Variational Methods and Dynamics

这个项目的重点是在变分和偏微分方程的微积分问题。查尔斯·莫雷在上世纪中叶提出了一种理论，现在被称为变分法的直接方法，他在其中介绍了凸性的各种基本概念。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（John Ball）发表了一系列变分学中的突出问题，这些问题的解决方案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，首席研究员已经开始开发工具来处理该列表中的具体问题。他希望，这将引出一般理论。该项目将扩展这项研究，包括一些重要的、与现有理论相悖的变分问题，比如奥格登材料的储能函数，它出现在弹性中。另一类需要研究的问题出现在大众运输理论中。它最初是由Monge在1771年作为一个简单的几何问题表述的，后来由Kantorovich扩展到更一般的成本函数。它包括寻找将一堆土从一个位置运送到另一个位置的挖掘地点的最佳（最小工作量）方法。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，特别关注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，初步计算表明这是瓦瑟斯坦空间上的哈密顿系统。首席研究员已经将这些流动的研究与一类鲜为人知的（具有挑战性的）参数依赖的蒙日-安培方程建立了联系。他将研究满足一定基本稳定性判据并连接两个规定辛形式的路径的存在性。这些问题本身就很有趣，包括在辛形式集合中寻找测地线。该项目将包含许多重要的科学和教育组成部分。首先，理论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，在他们的算法中使用变分方法。其次，首席研究员将继续通过在国际会议、讲习班和研讨会上授课和演讲来传播研究成果。最后，除了继续指导他所在大学的本科生和博士生，他还将积极支持数学领域代表性不足的群体，部分方式是指导当地HBCU斯佩尔曼学院（Spelman College）的学生和教师。目前的项目包括Spelman的Yewande Olubummo教授，他和首席研究员一起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.