Rational Points and Heights

该项目研究算术代数几何中的问题，研究数域上代数簇上有理点的分布，并探索整体几何不变量与簇的算术性质之间的相互作用。这项工作的重点是证明变种的族的势密度，如Fano变种或Calabi-Yau变差变种，并建立群和齐次空间的等变紧化上有界高度有理点个数的渐近公式。建议的研究范围从显式的数值实验到现代快速发展的数学领域中的高级问题。这项工作将对学生的教育和年轻数学家的培养产生更广泛的影响，因为有丰富的具体问题和例子，巩固了我们对主题的理解，并导致了新的结构。算术代数几何的应用正日益扩展到信息存储、处理和传输的重要领域；这些应用依赖于本建议中要解决的基本问题的进一步进展。

The project addresses questions in Arithmetic Algebraic Geometry.It is proposed to study the distribution of rational points onalgebraic varieties over number fields, and to explore the interplaybetween global geometric invariants and arithmetic properties ofvarieties. The focus is on proving potential density for familiesof varieties, such as Fano varieties or Calabi-Yau varieties, andon establishing asymptotic formulas for the number of rational pointsof bounded height on equivariant compactifications of groups and homogeneous spaces.The proposed research ranges fromexplicit numerical experiments to advanced problems inmodern and rapidly developing areas of mathematics.This work will have a broader impact on the education of studentsand training of young mathematicians, as there is a rich supply ofconcrete questions and examples, consolidating our understanding of thesubject and leading to new structures.Increasingly, applications of Arithmetic Algebraic Geometryare spreading to vital areas of information storage, processing andtransmission; these applications rely on further advances onfundamental problems to be addressed in this proposal.