10h30 Esther ELBAZ
Having an o-minimal open core is not elementary.
The open core of a first-order structure expanding a dense totally ordered set is the definable reduct whose primitives are the definable sets, in all dimensions, that are open for the order topology. Having an o-minimal open core is a tameness property that is strictly weaker than being o-minimal. Dolich, Miller and Steinhorn and asked if this property is elementary, i.e. if models of the complete theory of a structure with an o-minimal open core must have an o-minimal open core.
I will answer the question in the negative.
This is based on joint work with Alexi Block Gorman.
14h Pietro FRENI (Institute of Mathematics of the Czech Academy of Sciences, Prague, Tchéquie)
Cuts and derivations in exponential o-minimal structures.
After a brief reminder on Hardy fields, I will describe how some axioms of the theory of H-fields can be generalized so as to cover the case of differential fields of germs at a non-principal cut in an o-minimal ordered field. I will then show how this can be used to answer some questions about cuts in o-minimal structures: I will sketch the proof that in polynomially bounded structures expanded by an exponential function, a symmetric cut is weakly orthogonal to its invariance group, answering a question of Tressl; and I will characterize exponential o-minimal theories where the definable image of a pseudolimit (wim-cut) over a model is always an image by a composition of exponentials, translations and sign-changes of another pseudolimit.
16h Rosario MENNUNI (Universita di Pisa, Italie)
Ordered abelian groups, valued fields, and generic automorphisms.
I will talk about joint work with Jan Dobrowolski and Francesco Gallinaro on existentially closed valued difference fields, with no restriction imposed on the behaviour of the automorphism on the value group. We prove that such structures admit an equivariant section of the valuation, hence an equivariant angular component. In the language with such an angular component, assuming residue characteristic 0, we prove transfer of amalgamation of substructures from the residue field, characterise the existentially closed models, and deduce that they are NTP2 in the sense of positive logic.
