Objective The analysis of fields is one of the most active branches of Model Theory, which has found its most spectacular applications in Hrushovski's proofs of the Mordell-Lang and Manin-Mumford conjectures.There are three principal aspects:- ways of interpreting a field,- studying the general properties of fields thus obtained, and- determining the properties of particular theories of fields (with additional algebraic structure like a derivation or an automorphism).All three aspects are closely interrelated. This proposal concerns mainly part (b). A theorem of Macintyre, Cherlin and Shelah states that a super-stable field is algebraically closed; this theorem is at the basis of many applications. Recently Kim and Pillay have extended the apparatus of stability theory to a wider class: simple theories; Pillay has conjectured that super-simple fields are perfect, bounded and pseudo-algebraically closed (the converse was shown by Hrushovski).A positive answer to this conjecture should play a role similar to Macintyre's theorem. Since Pillay and Poizat have shown super-simple fields to be perfect and bounded, only the PAC condition that every absolutely irreducible variety has a rational point needs to be checked; this can be reduced to the consideration of plane curves.The case of elliptic and hyperelliptic curves with generic modulus has already been dealt with; however, attempts to treat the non-generic case have met with considerable difficulty. We propose to prove triviality of the first cohomology group in order to treat the non-elliptic genus 1 case. We shall also consider isogenies between elliptic curves defined over our field, in order to treat the case of non-generic j-invariant.Finally, we want to study the question whether super-simple fields are C_1 (related to a question of Ax). A natural approach here will be to study cubic surfaces over a super-simple field. This programme interrelates Algebraic Geometry, Field Theory and Model Theory. Fields of science natural sciencesmathematicspure mathematicsdiscrete mathematicsmathematical logicnatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsalgebraalgebraic geometry Keywords An interaction between Algebraic Geometry Model Theory and Field Theory Programme(s) FP6-MOBILITY - Human resources and Mobility in the specific programme for research, technological development and demonstration "Structuring the European Research Area" under the Sixth Framework Programme 2002-2006 Topic(s) MOBILITY-2.1 - Marie Curie Intra-European Fellowships (EIF) Call for proposal FP6-2002-MOBILITY-5 See other projects for this call Funding Scheme EIF - Marie Curie actions-Intra-European Fellowships Coordinator CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE - DELEGATION RHÔNE ALPES - SITE VALLEE DU RHÔNE EU contribution No data Address 2 Avenue Albert Einstein VILLEURBANNE France See on map Links Website Opens in new window Total cost No data