RII Track-4: Applied Symplectic Topology

RII Track-4：应用辛拓扑

• 批准号: 1929176
• 负责人: Samuel Lisi
• 金额: $ 19万
• 依托单位: University of Mississippi
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-12-01 至 2023-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1929176&HistoricalAwards=false
• 关键词: RII; Track; Applied; Symplectic; Topology

Topology is the branch of mathematics that, broadly speaking, studies the shape of space. This notion of "shape" can be very abstract: symplectic topology is the study of the "shape" of energy surfaces of physical systems with conservation of energy. An important question in statistics is to determine the "shape" of data. Classical tools include, for instance, linear regression to find the line of best fit -- but what if the data doesn't have such a simple shape? Recent developments have brought powerful tools and ideas from abstract topology to bear on problems in statistics. This fellowship will develop a new collaboration between the University of Mississippi and the TGDA@OSU interdepartmental research group on applied topology at the Ohio State University in order to adapt these methods to problems in symplectic topology, thus opening up a new field of study: probabilistic symplectic topology. The methods developed will be among the first probabilistic tools used in symplectic topology and will open a new vista of questions to consider. This project will expand the PI's ability to collaborate with colleagues in engineering and the sciences, strengthen the UM's research program in topology and dynamics, and enhance graduate and undergraduate education.The goal of this project is to combine the techniques and ideas of applied topology with problems of symplectic topology. One of the key problems in symplectic topology concerns the existence of periodic orbits of a Hamiltonian system within a given range of actions and indices, or to be more precise, from filtered chain complexes. From this are constructed a large number of symplectic invariants such as symplectic capacities. The PI will investigate the potential to construct and compute new symplectic invariants using the tools of applied topology, such as persistent homology. The PI will also study the distribution of symplectic capacities for a (suitably defined) random convex domain in 4-space, both numerically by using the computational methods of applied topology, and theoretically, using the techniques developed by Kahle to study random complexes. This will open up many questions in the new area of probabilistic symplectic topology. Furthermore, a development of the framework of discretized symplectic topology will enable the effective computation of invariants, allowing for applications in the study of concrete dynamical systems. This project, through the PI and his graduate student's exposure to the techniques of applied topology, will additionally bring a new area of expertise to Mississippi.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

拓扑学是数学的一个分支，广义上讲，研究空间的形状。这种“形状”的概念可能非常抽象：辛拓扑是对具有能量守恒的物理系统的能量表面“形状”的研究。统计学中的一个重要问题是确定数据的“形状”。例如，经典工具包括线性回归来找到最佳拟合线，但如果数据没有如此简单的形状怎么办？最近的发展带来了来自抽象拓扑的强大工具和思想来解决统计问题。该奖学金将在密西西比大学和俄亥俄州立大学应用拓扑学 TGDA@OSU 跨部门研究小组之间开展新的合作，以使这些方法适应辛拓扑问题，从而开辟一个新的研究领域：概率辛拓扑。 所开发的方法将成为辛拓扑中最早使用的概率工具之一，并将开启一个需要考虑的问题的新前景。该项目将扩大 PI 与工程和科学界同事合作的能力，加强 UM 在拓扑和动力学方面的研究计划，并加强研究生和本科生教育。该项目的目标是将应用拓扑的技术和思想与辛拓扑问题结合起来。辛拓扑的关键问题之一涉及哈密顿系统在给定的作用和指数范围内（或者更准确地说，来自过滤链复合体）的周期轨道的存在性。由此构造了大量辛不变量，例如辛容量。 PI 将研究使用应用拓扑工具（例如持久同调）构建和计算新辛不变量的潜力。 PI 还将研究 4 空间中（适当定义的）随机凸域的辛容量分布，无论是在数值上使用应用拓扑的计算方法，还是在理论上使用 Kahle 开发的技术来研究随机复数。这将在概率辛拓扑的新领域提出许多问题。此外，离散辛拓扑框架的发展将能够有效计算不变量，从而在具体动力系统的研究中得到应用。该项目通过 PI 及其研究生对应用拓扑技术的接触，还将为密西西比州带来一个新的专业领域。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。