Between rational and integral points

数学无疑是科学和自然界的通用语言，其过程通常由方程控制。这个proposalcentres系统的方程涉及多项式与整数系数。丢番图方程的有理解或积分解的研究是一个既古老又困难的课题，自从近2000年前的古希腊时代以来就一直吸引着我们的注意力。它与许多学科领域有着深刻的互动，从代数几何到复杂的分析，通过数学逻辑以及介于两者之间的一切。在其核心，这个建议使用分析方法来解决一系列长期存在的关于丢番图方程解的存在性和分布的问题，在有理点和积分点以及在两者之间插值的“坎帕纳点”的背景下工作。在相反的方向，通过研究丢番图方程的适当系统上的有理点和整点的分布，PI希望阐明其他基本研究领域，包括朗兰兹对应和马勒关于具有指定伽罗瓦群的数域密度的猜想。

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.