RII Track-4: Applied Symplectic Topology

拓扑学是数学的一个分支，广义地说，它研究空间的形状。这个概念的“形状”可以是非常抽象的：辛拓扑学是研究“形状”的能量表面的物理系统的能量守恒。统计学中的一个重要问题是确定数据的“形状”。例如，经典的工具包括线性回归来找到最佳拟合线-但是如果数据没有这样简单的形状呢？最近的发展带来了强大的工具和思想，从抽象的拓扑承担统计问题。该奖学金将发展密西西比大学和TGDA@OSU跨部门研究小组之间的新合作，在俄亥俄州州立大学应用拓扑，以适应这些方法的问题辛拓扑，从而开辟了一个新的研究领域：概率辛拓扑。 所开发的方法将是辛拓扑中使用的第一批概率工具之一，并将开辟一个新的前景的问题考虑。该项目将扩大PI与工程和科学领域的同事合作的能力，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学的技术和思想与辛拓扑学问题相结合。辛拓扑中的一个关键问题是在给定的作用量和指数范围内，或者更精确地说，从过滤链复形中，哈密顿系统的周期轨道的存在性。由此构造了大量的辛不变量，如辛容量。PI将研究使用应用拓扑学的工具（如持久同源性）来构建和计算新的辛不变量的潜力。 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开发的研究随机复形的技术。这将在概率辛拓扑的新领域中开辟许多问题。此外，离散辛拓扑框架的发展将使有效的计算不变量，允许在具体的动力系统的研究中的应用。这个项目，通过PI和他的研究生的应用拓扑学的技术曝光，将另外带来一个新的专业领域的密西西比。这个奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.