RII Track-4: Applied Symplectic Topology

拓扑学是广义上研究空间形状的数学分支。这个“形状”的概念可以是非常抽象的：辛拓扑是研究具有能量守恒的物理系统的能量表面的“形状”。统计学中的一个重要问题是确定数据的“形状”。经典的工具包括，例如，线性回归，以找到最适合的线，但如果数据不是这样一个简单的形状呢？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俄亥俄州立大学应用拓扑跨部门研究小组TGDA@OSU之间的新合作，以便将这些方法应用于辛拓扑问题，从而开辟一个新的研究领域：概率辛拓扑。所开发的方法将是在辛拓扑中使用的第一批概率工具之一，并将打开一个新的问题要考虑的前景。该项目将扩大PI与工程和科学同事合作的能力，加强UM在拓扑和动力学方面的研究计划，并提高研究生和本科生的教育水平。这个项目的目标是将应用拓扑的技术和思想与辛拓扑的问题结合起来。辛拓扑中的一个关键问题是在给定的作用和指标范围内，或者更准确地说，在过滤链配合物中，哈密顿系统周期轨道的存在性。由此构造了大量辛不变量，如辛容量。PI将研究使用应用拓扑工具（如持久同调）构建和计算新的辛不变量的潜力。PI还将研究（适当定义的）随机凸域在4-空间中的辛容量分布，包括数值上使用应用拓扑的计算方法，以及理论上使用Kahle开发的技术来研究随机复合体。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不变量的有效计算成为可能，从而可以应用于具体动力系统的研究。这个项目，通过PI和他的研究生对应用拓扑技术的接触，将为密西西比带来一个新的专业领域。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.