Local positivity and Diophantine applications

代数几何是数学的一个丰富的分支，它统一了代数学、拓扑学、数论和微分几何。它允许一个重新制定的问题，似乎是关于代数或数字到几何问题的形状，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描述的形状的内在几何性质密切相关。 丢番图几何是这一哲学最深远的实现之一。在一般情况下，它连接的整体行为的有理数解决方案的一个系统的多项式方程的整体行为的曲率对相应的形状所定义的这些方程。 申请人先前的研究已经将线上的经典近似结果扩展到任意投影簇。其中一个新的想法介绍是考虑当地的影响曲率问题的当地近似合理的点。这个建议的目的是联合收割机这一见解与新的想法，从双有理和渐近代数几何建立丢番图几何的新成果。 该提案还旨在为代数几何中的积极性问题提供新的结果。 本研究将培养4名本科生、3名硕士生、3名博士生和2名博士后。

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.