Between rational and integral points

无可否认，数学是科学和自然的通用语言，其过程往往受方程式的支配。这一建议集中在含有整系数多项式的方程组。丢番图方程的有理或整体解的研究是一个既古老又困难的课题，自近2000年前的古希腊时代以来就一直受到我们的关注。它与许多学科领域有着深刻的互动，从代数几何到通过数理逻辑进行的复杂分析，以及介于两者之间的一切。这一提议的核心是使用分析方法来解决关于丢番图方程解的存在和分布的一系列长期猜想，这些猜想既适用于有理点，也适用于积分点，以及介于两者之间的“坎帕纳点”。在相反的方向上，PI希望通过研究适当的丢番图方程组上有理点和积分点的分布来阐明其他基本研究领域，包括朗兰兹对应和Malle关于具有指定伽罗瓦群的数域的密度的猜想。

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.