Dynamics, Embeddings, and Continuous Symplectic Geometry

这是一个关于随时间进化的系统的项目，叫做动力系统。动力系统是所有科学的基础，但其研究背后的数学是非常困难的，许多重要的问题仍然是开放的，需要新的想法。该项目的一个主题是使用几何方法为该领域带来新的见解。最近，PI和合作者使用辛几何的技术解决了一个关于某些动力系统的长期问题。他们证明了一个特定的动力系统需要无限的能量才能产生，并以此在系统集合中建立了一个新的二分法：有限能量的系统，以及其他的。这种二分法将使人们对动力系统的分类有更深入的了解，并应允许解决几个引起广泛兴趣的问题。这个项目涉及这个方向的进一步研究。该项目的一个目标是找到有限能量系统的进一步二分法。通过引入几何方法来解决动力系统的问题，这些问题无法用其他方法解决，该项目旨在展示这类技术的力量。同时，该方法的核心思想来自于研究一种辛形状何时可以变形为另一种辛形状的子集，在这些方向上的研究应该加深我们对辛几何的理解。该项目还包含大量的教育内容。在此之前，PI一直致力于通过监狱教育、高中和本科阶段的几何研究培训以及课程设计来扩大数学的参与，促进公平，并支持该领域的教育工作。该项目支持这项工作和相关活动。本项目涉及辛动力学、辛嵌入和连续辛几何等相关领域的一系列研究调查。主要的重点是在低维的问题，其中强大的规范理论技术是可用的。PI和合作者最近解决了长期存在的“简单性猜想”，部分原因是他们提出了周期花同调谱不变量理论。该项目旨在进一步发展这一理论，并解决几个长期存在的问题；例如，证明高等属的简单性猜想的类似物。该项目还旨在澄清Reeb流的动力学，研究辛嵌入问题中的无限阶梯，研究拓扑辛流形问题，并研究高维辛填充问题。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。

This is a project about systems that evolve in time, called dynamical systems. Dynamical systems are fundamental across the sciences, but the mathematics underlying their study is very difficult, many important questions remain open, and new ideas are needed. A main theme of this project is to use a geometric approach to bring fresh insight to this field. Recently, the PI and collaborators used techniques from symplectic geometry to settle a longstanding question about certain dynamical systems. They showed that a particular dynamical system would take an infinite amount of energy to produce and used this to establish a new dichotomy in the set of systems: the finite-energy ones, and the rest. This dichotomy will enable a deeper understanding of the classification of dynamical systems and should allow for the resolution of several questions that have attracted wide interest. This project involves further research in this direction. One goal of the project is to find a further dichotomy for the finite-energy systems. By bringing a geometric approach to bear on questions about dynamical systems that have resisted solution by other methods, the project aims to showcase the power of these kinds of techniques. At the same time, an idea at the heart of the approach comes from studying when one symplectic shape can be deformed into a subset of another, and research in these directions should deepen our understanding of symplectic geometry. The project also has a substantial education component. The PI has previously worked to broaden participation in mathematics, foster equity, and support educational efforts in the field through prison education, research training in geometry at the high school and undergraduate levels, and course design. The project supports this work and related activities. This project involves a series of research investigations touching on the related fields of symplectic dynamics, symplectic embeddings, and continuous symplectic geometry. The main emphasis is on questions in low dimensions where powerful gauge theoretic techniques are available. The PI and collaborators recently settled the longstanding "Simplicity Conjecture" partly by contributing to the theory of periodic Floer homology spectral invariants. The project aims to further develop this theory and solve several longstanding questions; for example, to prove an analogue of the Simplicity Conjecture in higher genus. The project also aims to clarify the dynamics of Reeb flows, study infinite staircases in symplectic embedding problems, investigate questions about topological symplectic manifolds, and study higher-dimensional symplectic packing problems.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.