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.

Publications

10 25 50

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509589/1 01/10/2016 30/09/2021
1879285 Studentship EP/N509589/1 18/09/2017 30/09/2021 Samuel Streeter
EP/N509565/1 01/10/2016 30/09/2021
1879285 Studentship EP/N509565/1 18/09/2017 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