I will start by explaining a possible strategy for using o-minimality to prove (the second part of) Hilbert’s 16th problem (H16). This would involve establishing the o-minimality of a particular expansion of the real field, which is currently a wide open question.

The course deals with what we know to date about a particular reduct of this structure, in which certain maps crucial to counting limit cycles are definable. This reduct is generated by a particular transserial Hardy field, and the course’s focus is on the construction of this Hardy field. It consists of two main steps:

1) a description of the complex analytic continuations of all one-variable germs definable in the o-minimal structure R_an,exp; and

2) an extension, based on 1), of Ilyashenko’s construction of the quasianalytic class of “almost regular” germs.

Students in this class are expected to be familiar with the following:

- basic model theory and o-minimality;

- graduate-level real and complex analysis;

- the structure R_an,exp as discussed in the papers by Van den Dries, Macintyre and Marker;

- the field of trasnseries as discussed by Berarducci and Mantova in their course module.