  Geometric model theory grew out of Michael Morley’s 1965 paper, but in a different direction from Shelah’s work (though today it makes regular use of technical tools developed by Shelah in his classification programme). Morley had shown that models of an uncountably categorical theory have structural properties that are interesting in their own right, regardless of the complete theory of the structure; so it became the custom to talk of uncountably categorical structures, meaning models of uncountably categorical theories. (And likewise totally categorical structures.) Independently Boris Zilber in Siberia and Greg Cherlin in the United States noticed that any infinite group that is definable in an uncountably categorical structure must have many features in common with the algebraic groups studied by algebraic geometers. Zilber in particular showed that many methods from algebraic geometry generalise to the model-theoretic case. His secret weapon was Bezout’s Theorem from geometry, which he used to guide him to solutions of very difficult model theoretic problems; for example his theorem that no totally categorical theory can be axiomatised by a finite number of axioms. (It was secret in the sense that it guided his intuition but never appeared explicitly in his results.) Zilber also noticed an important difference between the first and second of Łoś’s examples above. Namely, in a vector space the subspaces (i.e. the subsets closed under linear dependence) form a modular lattice; but the algebraically closed subsets of an algebraically closed field form a lattice that is not modular. 