Séminaires : Séminaire Général de Logique

Equipe(s) : lm,
Responsables :S. Anscombe, A. Khélif, A. Vignati
Email des responsables : sylvy.anscombe@imj-prg.fr, vignati@imj-prg.fr
Salle : 1013
Adresse :Sophie Germain

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Orateur(s) Marcus Tressl - University of Mancherster,
Titre First order theory of semi-algebraic sets and continuous functions.
Horaire16:00 à 17:15
Diffusion https://u-paris.zoom.us/rec/share/NlDokdLxWhdCJ5_cp888uvKTKE9WVr3HbgTAuwrxI1HSj_0hxRt-218Bg2VlCod1.yKfbETnFiegJNwLs
RésumeI will give an overview - and report some recent progress - on first order structures attached to collections of semi-algebraic sets and functions. Semi-algebraic here means definable in a field K that is real closed (i.e. elementary equivalent to the real field) or p-adically closed (elementary equivalent to the p-adics). For a semi-algebraic subset X of Kn let C(X) be the set of continuous semi-algebraic functions on X with values in K. There are various structures on (or derived from) C(X): We will mainly look at C(X) as a ring with pointwise operations, or as lattice ordered group with pointwise comparison (when K is real closed), as well as the lattice L(X) of closed definable subsets of X (zero sets of functions in C(X)). Key results are (in slightly paraphrased form):
(1) ( with Luck Darnière ) the ring C(X) defines true arithmetic, unless X is discrete, or K is real closed and the dimension of X is 1. This result can be used to classify the homeomorphism type of X in terms of the first order theory of the ring C(X).
(2) If K is real closed, then the theory of the lattice ordered group C(X) can be interpreted in the theory of the lattice L(X) (in a certain non-standard form), which e.g., transfers decidability results from L(X) to C(X).
(3) The remaining case in (1), when K is real closed and X is of dimension 1 is unsolved yet, but a very recent result by Deacon Linkhorn indicates a path to a model completeness result for the ring of semi-algebraic curves in a small extension of the ring language.
AdresseSophie Germain