Schanuel's conjecture and excellence

This is the title of a talk given at the CMU graduate student seminar on September 30, 2014.


Schanuel's conjecture is a very general statement of number theory which implies in particular that e + π is transcendental. Excellence is a tool of model theory introduced by Saharon Shelah in 1983. Rouhgly speaking, a class of structures (like groups or fields) is excellent if any n-dimensional cube of countable structures with a corner missing can be "completed" in a unique way.

In 2003, Boris Zilber used excellence to shed some light on Schanuel's conjecture: he constructed an exponential field of cardinality continuum that satisfies Schanuel's conjecture and proved that it was (in some sense) the only one. It has been conjectured that Zilber's field is isomorphic to the complex numbers, and Schanuel's conjecture would follow. This talk will attempt to give a taste of what excellence is and explain its relevance to Zilber's proof.