Differential Structures

摘要/ abstract摘要：dms -9802646首席研究员：Eugenio Calabi Eugenio Calabi计划继续他在流形几何各方面引起的变分问题和偏微分方程方面的研究。赫尔曼·格拉克建议继续他的研究：关于扭结和矢量场螺旋度的谱几何，关于三维空间中正曲率曲面的嵌入和结，关于格拉斯曼流形中体积最小化环的存在性和唯一性。JuliusShaneson计划将高维欧拉-麦克劳林公式（Euler-MacLaurin）与余式应用于数论中与点位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打算继续他关于正弯曲同齐次- 1流形、最小等长球体浸没、弱对称空间和李群的原始子群的研究。卡拉比工作的一个方面是通过一种不干扰由于投影到照相板上而产生的光学畸变的方法来应用计算机图像增强。格拉克工作的一个方面适用于DNA的扭曲和盘绕，以及蟹状星云中磁力线的扭曲。shaneson计划用带余数的欧拉-麦克劳林公式来研究平面弯曲区域和高维空间中点位的数量，以期在数论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。齐勒计划继续与马里兰大学的卡斯滕·格罗夫（Karsten Grove）合作，研究同质性——一个具有正或非负截面曲率的指标；寻找具有这种性质的新例子对几何学来说很重要，但也很困难，因为我们可以看到，每隔15年才会发现一个新的例子。

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.