by describing Shelah’s

solution of the categoricity transfer problem for 𝐿𝜔1,𝜔 and placing Part 1 in that

context.

One reason to study AEC is that first order logic is inadequate to describe

certain basic mathematical structures. One example is complex exponentiation;

the ring of integers is interpretable in (ℂ, ⋅, +, exp) with domain the zeros of the

sine function and so the resulting theory is unstable. Zilber suggested investigating

this structure in 𝐿𝜔1,𝜔. A somewhat simpler object, the covers of the multiplica-

tive group of ℂ, is a concrete example of a quasiminimal excellent classes. This

is a specific example of the notion of excellence discussed in Part 4. A notion of

excellence presupposes a notion of dependence. In the case of quasiminimality, this

dependence notion determines a geometry. We prove that a quasiminimal excellent

class is categorical in all uncountable powers. We expound several papers of Zilber

and explain the fundamental notion of excellence in the ‘rank one’ case. We study

the ‘covers’ situation in detail. It provides a ‘real’ example of categorical struc-

tures which are not homogenous and thus shows why the notion of excellence must

be introduced. We briefly discuss (with references) at the end of Chapter 3 the

extensions of these methods to other mathematical situations (complex exponen-

tiation and semi-abelian varieties). We return to the more general consideration

of categoricity of 𝐿𝜔1,𝜔 in Part 4 and discuss the full notion of excellence. After

Proposition 25.20, we examine which parts of Zilber’s work are implied by Shelah

[She83a, She83b] and the sense in which Zilber extends Shelah.