Singularity Behavior in Some Geometric Variational Problems Sciences

摘要为NSF提案DMS-0306294，“奇点行为在一些几何变分问题”，罗伯特·哈特，P.I.该项目位于几何变分领域，处理奇点和能量集中的行为，为各种最佳或固定的功能，领域，或几何结构受几何或分析约束。指定的调查集中在能源和拓扑障碍之间的关系，流形之间的映射，液晶材料，不适当的切片多项式品种，运输问题，和放松energyminimizers的规律性。 在各种高维情形下，光滑映射极限的能量集中可能沿着无穷测度集发生，并与空间的同伦非平凡映射有关。对应于目标映射的任何有理同伦不变量的这种集中行为现在可以被描述。 液晶材料中的奇异性也将在几个方面进行研究。 本课程将探讨代数几何中多项式零集的非正交理论，以了解分析与偏微分方程中的相关行为。 联合运输系统，如邮路、循环系统和植物根系，都可以用最近发现的几何结构来描述。许多物理现象的基础是能量最小原理，根据这一原理，某些构型、场或几何形状的特征在于它们比竞争物体具有更少的能量或面积。 外部约束往往导致奇异性，其特征是在非常小的空间区域内发生结构的快速变化。 例如，人们观察到固体在应力下的位错缺陷，磁化材料中的畴壁，肥皂膜中的液体边缘和角落，以及各种液晶材料中的点，曲线和表面缺陷。 我们处理新的数学结构和必要的理论来解释和预测这种现象。 在这些问题中，纯数学的理论研究、应用数学的数学计算研究、观察和实验的现象学研究都是相辅相成的，都具有重要的科学作用。目前提出的研究有许多问题的动机和适用于科学和工程。新的几何变分工具目前在更广泛的科学界还不为人所知，而这些学术主题和技术的交流将是非常有用的。

Abstract for NSF Proposal DMS-0306294, "Singularity Behavior inSome Geometric Variational Problems", Robert Hardt, P.I.This project lies in the area of geometric variational calculus,treating the behavior of singularities and energy concentrationfor various optimal or stationary functions, fields, or geometricstructures subject to geometric or analytical constraints.Specified investigations focus on the relation between energy andtopological obstruction in mappings between manifolds, liquidcrystal materials, improper slicing of polynomial varieties,transport problems, and the regularity of relaxed energyminimizers. In various higher dimensional cases, energyconcentration of limits of smooth mappings may occur along setsof infinite measure and is related to homotopically nontrivialmappings of spaces. This concentration behavior corresponding toany rational homotopy invariant of the target mapping can now bedescribed. Singularities in liquid crystal materials will be alsostudied in several contexts. The theories of improperintersections of polynomial zero sets from algebraic geometrywill be investigated to understand related behavior in analysisand partial differential equations. Combined transport systemssuch as occurs in postal routes, the circulatory system, andplant root systems can be described variationally with recentlydiscovered geometric structures.Underlying many physical phenomena is a least energy principlewhereby certain configurations or fields or geometric shape aredistinguished by their property of having less energy or area thancompeting objects. The external constraints often lead tosingularities, which are characterized by rapid changes ofstructure occurring in very small spatial regions. For example,one observes dislocation faults in solids under stress, domainwalls in magnetized materials, liquid edges and corners in soapfilms, and point, curve, and surface defects in various liquidcrystal materials. We deal with new mathematical structures andtheories necessary to explain and predict such phenomena. Inthese problems, the theoretical studies of pure mathematics, thenumerical computational studies of applied mathematics, and thephenomenological studies from observation and experiment allbenefit each other and all have a crucial scientific role. Thepresent proposed research has many problems motivated by andapplicable to science and engineering. The new geometricvariational tools are not presently well-known in the broaderscientific community, and communication of these mathematicaltopics and techniques will be very useful.