Singularity Behavior in Some Geometric Variational Problems

罗伯特·哈特（美国莱斯大学）本课题属于几何变分学领域，研究各种可能受约束的最优或平稳函数、场、测度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p次幂能量之间的关系。在各种高维情况下，流形光滑映射极限的能量集中可能产生新的几何拓扑非平凡对象，这与流形的非消失同伦有关。这种与任何非扭转同伦不变量相对应的集中行为现在可以被描述，并且与扭转不变量相关的冒泡正在研究变分问题。我们也在研究高阶Sobolev空间，它对某些同伦类来说似乎更自然，但它的基本近似结果和构造以前还没有研究过。与Thierry De Pauw的工作涉及链，共链，电荷，以及具有一般系数群的一般度量空间的高维变分的研究。我们考虑了各种质量型泛函和推广了Ambrosio-Kirchheim的有限质量度量空间流和B.White的可整流、平坦欧几里得空间G链的平坦链的概念。半代数映射、链、形式和从几何测度理论推广出来的各种结构，在Pascal Lambrechts关于代数变体拓扑的工作中，包括实同伦理论，继续被证明是有用的。此外，用特殊的度量链和协链来研究品种的度量性质。其他研究包括微观结构计算，结合传输形状问题与成像的应用，以及最优桁架的存在性和规律性。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点，也就是说，涉及发生大振荡的区域。例如，在球形容器中的向列液晶材料，其光轴在容器上被迫指向外，必然会在内部具有奇点（通过交叉偏振器或x射线衍射观察到）。在这个例子中，光轴有一个能量密度，它测量其局部变化率，其积分在所有可能的构型中趋向于具有最小值。我们的研究旨在了解这些变分问题中的能量与这些问题的物理所施加的拓扑障碍之间的关系。我们已经得到了新的概念，这些概念允许处理和精确的几何和分析描述各种各样的问题，从肥皂膜（局部面积最小化）及其高维推广到各种复杂介质中的最佳传输路径。这些几何变分学应用的目标是开发足够的数学工具来建模、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。具体来说，我们正在研究的三个领域涉及图像处理中的流动问题、某些晶体材料中的微观结构形成以及大型数据集的几何分析。

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.