Local positivity and Diophantine applications

代数几何是一个丰富的数学分支，它统一了代数、拓扑学、数论和微分几何。它允许人们将看似关于代数或数字的问题重新表述为关于形状的几何问题，反之亦然。它的起源在于认识到，寻找代数方程的解或理解其解的性质与这些方程所描述的形状的内在几何性质密切相关。丢番图几何学是这一哲学最深远的实现之一。一般而言，它将多项式方程组的有理数解的整体性质与这些方程所定义的相应形状上的曲率的整体性质联系起来。申请人先前的研究已将该直线上的经典逼近结果推广到任意射影变种。引入的新思想之一是考虑曲率对有理点局部逼近问题的局部影响。这一建议旨在将这一见解与二次代数几何和渐近代数几何的新思想相结合，以建立丢番图几何的新结果。该提案还旨在为代数几何中围绕正性的问题贡献新的结果。这项拟议的研究将培养四名本科生、三名硕士、三名博士生和两名博士后研究员。

Algebraic geometry is a rich branch of mathematics which unifies algebra, topology, number theory, and differential geometry. It allows one to reformulate questions which seem to be about algebra or numbers into geometrical questions about shape, or vice versa. Its origins lie in the realization that finding solutions to algebraic equations or understanding the properties of their solutions is intimately tied to the intrinsic geometric nature of the shapes those equations describe. Diophantine geometry is one of the most far-reaching realizations of this philosophy. In general terms it connects the global behaviour of rational number solutions to a system of polynomial equations with the global behaviour of the curvature on the corresponding shape defined by those equations.Previous research by the applicant has extended classic approximation results on the line to arbitrary projective varieties. One of the new ideas introduced was to consider the local influence of curvature on questions of local approximation by rational points. This proposal aims to combine this insight with new ideas from birational and asymptotic algebraic geometry to establish new results in Diophantine geometry. The proposal also aims to contribute new results to questions surrounding positivity in algebraic geometry. The proposed research will train four undergraduate students, three master's students, three doctoral students, and two postdoctoral fellows.