Problems in Model Theory, Diophantine Geometry, and Functional Transcendence

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The project will study the interaction of Schanuel-type conjectures and related Diophantine conjectures with model theory.The context to this project will be the Zilber-Pink conjecture in diophantine geometry and its connection with model theory and Schanuel-type conjectures. Schanuel's conjecture is a conjecture of transcendental number theory relating to the transcendence properties of the exponential function. It encapsulates the classical results of Lindemann-Weierstrass and Hilbert's 7th Problem. Though Schanuel's conjecture remains widely viewed as inaccessible using currnet methods, certain model theoretic approaches, such as applications of O-minimality and the study of exponential fields, are some of the more recent developments in the area. The Zilber-Pink conjecture forms a generalization of the Andre-Oort conjecture, both remaining open in the general form, though methods from O-minimal geometry have been applied to the Andre-Oort conjecture to provide results for certain cases. Certain examples of recent work in classical and model theoretic terms in this area are investigations of the model theoretic properties of, and Schanuel-type analogues for, the j-function and its analogues for abelian varieties of higher dimension, the relation between exponentiation and pseudo-exponentiation, and the model-theoretic properties of pseudo-exponential fields in first-order and infinitary logics, and the Pila-Wilkie counting theorem.More specifically, the project will involve studying specific problems within the Zilber-Pink conjecture in either classical or model-theoretic settings, with the primary aim being to in some way increase the tractability of, or investigate certain special cases of, the Zilber-Pink conjecture and related Schanuel-type conjectures.This will involve combining methods from O-minimality or other parts of model theory, arithmetic, and differential algebra. With regards to O-minimality in particular, the recent developments in its application to Diophantine problems and in particular the Andre-Oort Conjecture have developed essentially into a general strategy of tackling varied cases of the Zilber-Pink conjecture which has achieved significant results which are likely not yet exhausted. Another area of model theory that has been applied successfully to Diophantine problems by Hrushovski and others recently has been the field of geometric stability theory.This project falls within the EPSRC Logic and Combinatorics research area