Between rational and integral points

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Mathematics is undeniably the universal language of science and nature, whose processes are often governed by equations. This proposalcentres 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.