Problems in Model Theory, Diophantine Geometry, and Functional Transcendence
Lead Research Organisation:
University of Oxford
Department Name: Mathematical Institute
Abstract
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
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
Organisations
People |
ORCID iD |
| John Armitage (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509711/1 | 01/10/2016 | 30/09/2021 | |||
| 1789662 | Studentship | EP/N509711/1 | 01/10/2016 | 02/06/2020 | John Armitage |
| Description | We have obtained so far, For Gauss' one class per genus problem, we have improved computationally the known lower bound for such a discriminant, and obtained a large intermediate region in which the discriminant cannot lie, and obtained a new result showing that such discriminants canot have a proportionally very large prime factor. New bounds on zeros of polynomials P(z,j(z)) and P(z,p(z)), have been obtained, where j is the j-invariant, and p is Weierstrass' p-function. A competitive algorithm has been developed for the combination of linear congruences. A new bound has been obtained for the product of modular degrees of elliptic curves of the same conductor. New results have been obtained by Jonathan Pila's "mildness" method to bound the number of algebraic points on the graphs of the Gamma function, and Riemann's Zeta function. |
| Exploitation Route | The algorithm for combining congruences would likely have application to other problems in number theory, especially those requiring the computational lower bounds. |
| Sectors | Other |