| Résume | I will present a few ideas concerning a possible direction for future research on the analytic side of model theory. They are based on two pieces of work, one celebrating its 30th anniversary this year and the other its 50th. Roughly stated the first, the Marker-Steinhorn Theorem, asserts that the standard part of an o-minimally definable function is definable and the second, extracted from Abraham Robinson's book “Non-Standard Analysis” (Chapter VI-well worth revisiting) that the standard part of a complex holomorphic function is also holomorphic.
I begin by giving precise statements of these results and then go on to combine them. Of course, there is a wonderful development of complex analysis within an arbitrary o-minimal structure by Peterzil and Starchenko but here we will always assume that our models are elementary extensions of an expansion of the standard real field thereby giving us access, via Robinson's theory, to all of complex analysis. |
