Holomorphic Curves in Embeddings and Dynamics

随着时间的推移而演化的系统被称为动力系统，出现在包括生物、经济、物理、科学和工程在内的广泛领域。描述这些系统的方程通常很难直接求解，因此需要新的工具和技术来研究它们。这个项目围绕着一种名为接触同调的工具展开，该工具近年来对许多类型的动力系统产生了强大的见解。接触同调提供了一个框架，用于寻找即使动力系统的参数允许变化也是守恒的动力系统的特征。通过这种方式，通过不断地将系统改变成更基本的形式，然后在这种更简单的状态下利用动力学知识来了解原始系统的结构，就可以研究复杂的系统。这个项目旨在进一步发展接触同调的基础，并利用它来产生对动力学的新见解。一个预期应用的例子是发现许多动力系统的新的非周期轨迹，这些轨迹是系统的配置，在时间上无限频繁地重新出现。预期的教育影响是为本科生的研究培训创造新的机会。在这一领域，PI计划继续他的努力，在过去的几年里，这些努力已经导致本科生在同行评议期刊上发表了两篇文章。*PI还计划继续致力于接触到更广泛的公众群体，例如通过参与监狱教育倡议和向当地高中进行外联。在不同的方向上，PI计划继续与其他科学领域的研究人员对话，以寻找新的合作机会。PI将尽可能广泛地传播项目的结果，例如通过组织会议和有效使用在线工具。为了详细说明该项目的科学价值，PI以前的工作使用了一种特殊的接触同调，称为嵌入接触同调(ECH)，以表明封闭三流形上的任何Reeb型向量场至少有两个不同的周期轨道。事实上，证据表明更强的结果成立，在目前的项目中，PI和合作者计划在许多情况下证明一个具有两个以上闭合轨道的Reeb向量场有无穷多个。对Reeb动力学的洞察也来自与辛嵌入问题密切相关的领域，这个项目将在这个方向上寻求几条线索，例如，一条线索涉及更好地理解四维嵌入问题的数论方面，而另一条线索涉及找到更高维辛嵌入的新障碍。新的组合工具有望对这些研究有用。因此，PI计划继续发展经典Ehrhart理论的无理版本，该理论熟悉于格点计数理论。该项目的更多推测方向涉及建立ECH谱的渐近性的新公式，以及通过应用新的工具来正则化伪全纯曲线的模空间来进一步发展嵌入接触同调的基础。

Systems that evolve over time are known as "dynamical systems" and appear in a wide range of fields including biology, economics, physics, and engineering.  The equations that describe these systems are often difficult to solve directly, so new tools and techniques are needed to study them.  This project is centered around a tool called "contact homology" that has produced powerful insights into many kinds of dynamical systems in recent years.  Contact homology provides a framework for finding features of a dynamical system that are conserved even as the parameters of the dynamical system are allowed to vary.   In this way, complicated systems can be studied by continuously changing the system into a more basic form, and then using knowledge of the dynamics in this simpler state to learn about the structure of the original system.  This project aims to further develop the foundations of contact homology and to use it to produce new insights about dynamics.  An example of an expected application is the discovery of new periodic trajectories for many dynamical systems, which are configurations of the system that re-occur infinitely often in time. An expected educational impact is the fostering of new opportunities for research training for undergraduates. In this area the PI plans to continue his efforts, which have led to two publications by undergraduates in peer-reviewed journals in the last several years.  The PI also plans to continue his commitment to reach broader segments of the public, for example through participation in prison education initiatives and through outreach to local high schools. In a different direction, the PI plans to continue conversations with researchers in other fields of science to find new collaborative opportunities. The PI will disseminate the results of the project as broadly as possible, for example through organizing conferences and through effective use of online tools. To elaborate on the scientific merit of the project, previous work of the PI used a special kind of contact homology, called embedded contact homology (ECH), to show that any vector field of Reeb type on a closed three-manifold has at least two distinct periodic orbits.  In fact, evidence suggests that stronger results hold, and in the current project the PI and collaborators plan to show in many cases that a Reeb vector field with more than two closed orbits has infinitely many.  Insights into Reeb dynamics also come from the closely related field of symplectic embedding problems, and this project will pursue several lines of inquiry in this direction, for example one thread involves better understanding the number theoretic aspects of four-dimensional embedding problems while another involves finding new obstructions to symplectic embeddings in higher dimensions.  New combinatorial tools are expected to be useful for these investigations, so the PI plans to continue developing an irrational version of the classical Ehrhart theory familiar from the theory of lattice point enumeration.  More speculative directions of the project involve establishing new formulas for the asymptotics of the ECH spectrum and further developing the foundations of embedded contact homology by applying new tools for regularizing moduli spaces of pseudoholomorphic curves.