Geometric Applications of Exterior Differential Systems

提案DMS-0305829PI：Joseph M.Landsberg标题：外微分系统的几何应用摘要。Landsberg将致力于两个项目：对偶缺陷猜想以及Legendrian簇和接触Fano流形的分类。第二个项目起源于微分几何中的一个问题，紧致四元数-Kahler流形的分类，由于LeBrun和Salamon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一个浸入的Legendrian簇，关于这种簇的知识对于接触Fano问题是很重要的。Salamon猜想是所有接触Fano流形都是齐次的。第一个项目是证明余维小于一半的光滑簇的对偶簇是一个超曲面。这个问题在代数几何中是重要的，因为它是由HartShornes关于完全交的猜想所激发的研究的一部分。为解决这一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将对几何和其他科学领域的其他问题有用，如计算复杂性和代数统计(贝叶斯网络)。Landsberg将使用偏微分方程组(更准确地说是外微分系统)和表示理论(单李代数)中的技巧来研究几何中的两个问题，分别与对偶簇和接触Fano流形有关。除了这些问题对数学界的重要性外，将要开发的技术还将适用于其他领域的问题，如计算复杂性和代数统计(贝叶斯网络)。

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).