Yvon Bossut - On Definable Groups in Omega-Free PAC Fields

jeudi 12 juin 2025, 16:00 → 17:00 Europe/Paris

112 (ICJ)

PAC fields have long been studied by model theorists. Hrushovski showed that any group definable in an e-free PAC field (for finite e) is definably isogenous to the rational points of an algebraic group. Chatzidakis and Ramsey also defined a measure on definable 
subsets of an irreducible variety in e-free PAC fields. Using Hrushovski’s result, they deduced that groups definable in e-free PAC fields are definably amenable. Omega-free PAC fields can be obtained as non-principal ultraproducts of e-free PAC fields; therefore, groups definable in omega-free PAC fields are also definably amenable. A natural question is whether these groups are also definably isogenous to the rational points of an algebraic group. In this talk, I will present a result that partially answers this question. I will briefly introduce PAC fields and some relevant model-theoretic results. I will then present two different notions of stabilizer that appear in the proof, and discuss the contexts in which each may be relevant.