Geometric Applications of Exterior Differential Systems

提案DMS-0305829 PI：Joseph M.外微分系统的几何应用。兰茨贝格将工作的两个项目：双重缺陷的contradure和分类的勒让德品种和接触法诺流形。第二个项目起源于一个问题，在微分几何，分类的紧凑四元数卡勒流形，这是由于工作的勒布朗和Salamon，基本上是等同于分类的接触法诺流形。在切空间内的一般点的接触法诺流形是一个沉浸勒让德品种和知识，这种品种是重要的接触法诺问题。Salamon猜想是所有的切触Fano流形都是齐次的。 第一个项目是证明余维数小于其维数一半的光滑簇的对偶簇是超曲面。这个问题是重要的algebgraic几何的一部分，调查的动机Hartshornes的猜想完全交叉。解决这一问题的技术将与问题本身一样重要--外微分系统和表示论的结合将有助于解决几何学和其他科学领域的其他问题，如计算复杂性和代数统计兰茨贝格博士将使用偏微分方程的技术（更确切地说，外微分系统）和（单李代数的）表示论来研究分别与对偶簇和切触Fano流形有关的两个几何问题。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).