Singularity Behavior in Some Geometric Variational Problems

摘要奖：DMS-0072486首席研究员：罗伯特·M·哈特这个项目涉及几何变分领域，研究几何或解析约束下各种最优或静止的函数、场或几何结构的奇点行为和能量集中。具体的研究集中在流形、铁磁材料和液晶材料之间映射中的能量与拓扑阻塞的关系，多项式簇的不适当切片，卡诺群中电流的紧性，以及松弛能量极小子的正则性。我们将研究p-能量最小化映射的奇点极限在p接近临界幂时沿面积最小化集合的能量集中。在各种高维情形下，光滑映射极限的能量集中沿无穷大测度集运动，并与球面上的同伦非平凡映射有关。奇异性、亚磁性和液晶材料也将在静态和动态环境中进行研究。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。研究卡诺群中的电流理论将着眼于变分问题的应用。在许多物理现象的背后是一个最小能量原理，根据该原理，某些构型、场或几何形状是通过它们的能量较少或面积较小的竞争物体的性质来区分的。外部约束往往导致单一性，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，人们可以观察到应力作用下固体中的位错错，磁化材料中的磁化壁，超导材料中的涡旋，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物理学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。

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.