Rational points on algebraic varieties
Lead Research Organisation:
University of Bath
Department Name: Mathematical Sciences
Abstract
This project will focus on the study of rational points on algebraic varieties. Given an algebraic variety over a number field, natural questions are: Is there a rational point? If yes, are there infinitely many? If also yes, can one obtain a finer quantitative description of the distribution of the rational points? These problems are very difficult in general, but in this project the aim is to make some progress for some special classes of varieties (in particular solve some new cases of Manin's conjecture). It also expects to contribute to a popular current research theme, that of considering these problems in families, such as studying the distribution of varieties in a family with a rational point, or controlling failures of the Hasse principle in families. To solve such problems one normally uses a combination of techniques from algebraic geometry, algebraic number theory and analytic number theory.
Organisations
Publications
Streeter S
(2021)
Campana points and powerful values of norm forms
in Mathematische Zeitschrift
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509565/1 | 01/10/2016 | 30/09/2021 | |||
| 1879285 | Studentship | EP/N509565/1 | 01/10/2017 | 30/09/2021 | Samuel Streeter |
| EP/N509589/1 | 01/10/2016 | 30/09/2021 | |||
| 1879285 | Studentship | EP/N509589/1 | 01/10/2017 | 30/09/2021 | Samuel Streeter |
| Description | During the first eighteen months of my PhD, I completed a short research paper with several theorems which prove, roughly speaking, that, for certain well-studied families of equations, there are (in a geometric sense) many solutions in the rational numbers, i.e. rational points. This paper was submitted as a preprint online and was subsequently approved for publication in the journal Mathematical Research Letters. Since then, I have been working on a second paper with the same goal of assessing the number of solutions to certain systems of equations, but the equations considered are different, the solutions are certain "special" rational solutions and my approach is more quantitative. I now have finished my second draft of this second paper and hope to upload it as a preprint soon. |
| Exploitation Route | My work is in two rapidly expanding areas of research, namely the Hilbert property and Campana points, and I hope that my work will provide others with a platform to prove further results in these areas. |
| Sectors | Other |
| URL | https://arxiv.org/pdf/1812.05937.pdf |
| Description | Conference organisation - Mathematical Research Students Conference 2018 |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | Local |
| Primary Audience | Postgraduate students |
| Results and Impact | Co-organiser of conference welcoming new mathematics PhD students to the University of Manchester. |
| Year(s) Of Engagement Activity | 2018,2019 |
| Description | Research talks |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Postgraduate students |
| Results and Impact | Four research talks given at: Institut Henri Poincaré, Paris, France - May 2018 University of Manchester (Pure Postgraduate Seminar), Manchester, UK - June 2018 University of Bath (Algebra, Geometry and Number Theory Seminar), Bath, UK - October 2019 University of Bristol (Linfoot Number Theory Seminar), Bristol, UK - November 2019 |
| Year(s) Of Engagement Activity | 2018,2019 |