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）资助的。PI（约翰·米尔森）提出了三条主要的研究路线，都是在简化代数群和几何的一般框架内进行的。在第一个主线中（与Jens Funke一起），PI继续他的项目，使用theta对应和微分几何来构建siegel(参见。在正交的局部对称空间中，为某些（特殊）环的交数生成函数的厄密模形式。酉)组。第二条主线处理射影线上n个有序点的射影不变量的环R。在18世纪末和20世纪初，不变量理论家对这个环进行了大量研究。1894年A. Kempe发现R是由某些最低次不变量产生的。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，计算了肯普发电机之间的关系。它们是二次的，二项式的，并且有一个简单的图形描述。第三条主线是项目的延续，处理广义三角形不等式和代数相关（饱和）问题。该项目是PI最近获得的NSF资助的主题，是Prakash Belkale， Thomas Haines， Misha Kapovich和Shrawan Kumar获得的FRG资助。这个项目是与托马斯·海恩斯和米沙·卡波维奇合作的。PI（约翰·米尔森）提出了三条主要的研究路线，都是在简化代数群和几何的一般框架内进行的。该提议的第一部分应该适用于S. Kudla在北京ICM演讲中描述的数论以及弦理论（根据Ai-Ko Lu的预印本）。这个提议的第二部分的动机部分是因为这个问题完成了数学家在19世纪末和20世纪初的工作，解决了一个百年历史的问题。第三部分讨论了表示理论中的一些基本问题，如分解张量积和分支公式，这些问题的解决将在许多学科中得到应用。以上所有项目都是与美国或国外的其他数学家合作完成的。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).