Singularity Behavior in Some Geometric Variational Problems Sciences

美国国家科学基金会建议DMS-0306294，“某些几何变分问题中的奇点行为”，Robert Hardt，P.I.这个项目位于几何变分领域，处理各种受几何或分析约束的最佳或稳定的函数、场或几何结构的奇点和能量集中的行为。具体的研究集中在流形、液晶材料、多项式簇的不正确切片、输运问题以及松弛能量最小化的正则性等方面。在各种高维情形下，光滑映射的极限的能量集中可能发生在无穷测度集上，并且与空间的同伦非平凡映射有关。现在可以描述对应于目标映射的任何有理同伦不变量的这种浓度行为。液晶材料中的奇点也将在几种情况下被研究。我们将研究代数几何中多项式零点集的不相交理论，以了解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。在许多物理现象下面是最小能量原理，根据该原理，某些结构或场或几何形状通过其具有比竞争物体更小的能量或面积的特性来区分。外部约束经常导致单一性，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，人们观察到在应力作用下固体中的位错错，磁化材料中的磁化壁，肥皂膜中的液体边和角，以及各种液晶材料中的点、曲线和表面缺陷。我们讨论解释和预测这种现象所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、观察和实验的现象学研究都是相辅相成的，都具有至关重要的科学作用。本文提出的研究有许多问题是出于科学和工程的动机和适用。新的几何变分工具目前在更广泛的科学界并不广为人知，这些数学主题和技术的交流将非常有用。

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.