Schanuel's Conjecture - Zilber's Pseudo-exponentiation

Zilber's Pseudo-exponentiation

While a proof of Schanuel's conjecture with number theoretic tools seems a long way off, connections with model theory have prompted a surge of research on the conjecture.

In 2004, Boris Zilber systematically constructs exponential fields K_exp that are algebraically closed and of characteristic zero, and such that one of these fields exists for each uncountable cardinality. He axiomatises these fields and, using Hrushovski's construction and techniques inspired by work of Shelah on categoricity in infinitary logics, proves that this theory of "pseudo-exponentiation" has a unique model in each uncountable cardinal. Schanuel's conjecture is part of this axiomatisation, and so the natural conjecture that the unique model of cardinality continuum is actually isomorphic to the complex exponential field implies Schanuel's conjecture. In fact, Zilber shows that this conjecture holds iff both Schanuel's conjecture and another unproven condition on the complex exponentiation field, which Zilber calls exponential-algebraic closedness, hold.