On the quasi-minimality of certain expansions of the complex field
Seminar Room 1, Newton Institute
I shall consider various natural pre-geometries on expansions of the complex field inspired by the work of Peterzil and Starchenko on the development of complex analysis within an o-minimal structure. Unfortunately, I still cannot realise my original aim of using such methods to show that the complex exponential field is quasi-minimal (ie every definable subset of the complex numbers is either countable or co-countable) but I can at least show that we do have quasi-minimality if we only allow the operations of raising to real powers. (I should point out, however, that if one assumes postive answers to certain conjectures from diophantine geometry and transcendence theory then Zilber has already shown this,and more. See 'Raising to powers in algebraically closed fields', J Math Logic vol 3(2), 2003, 217-238.)
Another aspect of the talk is that it gives some sort of answer to a question of Hrushovski (private communication a couple of years ago) which asks whether elimation of quantifiers for algebraically closed fields may be naturally deduced from elimination of quantifiers for real closed ordered fields. I show that even though the definable closure operator on an o-minimal structure (expanding a real closed field) does not satisfy the modular law, it may nevertheless be linearised and thereby induce a pregeometry on an expansion of its algebraic closure. The Cauchy-Riemann equations play a role here so that, for example, in the pure field case this pregeometry IS algebraic closure (and not,say, "algebraic closure of the set of real and imaginary parts").