| EPSRC Reference: |
EP/T01170X/1 |
| Title: |
Pseudorandom majorants over number fields with applications in arithmetic geometry |
| Principal Investigator: |
Frei, Dr C |
| 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: |
New Investigator Award |
| Starts: |
27 November 2019 |
Ends: |
31 August 2020 |
Value (£): |
124,537
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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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Summary on Grant Application Form |
A Diophantine equation, named after the ancient Hellenistic mathematician Diophantus of Alexandria, is a polynomial equation in which all the coefficients are integers (whole numbers) or rational numbers (fractions). The most fundamental question, given a Diophantine equation, is whether it has a solution, that is a collection of integers or rational numbers which satisfy this equation. To decide whether a given Diophantine equation has a solution can be extremely hard, in spite of extensive mathematical machinery that was developed over centuries to attack these questions. A famous example is Fermat's Last Theorem. Despite the relative simplicity of its statement that for any integer n greater than two, the sum of two positive nth powers can not be an nth power, a proof has eluded the efforts of mathematicians for more than 350 years. It has spawned numerous new developments and was finally completed by Andrew Wiles at the end of the 20th century.
Equations define not just number theoretic, but also geometric objects. A particularly successful approach, developed in the 20th century, tries to investigate solutions to Diophantine equations via the corresponding geometric objects. The modern study of Diophantine equations using these geometric techniques is called arithmetic (or Diophantine) geometry.
Another branch of number theory, in which UK mathematicians play a world leading role, is called additive combinatorics. One of the aims of this discipline is to understand subsets of the integers by decomposing them into structured and random looking parts, with the main challenge arising from the fact that this is usually not a clean dichotomy, but rather a full spectrum.
Extremely fruitful connections between these two fields were initiated very recently by applying certain results and techniques from additive combinatorics to questions in arithmetic geometry, thus expanding our knowledge of Diophantine equations significantly. The central aim of this project is to enhance the impact of these techniques by making them available in a much wider context that is natural in arithmetic geometry.
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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 |