Between rational and integral points

Mathematics is undeniably the universal language of science and nature, whose processes are often governed by equations. This proposal
centres on systems of equations involving polynomials with integer coefficients. The study of rational or integral solutions to Diophantine equations is a subject that is both ancient and difficult, having commanded our attention since the time of the ancient Greeks nearly 2000 years ago. It has had profound interactions with a host of subject areas, ranging from algebraic geometry to complex analysis via mathematical logic and everything in between. At its core this proposal uses analytic methods to tackle a range of long-standing conjectures of about the existence and distribution of solutions to Diophantine equations, working in the context of both rational and integral points, as well as 'Campana-points', which interpolate between the two. In the reverse direction, by studying the distribution of rational and integral points on appropriate systems of Diophantine equations, the PI hopes to shed light on other fundamental areas of research, including the Langlands correspondence and Malle's conjecture about the density of number fields with prescribed Galois group.

The main beneficiaries of the research will be the academic community. The proposal contains novel and surprising uses of analytic methods to make inroads into very central conjectures in number theory. Once the work is disseminated in conferences and seminars, this is likely to open the door to a flurry of further activity in the field by other researchers in number theory.

One impact of the project is the raised profile of the School of Mathematics at the University of Bristol and an increased activity of the number theory group. This, together with links and exchanges with top international universities will help increase the global standing and prestige of UK science. In turn, this will lead to more applications for undergraduate and PhD places from overseas students (bringing fees), and an increased number of research fellowships funded from outside the UK, such as EU Marie-Curie Fellowships and European research grants. This results in a direct income increase for the relevant mathematics departments and local economies.

This proposal features topics which have been found to captivate the imagination of the general public, with Diophantine equations proving a important bridge to frontline mathematical research. Many number-theoretical results (such as, notably, the proof of Fermat's Last Theorem) have caused great excitement and increased awareness of the general public in mathematical research. Advertising exciting and current mathematical
research in this way will inspire the next generation of mathematicians, the impact of which on future technology and development should not be ignored. As described in the Pathways to Impact document, the reach of the project will be maximised through the PI's close contact with numberphile, in order to make a short Youtube video about an appropriate aspect of the planned research.