Variational Methods and Dynamics

这个项目关注的是变分和偏微分方程中的问题。查尔斯·莫里在上个世纪中叶提出了一种理论，现在被称为变分的直接方法，他在其中引入了各种基本的凸性概念。这一理论在许多领域都有广泛的应用，包括约翰·鲍尔在弹性理论方面的开创性工作。最近，约翰·鲍尔发表了一份变分中突出问题的清单，这些问题的解决不仅将推动该领域的发展，而且将对弹性理论产生重大影响。在最近的工作中，首席调查员已经开始开发工具来处理该清单中的具体问题。他希望，这将导致普遍的理论。该项目将扩展这项研究，以包括挑战现有理论的重要类别的变分问题，例如Ogden材料的储存能量泛函，它出现在弹性中。另一类需要研究的问题出现在大众交通理论中。它最初是由Monge在1771年提出的，是一个简单的几何问题，后来Kantorovich将其推广到更一般的成本函数。它包括找到最佳的(最小工作量)方法，将一个位置的一堆土运输到另一个位置的挖掘现场。这些经典的结果已经被首席研究员、他的学生、博士后、合作者和许多其他人扩展到更一般的成本函数，并特别注意Wasserstein(或Kantorovich)度量。在这些结果的基础上，该项目将研究所谓的轴对称流动，初步计算表明，这是沃瑟斯坦空间上的哈密顿系统。首席研究员建立了对这些流动的研究与一类知之甚少(具有挑战性)的参数相关的Monge-Ampere方程之间的联系。他将研究满足某种基本稳定性准则并连接两个指定辛形式的路径的存在性。这些问题本身就很有趣，包括寻找一组辛形式的测地线。该项目将有许多重要的科学和教育组成部分。首先，理论的进步将直接帮助其他科学研究领域，包括气象学、弹性理论和其他应用科学，这些领域在算法中使用变分方法。第二，首席调查员将继续通过在国际会议、讲习班和研讨会上举办课程和讲座来传播研究成果。最后，除了继续在他的母校指导本科生和博士生之外，他还将通过指导当地HBCU斯佩尔曼学院(Spelman College)的学生和教职员工，积极支持数学界代表性不足的群体。目前的项目包括斯佩尔曼的耶万德·奥卢布布莫教授，他和首席研究员计划开发用于形状识别研究的工具。

This project focuses on problems in the calculus of variations and partial differential equations. A theory, now called direct methods of the calculus of variations, was developed by Charles Morrey in the middle of the last century wherein he introduced various fundamental concepts of convexity. This theory has wide application to many fields, including John Ball's groundbreaking work in elasticity theory. Recently, John Ball published a list of outstanding problems in the calculus of variations whose solutions will not only advance that field but will also have major impacts in elasticity theory. In recent work, the principal investigator has started to develop tools to handle specific concrete problems from that list. This will, he hopes, lead to general theories. The project will extend this study to include important classes of variational problems that have defied existing theory, such as the stored energy functional of Ogden material, which appears in elasticity. Another class of problems to be studied arises in mass transportation theory. It was first formulated by Monge in 1771 as a simple geometry problem and later extended by Kantorovich to more general cost functions. It consists in finding the optimal (minimal work) method for transporting a pile of earth at one location to an excavation site at another location. These classical results have been extended to more general cost functions by the principal investigator, his students, postdocs, collaborators and many others, with special attention to the Wasserstein (or Kantorovich) metric. Building on these results, the project will investigate so-called axi-symmetric flows, which preliminary computations indicate are Hamiltonian systems on the Wasserstein space. The principal investigator has established a link between the study of these flows and a poorly understood (challenging) class of parameter-dependent Monge-Ampere equations. He will study the existence of paths satisfying a certain basic stability criterion and connecting two prescribed symplectic forms. These problems, interesting in their own right, include searching for geodesics in the set of symplectic forms.Tproject will have a number of important scientific and educational components. First, the theoretical advances will directly aid other scientific research areas, including meteorology, elasticity theory, and other applied sciences, that use variational methods in their algorithms. Second, the principal investigator will continue to disseminate the research by giving courses and lectures in international conferences, workshops, and seminars. Finally, in addition to continuing to mentor undergraduates and Ph.D. students at his home institution, he will extend his active support of underrepresented groups in mathematics, in part by mentoring students and faculty at Spelman College, a local HBCU. Current projects involve Spelman's Professor Yewande Olubummo, who together with the principal investigator plans to develop tools for use in the study of shape recognition.