Diophantine geometry from model theory (Q2732528)

This paper consists of 8 sections: Introduction, Background and notation; From algebra to geometry: the Mordell-Lang conjecture; From geometry to model theory: weakly normal groups; From enriched fields to pure fields: prolongations; The socle; From exact solutions to approximations: specializations; Concluding remarks. The author recalls: in Section 1 a history of the Mordell-Lang problem, in Section 2 some definitions from algebraic geometry and model theory, in Section 3 the Mordell-Lang conjecture and in Section 4 the theory of weakly normal groups and their relevance to Mordell-Lang-like problems. Also in Section 4 is a sketch of a proof of the positive characteristic Manin-Mumford conjecture. In Section 5 the author discusses the theory of prolongations of definable sets in difference and differential fields indicating how this theory may be used to compute upper bounds for the number of solutions to various diophantine problems, in Section 6 he discusses the theory of the semipluriminimal socle of a group of finite Morley rank and sketches how this theory has been used to prove some uniform finiteness theorems, in Section 7 he discusses a general specialization result on diophantine approximations and he concludes in Section 8 with some questions on extensions of these results and with some remarks on how these theorems have fed back into pure model theory.