# Schanuel's conjecture and excellence

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

## Abstract

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.

## References

- Boris Zilber.
*Pseudo-exponentiation on algebraically closed fields of characteristic zero*, Annals of Pure and Applied Logic 132 (2004), no. 1, 67–95.
- Martin Bays, Bradd Hart, Tapani Hyttinen, Meeri Kesälä, Jonathan Kirby.
*Quasiminimal structures and excellence*, Bulletin of the London Mathematical Society 46 (2014), no. 1, 155-163.
- Saharon Shelah.
*The number of uncountable models of Ψ ∈ L*_{ω1,ω}. Part A, Israel Journal of Mathematics 46 (1983), no. 3, 212-240.
- Saharon Shelah.
*The number of uncountable models of Ψ ∈ L*_{ω1,ω}. Part B, Israel Journal of Mathematics 46 (1983), no. 4, 241-273.