Differential Structures

摘要建议：DMS-9802646首席研究员：Eugenio Calabi Eugenio Calabi建议继续他对流形几何各个方面产生的变分问题和偏微分方程的研究。Herman Gluck建议继续研究纽结扭曲和矢量场的螺旋性的谱几何，研究三维空间中正曲率曲面的嵌入和纽结，以及Grassmann流形中体积最小化圈的存在和唯一性。Julius Shaneson计划将带余项的高维Euler-MacLaurin公式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建议继续他对正曲率上齐性-1流形、极小等距浸入到球面、弱对称空间和Lieg群的本原子群的研究。卡拉比工作的一个方面是应用于计算机图像增强，这种方法不会干扰由于投射到照相底片上而造成的光学失真。格鲁克的工作的一个方面适用于DNA的扭曲和卷曲，以及蟹状星云中磁力线的扭曲。Shaneson计划使用带余数的Euler-MacLaurin公式来研究平面和高维空间中曲线区格点的计数，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有一些实际应用。齐勒计划继续与马里兰大学的卡斯滕·格罗夫合作，研究上齐性--具有正或非负截面曲率的一种度量；寻找具有这种性质的新例子对几何学来说很重要，也非常困难，因为人们可以从大约每15年才发现一次新例子的事实中看出。

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.