Singularity Behavior in Some Geometric Variational Problems

一些几何变分问题中的奇异性行为

• 批准号: 1207702
• 负责人: Robert Hardt
• 金额: $ 24.22万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207702&HistoricalAwards=false
• 关键词: Singularity; Behavior; Some; Geometric; Variational

This project lies in the area of geometric calculus of variations, which treats the formation and behavior of singularities for various optimal or stationary functions, fields, measures, or geometric structures, possibly subject to constraints, both deterministic and stochastic. The first specific class of projects involves continuing work with Thierry De Pauw (Paris VI) concerning 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 and Fleming, Homology theories defined with such chains give an interplay between the topology and geometry of a metric spaces. For example, preliminary work as shown how Lipschitz path connectedness or the existence of finite mass spanning surfaces may be characterized by a suitable 0 and 1 dimensional flat homology groups. Variational cohomology may be treated by charges, which are cochains dual to normal currents, suitably topologized, and which often admit representation by pairs of continuous forms. Second we are continuing work with T. Riviere (ETH), on relations between various energies of maps between Riemannian manifolds and the homotopy classes of the maps. Following our recent work, we will consider energies given by integrating powers of the norm of the differential, the Hessian, etc. A key general question for critical dimensions is the minimum energy required to produce maps with a given nontrivial topology, in particular, how this minimum grows asymptotically as the topology degenerates. There are interesting concrete problems here for the rational homotopy of 4 manifolds. Third,with Betul Orcan (Rice) we propose to initiate work on geometric structures arising in geometric measure theory from stochastic variational problems with noisy data or noisy dependence on data. The work should be based on quantitative low regularity results and quantitatively described probable higher regularity for most data.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 among the many classes of liquid crystals, the optical axis may be forced to oscillate rapidly near points or along curves or along walls between regions. Our research proposes to understand the relationship between energies in such variational problems and the topological barriers imposed by the physics of the problems. Geometric constraints which occur naturally in many physical problems have led to new mathematical issues, and we need to use the full language of geometric measure theory to have sufficiently general geometric structures and objects to treat these issues. Also in applications, the presence of impurities in materials or of noise in measurements and data acquisition is important to consider. So it would be quite useful to develop mathematical tools to incorporate noise and stochastic considerations with the geometric structures. Two particular problems that we propose studying involve flow problems in image processing and the growth of higher dimensional ramified structures in biology with noisy data.

该项目位于变分几何学领域，它处理各种最佳或固定函数，字段，措施或几何结构的奇点的形成和行为，可能受到确定性和随机性的约束。第一类具体的项目涉及继续工作与蒂埃里德Pauw（巴黎六）有关的研究链，cochains，收费，和高维微积分的变化一般度量空间与一般系数组。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的可求长平坦欧氏空间G-链。白色和Fleming，定义了这样的链的同调理论给出了度量空间的拓扑和几何之间的相互作用。例如，初步工作显示如何Lipschitz路径连通性或存在的有限质量的跨越表面可能是其特征在于一个合适的0和1维平坦的同调群。变分上同调可以用电荷来处理，电荷是正规流的对偶上链，适当地拓扑化，并且经常允许用连续形式对来表示。第二，我们继续与T。里维埃（ETH），关于黎曼流形之间的映射的各种能量与映射的同伦类之间的关系。根据我们最近的工作，我们将考虑能量的积分权力的规范的微分，海森等一个关键的一般性问题的关键尺寸是最低的能源需要产生地图与一个给定的非平凡拓扑结构，特别是，如何这个最低增长渐近拓扑退化。对于4流形的有理同伦，这里有一些有趣的具体问题。第三，与贝图尔Orcan（水稻），我们建议发起工作的几何结构所产生的几何测度理论随机变分问题的噪声数据或噪声依赖于数据。这项工作应该是基于定量低正则性的结果和定量描述可能更高的规律性为大多数data.Solutions的许多变分问题在纯数学和应用数学往往被迫有奇点，也就是说，涉及区域发生大的振荡。例如，在许多种类的液晶中，可以迫使光轴在点附近或沿着曲线或沿着区域之间的壁快速振荡。我们的研究提出了理解能量之间的关系，在这样的变分问题和拓扑障碍所施加的物理问题。在许多物理问题中自然出现的几何约束导致了新的数学问题，我们需要使用完整的几何测度理论语言来处理这些问题。此外，在应用中，材料中杂质的存在或测量和数据采集中的噪声的存在也是重要的考虑因素。因此，它将是非常有用的，以开发数学工具，将噪声和随机考虑的几何结构。我们提出研究的两个特殊问题涉及图像处理中的流问题和生物学中具有噪声数据的高维分枝结构的生长。