Geometric Applications of Exterior Differential Systems

外差速系统的几何应用

基本信息

批准号：
0305829
负责人：
Joseph Landsberg
金额：
$ 8.6万
依托单位：
Georgia Tech Research Corporation
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2003
资助国家：
美国
起止时间：
2003-08-01 至 2005-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0305829&HistoricalAwards=false
关键词：
Geometric Applications Exterior Differential Systems

项目摘要

PROPOSAL DMS-0305829PI: Joseph M. LandsbergTitle: Geometric applications of exterior differential systemsABSTRACTDr. Landsberg will work on two projects: the dual defect conjectureand the classification of Legendrian varieties and contact Fano manifolds.The second project originates with a question in differential geometry,the classification of compact quaternionic-Kahler manifolds, which, thanks to work of LeBrun and Salamon, is essentially equivalent to the classification of contact Fano manifolds. Inside the tangent space of a general point of a contact Fano manifold is an immersed Legendrian variety and knowledge about such varieties is important for the contact Fano problem. The Salamon conjecture is that all contact Fano manifoldsare homogeneous. The first project is to show that the dual variety of a smooth variety of codimension less than half its dimension is a hypersurface. This question is of importance in algebgraic geometry as part of the investigations motivated by Hartshornes's conjecture on complete intersections. The techniques that will be developed to solve this problem are as important as the problem iself- a combination of exterior differential systems and representation theory that will be useful for other problems in geometry and other areas of science such as computational complexity and algebraic statistics (Baysean networks).Dr. Landsberg will use techniques from partial differential equations(more precisely, exterior differential systems) and representationtheory (of simple Lie algebras) to study two problems in geometryrelated respectively to dual varieties and contact Fano manifolds. In addition to the importance of these questions to the mathematical community, the techniques that will be developed will be applicable to problems in other fields such as computational complexity and algebraic statistics (Baysean networks).

提案 DMS-0305829PI：Joseph M. Landsberg 标题：外部差速系统的几何应用摘要博士。兰兹伯格将致力于两个项目：对偶缺陷猜想以及勒让德簇和接触法诺流形的分类。第二个项目源于微分几何中的一个问题，紧致四元数-卡勒流形的分类，这要归功于勒布伦和萨拉蒙的工作，本质上等价于接触 Fano 流形的分类。在接触 Fano 流形的一般点的切线空间内是浸没的勒让德簇，有关此类簇的知识对于接触 Fano 问题非常重要。萨拉蒙猜想是所有接触法诺流形都是齐次的。第一个项目是证明小于其维数一半的光滑簇余维的对偶簇是超曲面。这个问题在代数几何中非常重要，是由哈茨霍恩斯完全交集猜想引发的研究的一部分。为解决这个问题而开发的技术与问题本身一样重要——外部微分系统和表示理论的结合，对于几何中的其他问题和其他科学领域，如计算复杂性和代数统计（贝叶斯统计）很有用网络）。兰兹伯格将使用偏微分方程（更准确地说，外微分系统）和表示论（简单李代数）的技术来研究分别与对偶簇和接触 Fano 流形相关的两个几何问题。除了这些问题对数学界的重要性之外，将开发的技术还将适用于其他领域的问题，例如计算复杂性和代数统计（贝叶斯网络）。