Singularity Behavior in Some Geometric Variational Problems

摘要奖：DMS-0072486主要研究者：Robert M. Hardt这个项目属于几何变分学领域，处理各种最佳或固定函数、场或几何结构在几何或分析约束下的奇点和能量集中行为。具体的研究集中在流形、铁磁和液晶材料之间映射的能量和拓扑阻塞之间的关系、多项式簇的不适当切片、卡诺群中电流的紧性以及松弛能量极小的正则性。我们将研究p-能量极小化映射的奇异性极限在p接近临界幂时沿沿着面积极小化集的能量集中。 在各种高维情形下，光滑映射极限的能量集中可能发生在沿着无穷测度集上，并且与球面上的同伦非平凡映射有关。奇异性、铁磁性和液晶材料也将在静态和动态的背景下进行研究。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。卡诺群中的电流理论将着眼于变分问题的应用进行研究。许多物理现象的基础是一个最小能量原理，根据这个原理，某些构型、场或几何形状的特征在于它们具有比竞争物体更少的能量或面积。外部约束往往导致奇点，这是一种特殊的点，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，人们可以观察到固体在应力下的位错缺陷，磁化材料中的畴壁，超导材料中的涡旋，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理论必要的解释和预测这样的现象。 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物理学的唯象研究都是相辅相成的，都具有重要的科学作用。

AbstractAward: DMS-0072486Principal Investigator: Robert M. HardtThis 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 analytic constraints. Specificinvestigations focus on the relation between energy andtopological obstruction in mappings between manifolds,ferromagnetic and liquid crystal materials, improper slicing ofpolynomial varieties, compactness of currents in Carnot groups,and the regularity of relaxed energy minimizers. We willinvestigate the energy concentration along area-minimizing setsfor limits of singularities of p-energy minimizing maps as papproaches a critical power. In various higher dimensionalcases, energy concentration of limits of smooth mappings mayoccur along sets of infinite measure and is related tohomotopically nontrivial mappings of spheres. Singularities inferromagnetic and liquid crystal materials will be also studiedin both stationary and dynamic contexts. The theories of improperintersections of polynomial zero sets from algebraic geometrywill be investigated to understand related behavior in analysisand partial differential equations. A theory of currents inCarnot groups will be studied with an eye on applications tovariational problems.Underlying many physical phenomena is a least-energy principlewhereby certain configurations or fields or geometric shapes aredistinguished by their property of having less energy or areathan competing objects. The external constraints often lead tosingularities, which are special points characterized by rapidchanges of structure occurring in very small spatial regions. Forexample, one observes dislocation faults in solids under stress,domain walls in magnetized materials, vortices in superconductingmaterials, liquid edges and corners in soap films, and point,curve, and surface defects in various liquid crystalmaterials. We deal with new mathematical structures and theoriesnecessary to explain and predict such phenomena. In theseproblems, the theoretical studies of pure mathematics, thenumerical computational studies of applied mathematics, and thephenomenological studies from physics all benefit each other andall have a crucial scientific role.