|Responsables :||S. Anscombe, O. Finkel, A. Khélif, A. Vignati|
|Email des responsables :||email@example.com, firstname.lastname@example.org|
|Adresse :||Sophie Germain|
|Orateur(s)||Marcus Tressl - University of Mancherster,|
|Titre||First order theory of semi-algebraic sets and continuous functions.|
|Horaire||16:00 à 17:15|
|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.