Singularity Behavior in Some Geometric Variational Problems

这个项目属于几何变分学的领域，它处理各种最优或平稳函数、场、测量或几何结构的奇点的形成和行为，可能受到确定性和随机约束。第一类具体的项目包括与Thierry De Pauw （Paris VI）继续合作，研究链、共链、电荷以及具有一般系数群的一般度量空间中的高维变分。我们考虑了各种质量型泛函和推广了Ambrosio-Kirchheim有限质量度量空间流的平坦链的概念，以及B.White和Fleming的可整流平坦欧几里得空间g链，用这种链定义的同调理论给出了度量空间拓扑和几何之间的相互作用。例如，初步的工作显示了如何用合适的0和1维平面同调群来表征Lipschitz路径连通性或有限质量跨越曲面的存在。变分上同调可以用电荷来处理，电荷是对正电流对偶的协链，适当地拓扑化，并且通常允许用连续形式对来表示。其次，我们继续与T. Riviere （ETH）一起研究黎曼流形之间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，我们将考虑由微分范数、黑森范数等的幂积分所给出的能量。关键维的一个关键问题是生成具有给定非平凡拓扑的映射所需的最小能量，特别是，当拓扑退化时，这个最小值如何渐近增长。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Betul Orcan （Rice）一起，我们建议开始研究几何测量理论中由噪声数据或噪声依赖数据的随机变分问题引起的几何结构。这项工作应该基于定量的低规律性结果和定量描述大多数数据可能的高规律性。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，在许多种类的液晶中，光轴可能被迫在点附近或沿着曲线或区域之间的壁快速振荡。我们的研究旨在理解这类变分问题中的能量与问题物理所施加的拓扑障碍之间的关系。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，我们需要使用几何测量理论的完整语言来获得足够一般的几何结构和对象来处理这些问题。同样在应用中，材料中杂质的存在或测量和数据采集中的噪声也需要考虑。因此，开发数学工具将噪声和随机因素与几何结构结合起来是非常有用的。我们提出研究的两个特殊问题涉及图像处理中的流动问题和生物学中具有噪声数据的高维分支结构的生长。

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.