Holomorphic Curves in Embeddings and Dynamics

随着时间的推移而进化的系统被称为“动力系统”，并出现在包括生物学、经济学、物理学和工程学在内的广泛领域。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。这个项目围绕着一个叫做“接触同源”的工具展开，这个工具近年来对许多种动力系统产生了强有力的见解。接触同源性提供了一个框架来寻找一个动力系统的特征，即使动力系统的参数允许变化也是守恒的。通过这种方式，可以通过不断地将系统转变为更基本的形式，然后利用这种更简单状态下的动力学知识来了解原始系统的结构来研究复杂系统。本项目旨在进一步发展接触同源性的基础，并利用它来产生关于动力学的新见解。预期应用的一个例子是发现许多动力系统的新周期轨迹，这些轨迹是系统在时间上无限频繁地重复出现的构型。预期的教育影响是为本科生提供研究训练的新机会。在这个领域，PI计划继续他的努力，在过去的几年里，他已经在同行评议的期刊上发表了两篇本科生的论文。监狱长还计划继续致力于接触更广泛的公众群体，例如通过参与监狱教育倡议和与当地高中的联系。在另一个方向上，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。项目负责人将尽可能广泛地传播项目成果，例如通过组织会议和有效利用在线工具。为了详细说明该项目的科学价值，PI先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），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，有证据表明，更强的结果是成立的，在目前的项目中，PI和合作者计划在许多情况下证明，具有两个以上闭合轨道的Reeb向量场具有无限多个。对Reeb动力学的见解也来自与之密切相关的辛嵌入问题领域，该项目将在这个方向上进行几条探究线，例如，一条线索涉及更好地理解四维嵌入问题的数论方面，而另一条线索涉及寻找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计划继续开发经典Ehrhart理论的非理性版本，与晶格点枚举理论相似。该项目的更多推测方向包括建立ECH谱渐近性的新公式，并通过应用正则化伪全纯曲线模空间的新工具进一步发展嵌入式接触同调的基础。

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.