Definable groups in topological differential fields.
KeywordsDefinable groups, differential topological fields, elimination of imaginaries
Generalizing results of M. Tressl we developped partly with N. Guzy, the model theory of topological differential fields. Recently we described under certain hypothesis definable groups in existentially closed such fields. Our results apply to closed ordered fields and use the fact that they have the elimination of imaginaries (in the langage of ordered differential fields). It would be interesting to examine other theories for which a langage has been explicitely identified in which they have e.i., for instance the theory of real-closed valued fields, p-adically closed valued fields or algebraically closed valued fields. The proofs that one has e.i. in these langages (usually called geometric langages) are difficult but have been recently simplified by E. Hrushovski and W. Johnson. The project would be to see whether it is now feasible to obtain interesting algebraic properties of definable groups in models of these theories and possibly to identify these groups with classical ones.