Singularity Behavior in Some Geometric Variational Problems Sciences

国家自然科学基金申请DMS-0306294，“一些几何变分问题中的奇点行为”，Robert Hardt, p.i.l。本项目属于几何变分微积分领域，研究受几何或解析约束的各种最优或平稳函数、场或几何结构的奇点行为和能量集中。特定的研究集中在流形之间映射的能量和拓扑障碍之间的关系，液晶材料，多项式变量的不适当切片，传输问题以及松弛能量最小化的规律性。在各种高维情况下，光滑映射极限的能量集中可能沿无穷测度集发生，并且与空间的同伦非平凡映射有关。这种与目标映射的任意有理同伦不变量相对应的集中行为现在可以被描述。液晶材料中的奇点也将在一些情况下进行研究。我们将研究代数几何中多项式零集的反常相交理论，以了解其在分析和偏微分方程中的相关行为。联合运输系统，如邮政路线、循环系统和植物根系，可以用最近发现的几何结构进行不同的描述。许多物理现象的基础是一个最小能量原理，即某些结构、场或几何形状的特征是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点，其特征是结构在很小的空间区域内发生快速变化。例如，人们可以观察到应力作用下固体中的位错缺陷，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。我们处理解释和预测这些现象所必需的新的数学结构和理论。在这些问题中，纯数学的理论研究、应用数学的数值计算研究和观察与实验的现象学研究相互促进，都具有至关重要的科学作用。目前提出的研究有许多问题是由科学和工程驱动和适用的。新的几何变分工具目前在更广泛的科学界还不为人所知，这些数学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.