Singularity Behavior in Some Geometric Variational Problems

DMS-0905909关于某些几何变分问题中的奇点行为罗伯特·哈特(赖斯大学)这个项目位于几何变分领域，研究各种可能受约束的最优或静态函数、域、测量或几何结构的奇点和集中结构的形成和行为。第一类具体的项目包括继续与T.Riviere合作，研究黎曼流形与其同伦类之间映射的第p次方能之间的关系。在各种高维情形下，光滑映射到流形的极限的能量集中可以产生新的几何拓扑非平凡对象，并且与流形的非零同伦有关。现在可以描述对应于任何非扭转同伦不变量的这种浓度行为，并且正在为变分问题研究与扭转不变量有关的气泡现象。我们还攻击高阶Soblev空间，它对于某些同伦类来说似乎更自然，但其基本逼近结果和构造以前还没有被研究过。与蒂埃里·德·保的合作涉及到研究具有一般系数组的一般度量空间中的链、余链、电荷和高维变分。我们考虑了各种质量型泛函和平坦链的概念，它推广了Ambrosio-Kirchheim的有限质量度量空间流和B.White的可直平坦欧几里德空间G链。从几何测度论中推广的半代数映射、链、形式和各种结构在与Pascal Lambrechts一起研究代数簇的拓扑(包括实同伦理论)中继续被证明是有用的。另外，品种的度量性质也要用特殊类别的度量链和余链来探讨。其他研究包括微结构计算、输运形状组合问题及其在成像中的应用，以及最优桁架的存在和正则性。在纯数学和应用数学中，许多变分问题的解经常被迫具有奇异性，即涉及发生大振荡的区域。例如，球形容器中的向列相液晶材料，其光轴被迫指向容器外，其内部必然会有奇点(通过交叉偏振器或x射线衍射可以观察到)。在该示例中，光轴具有能量密度，该能量密度测量其局部变化率，并且其积分在所有可能的构型中倾向于具有最小值。我们的研究建议理解这类变分问题中的能量之间的关系，以及这些问题的物理所施加的拓扑障碍。我们已经导出了新的概念，允许处理和精确地描述从肥皂膜(局部最小化面积)及其高维推广到各种复杂介质中的最佳传输路径的各种问题的精确几何和解析描述。几何变分在这些应用中的目标是开发足够的数学工具来模拟、计算和预测物理行为。在许多物理问题中自然出现的几何约束导致了新的数学和计算问题。特别是，我们现在正在研究的三个问题涉及图像处理中的流动问题，某些晶体材料中的微结构形成，以及大数据集的几何分析。

Abstract for DMS - 0905909Singularity Behavior in Some Geometric Variational Problems Robert Hardt (Rice University)This project lies in the area of geometric calculus of variations, which treats the formation and behavior of singularities and concentration structures for various optimal or stationary functions, fields, measures, or geometric structures, possibly subject to constraints. The first specific class of projects involves continuing work with T. Riviere, on relations between the pth power energy of a map between Riemannian manifolds and its homotopy class. In various higher dimensional cases, energy concentration of limits of smooth mappings to a manifold may produce new geometric topologically nontrivial objects and is related to the nonvanishing homotopy of the manifold. This concentration behavior corresponding to any nontorsion homotopy invariant can now be described, and bubbling related to torsion invariants is being investigated for variational problems. We also are attacking higher order Sobolev spaces which seem more natural for certain homotopy classes, but for which basic approximation results and constructions have not been previously studied. Work with Thierry De Pauw involves the study of chains, cochains, charges, and the higher dimensional calculus of variations in general metric spaces with general coefficient groups. We consider a variety of mass-type functionals and the notion of a flat chain which generalizes the finite mass metric-space currents of Ambrosio-Kirchheim and the rectifiable and flat Euclidean-space G- chains of B.White. Semi-algebraic maps, chains, forms, and various structures generalized from geometric measure theory continue proving useful in work with Pascal Lambrechts on the topology of algebraic varieties, including the real homotopy theory. Also metric properties of varieties are to be approached using special classes of metric chains and cochains. Other studies include microstructure computation, combined transport-shape problems with applications to imaging, and the existence and regularity of optimal trusses.Solutions to many variational problems in both pure and applied mathematics often are forced to have singularities, that is, to involve regions where large oscillations occur. For example a nematic liquid crystal material in a spherical container whose optical axis is forced to point outward on the container necessarily will have singularities inside (observable through cross-polarizers or x-ray diffraction). In this example the optical axis has an energy density, which measures its local rate of change and whose integral tends to have a minimum value among all possible configurations. Our research proposes to understand the relationship between energies in such variational problems and the topological barriers imposed by the physics of these problems. We have derived new notions which allow the treatment and precise geometric and analytic description of a wide variety of problems from soap films (which locally minimize area) and their higher dimensional generalizations to optimal transport paths in various complex media. The goal in these applications of geometric calculus of variations is to develop sufficient mathematical tools to model, compute, and predict physical behavior. Geometric constraints which occur naturally in many physical problems have led to new mathematical and computational issues. In particular, three that we are now studying involve flow problems in image processing, microstructure formation in certain crystalline materials, and geometric analysis of large data sets.