Equipe(s) : | lm, |

Responsables : | S. Anscombe, O. Finkel, A. Khélif, A. Vignati |

Email des responsables : | sylvy.anscombe@imj-prg.fr, vignati@imj-prg.fr |

Salle : | 1013 |

Adresse : | Sophie Germain |

Description |

Orateur(s) | Marcus Tressl - University of Mancherster, |

Titre | First order theory of semi-algebraic sets and continuous functions. |

Date | 31/03/2021 |

Horaire | 16:00 à 17:15 |

Diffusion | https://u-paris.zoom.us/rec/share/NlDokdLxWhdCJ5_cp888uvKTKE9WVr3HbgTAuwrxI1HSj_0hxRt-218Bg2VlCod1.yKfbETnFiegJNwLs |

Résume | I 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. |

Salle | 1013 |

Adresse | Sophie Germain |

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