Counting and Equidistribution on homogeneous spaces

本文提出了关于李群齐次空间上几何和算术对象分布的四个课题。它们是赫克对应的等分布、有界高度有理点的计数、给定分母有理点的分布和无理值形式的分布。他们提出利用调和分析、群作用动力学、遍历理论、半单代数群的自同构形式等不同领域的技术来证明（或证伪）数论和几何情况下出现的密集分布对象的均匀分布。现代数学的一大成就是利用遍历理论、齐次空间动力学、算术几何和自同构形式来解决数论中长期存在的开放问题。拟议的项目重点是进一步建立不同数学学科之间的联系。

DMS-0629322Hee OhThe proposal concerns four projects regarding the distribution of geometric and arithmetic objects on homogeneous spaces of Lie groups. They are equidistribution of Hecke correspondence, counting rational points of bounded height, distribution of rational points with given denominator and distribution of values of irrational forms. They propose to use techniques from various different fields such as harmonic analysis, dynamics of group actions, ergodic theory, automorphic forms of semisimple algebraic groups in order to prove (or disprove) the equidistribution of densely distributed objects arising in number theoretic and geometric situation.One of big achievements in modern mathematics is the use of ergodic theory, dynamics of homogeneous spaces, arithmetic geometry and automorphic forms in solving long standing open problems in number theory. The proposed projects are focused on establishing further these connections between different disciplines of mathematics.