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，都在还原代数群和几何的一般框架内。在第一条主线（与延斯芬克）的PI继续他的项目，使用θ对应和微分几何，以constructSiegel（resp.厄米特）模形式，是正交（分别）的局部对称空间中某些（特殊）圈的交数的生成函数。单一的）组。第二条主线研究射影直线上n个序点的射影不变量环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的研究。肯普发现R是由某些最低程度的不变量产生的。 去年，PI与本·霍华德、安德鲁·斯诺登和拉维·瓦基尔合作，计算了肯普生成器之间的关系。 它们是二次的，二项的，并且有一个简单的图形描述。 第三条主线是处理广义三角形不等式和相关的（饱和）问题的代数项目的延续。这个项目是PI最近的NSF资助的主题，FRG与Prakash Belkale，托马斯海恩斯，Misha Kapovich和Shrawan Kumar一起资助。这个项目是与托马斯海恩斯和米沙卡波维奇。PI（约翰·米尔森）提出了三条主要的研究路线，都在约化代数群和几何的一般框架内。建议的第一部分应该沿着S.库德洛在他的北京ICM演讲中也提到了弦理论（根据陆爱科的预印本）。该提案的第二部分的动机部分是因为这个问题完成了数学家在十九世纪末和二十世纪初解决一个百年老问题的工作。 第三部分涉及表示论中的基本问题，如张量积的分解和分支公式，这些问题如果得到解决，将在许多学科中得到应用。所有上述项目都是与美国或国外的其他数学家合作完成的。PI presentlyhas与八位数学家继续广泛合作的历史（超过40份联合论文）。

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).