Rational Points and Heights

该项目旨在研究数域上代数簇上有理点的分布，并探索簇的整体几何不变量与算术性质之间的相互作用。重点是证明品种家族的潜在密度，如Fano品种或Calabi-Yau品种，在群和齐次空间的等变紧化上建立了有界高有理点数的渐近公式.所建议的研究范围从显式数值实验到现代迅速发展的数学领域的高级问题.这项工作将对学生的教育和培训年轻的数学家，因为有丰富的具体问题和例子的供应，巩固我们对这个主题的理解，并导致新的结构。越来越多的算术代数几何的应用正在扩展到信息存储，处理和传输的重要领域;这些应用依赖于进一步的进展基本问题要解决这个建议。

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.