Lie Group and Their Discrete Subgroups

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。PI(John Millson)提出了三条研究主线，全部在约化代数群和几何的一般框架内。在第一条主线(与Jens Funke)中，PI继续他的项目，使用theta对应和微分几何来构造Siegel(分别。Hermite)模形式，它是正交局部对称空间中某些(特殊)圈的交数的生成函数。么正)组。第二条主线是关于投影线上n个有序点的射影不变量环R。这个环在18世纪末和20世纪初被不变理论家大量研究。1894年，A.Kempe发现R是由某些最低阶不变量生成的。去年，PI与本·霍华德(Ben Howard)、安德鲁·斯诺登(Andrew Snowden)和拉维·瓦基尔(Ravi Vakil)合作，计算了肯佩发电机之间的关系。它们是二次、二项式的，并且有一个简单的图形描述。第三条主线是从代数中处理广义三角形不等式和相关(饱和)问题的项目的延续。这个项目是国际和平协会最近一次NSF拨款的主题，是与Prakash Belkale、Thomas Haines、Misha Kapovich和Shrawan Kumar一起获得的FRG赠款。这个项目是与托马斯·海恩斯和米莎·卡波维奇共同完成的。PI(John Millson)提出了三条主要的研究路线，都是在约化代数群和几何的一般框架内进行的。建议的第一部分应该适用于库德拉在他的北京ICM演讲中所描述的数论，也应该应用于弦理论(根据陆爱柯的预印本)。提案的第二部分是因为这个问题完成了19世纪末和20世纪初数学家们解决一个一百年前的问题的工作。第三部分涉及表示论中的基本问题，如张量积分解和分支公式，如果解决了这些问题，将在许多学科中使用。所有上述项目都是与美国或国外的其他数学家合作的。PI目前与8位数学家合作，延续了广泛合作的历史(超过40篇联合论文)。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The PI (John Millson) proposes three main lines of research all within the general framework of reductive algebraic groups and geometry. In the first main line (with Jens Funke) the PI continues his project of using the theta correspondence and differential geometry to constructSiegel (resp. Hermitian) modular forms that are generating functions for the intersection numbers of certain (special) cyclesin locally symmetric spaces of orthogonal (resp. unitary) groups. The second main line deals with the ring R of projective invariants of n ordered points on theprojective line. This ring was much studied by the invariant theorists in the late eighteenth and early twentieth centuries.In 1894, A. Kempe found that R was generated by certain invariants of lowest degree. In the last year, working with Ben Howard, Andrew Snowden and Ravi Vakil, the PI computed the relations between the Kempe generators. They are quadratic, binomial and havea simple graphical description. The third main line is a continuation of the project dealing with the generalized triangle inequalities and related (saturation)problems from algebra. This project was the subject of the PI's most recent NSF grant, an FRG grant with Prakash Belkale, Thomas Haines, Misha Kapovich and Shrawan Kumar. This project is joint with Thomas Haines and Misha Kapovich. The PI (John Millson) proposes three main lines of research all within the general framework of reductive algebraic groups and geometry.The first part of the proposal should have applications to number theory along the lines described by S. Kudla inhis Beijing ICM talk and also to string theory (according to a preprint of Ai-Ko Lu).. The second part of the proposal is motivated in part becausethe problem completes work of mathematicians working in the late nineteenth and early twentieth centuries solving a one hundred year old problem. The third part deals with basic problems in representation theory e.g. decomposing tensor products and branching formulas which if solved would be of use inmany disciplines.All the above projects are in collaboration with other mathematicians from within the USA or abroad. The PI presentlyhas collaborations with eight mathematicians continuing a history of extensive collaboration (over forty joint papers).