Differential Structures

摘要建议：DMS-9802646首席研究员：Eugenio Calabi Eugenio Calabi建议继续他对变分问题和偏微分方程的研究，这些问题来自流形几何的各个方面。 赫尔曼格鲁克建议继续他的研究谱几何的扭曲的knots和螺旋的向量场，嵌入和knots的正曲率曲面在3-空间，并在存在性和唯一性的体积最小化循环在格拉斯曼流形。 JuliusShaneson计划将高维Euler-MacLaurin公式的余项应用于数论中与格点计数有关的一些问题，并继续他对奇异空间、特征类和相关分类问题的研究; Wolfgang Ziller计划继续他对正curvedcoordinity-one流形、极小等距浸入球面、弱对称空间和李群的原始子群的研究。卡拉比的工作的一个方面是应用计算机图像增强的方法，不干扰光学失真，由于投影到一个摄影板。 格鲁克工作的一个方面适用于DNA的扭曲和盘绕，以及蟹状星云中磁场线的扭曲。 Shaneson计划使用Euler-MacLaurin公式和余数来研究平面和高维空间的弯曲区域中的格点的分布，以期在数论中的一些基本问题上取得进展，并在涉及复杂网络的问题上取得一些实际应用。 Ziller计划与马里兰州大学的Karsten格罗夫联合继续研究具有正或非负截面曲率的共齐性一度量;寻找具有此属性的新例子对于几何学来说很重要，但也非常困难，正如人们从大约每15年才发现新例子的事实中可以看出的那样。

AbstractProposal: DMS-9802646Principal Investigator: Eugenio Calabi Eugenio Calabi proposes to continue his research on variationalproblems and partial differential equations arising from variousaspects of the geometry of manifolds. Herman Gluck proposes tocontinue his research on spectral geometries for the writhing of knotsand the helicity of vector fields, on embedding and knotting ofpositive curvature surfaces in 3-space, and on the existence anduniqueness of volume-minimizing cycles in Grassmann manifolds. JuliusShaneson plans to apply higher dimensional Euler-MacLaurin formulaewith remainder to some problems in number theory related to countingof lattice points, and to continue his research on singular spaces,characteristic classes, and related classification problems.Wolfgang Ziller proposes to continue his research on positively curvedcohomogeneity-one manifolds, minimal isometric immersions intospheres, weakly symmetric spaces, and primitive subgroups of Liegroups. One aspect of Calabi's work is an application to computerized imageenhancing by a method that does not interfere with the opticaldistortion due to the projection into a photographic plate. Oneaspect of Gluck's work applies to the writhing and coiling of DNA andto the writhing of magnetic field lines in the Crab Nebula. Shanesonplans to use Euler-MacLaurin formulae with remainder to study thenumer of lattice points in curved regions of the plane andhigher-dimensional spaces, with a view towards progress on some basicquestions in number theory and also some practical applications toproblems involving complex networks. Ziller plans to continue, injoint work with Karsten Grove of the University of Maryland, the studyof cohomogeneity-one metrics with positive or nonnegative sectionalcurvature; finding new examples with this property is important forgeometry and also very difficult, as one can see by the fact that newones are discovered only about every 15 years.