| EPSRC Reference: |
EP/J01933X/1 |
| Title: |
O-minimality and diophantine geometry |
| Principal Investigator: |
Wilkie, Professor A |
| Other Investigators: |
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| Researcher Co-Investigators: |
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| Project Partners: |
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| Department: |
Mathematics |
| Organisation: |
University of Manchester, The |
| Scheme: |
Standard Research |
| Starts: |
01 August 2012 |
Ends: |
31 July 2015 |
Value (£): |
327,547
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| EPSRC Research Topic Classifications: |
| Algebra & Geometry |
Logic & Combinatorics |
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| EPSRC Industrial Sector Classifications: |
| No relevance to Underpinning Sectors |
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| Related Grants: |
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| Panel History: |
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Summary on Grant Application Form |
The investigators on this project are Alex Wilkie (Manchester, PI), Jonathan Pila (Oxford, co-I) and Gareth Jones (Manchester, co-I). It is a
joint proposal between the Universities of Manchester and Oxford with Manchester being the lead institution.
The aim of the project is to further the links between mathematical logic, specifically the branch of model theory known as o-minimality, and
diophantine geometry (i.e. the study of points on curves, surfaces etc with integer or rational coordinates). The o-minimality axiom applies to
collections of subsets of real euclidean spaces ("structures") and, when satisfied, implies a variety of topological, analytic and geometrical
finiteness properties that fail in general for the more classical classifications of sets that are studied in mainstream topology and analysis (such as those having a differentiable, or even analytic, manifold structure). Further, many interesting examples of o-minimal structures are known.
Wilkie was the first to notice that there are diophantine consequences for sets occurring in an o-minimal structure. Pila's work on diophantine
problems started with his influential 1989 paper with Bombieri and he proceeded to develop the diophantine theory of the so called real analytic
sets in several dimensions. This culminated in the Pila-Wilkie theorem establishing a general result in the broader setting of o-minimal structures. The application of this result, in work of Masser, Pila and Zannier, has opened up a new connection between mathematical logic and diophantine geometry with great potential, the most remarkable example to date being Pila's unconditional proof of a special case of the long standing Andre-Oort conjecture.
It is our intention to further such application as well as to advance the pure theory of o-minimal structures with this in mind. One step on the way is a conjecture of the PI concerning rational points on sets in one particular o-minimal structure known as the real exponential field. To carry this out one will also need more precise results on the model theory of the real Pfaffian field (another structure known to be o-minimal). Jones is an internationally recognised expert in both these areas and has obtained (jointly with several other researchers who are named as visitors in the proposal) the best results to date. All three aspects of our project, including details of visits and research meetings, are described at length in the following text.
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Key Findings |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
| Summary |
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| Date Materialised |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.man.ac.uk |