Floer Homology, Concordance, and Complex Curves

该项目将研究拓扑和几何对象，使用广泛的现代数学工具。该项目的一个核心主题是加深我们对“打结”的四维方面的理解，这是一个试图理解和量化弦在空间中缠绕和打结的方法的领域。除了结理论在拓扑学中扮演的基本角色外，它还与许多看似无关的数学和物理领域，甚至与聚合物科学和生物学等领域有着深刻的相互作用。该项目的主要目的之一是了解结如何随着时间的推移而演变，以及它们在四维空间中结合的表面。想象一下，在一部电影中，一根绳子在空间中自由移动，随着时间的推移或多或少地缠在一起。此外，想象一下在电影的某些帧中出现了更激烈的现象，例如出现或消失了一个新的字符串循环或将字符串的两个部分粘合在一起。这样的电影被称为“共体”，利用这个概念，我们可以用对待数字的方式来对待打结的绳子；也就是说，我们可以用代数的方法加减结点。理解这个算法对四维空间的研究有着深远的影响，四维空间是数学中最困难和最不为人所知的领域之一。这方面的主要目标包括理解由复杂多项式产生的共轭曲线，以及使用受数学物理启发的工具探索上述代数结构。除了它的数学目标，该项目将有助于会议和研讨会的组织，并为学生提供关键的支持。结理论及其四维方面在整个项目中发挥着重要作用，可以视为其主要关注点。具体的目标包括确定协调群和协伴群之间的几何映射的可靠性，在Stein域中约束复杂曲线的结的拓扑特征，以及开发代数工具以方便计算由拉格朗日花同调和非阿贝尔规范理论定义的结、缠结和3流形的微妙不变量。该项目跨越了拓扑学、分析学、代数和组合学的界限，并使用了同调代数、规范理论、结理论、接触几何和辛流形中伪全纯曲线理论的技术。

This project will study topological and geometric objects, using a broad array of tools from modern mathematics. A theme central to the project is deepening our understanding of 4-dimensional aspects of "knotting," an area which attempts to understand and quantify the method by which strings become tangled and knotted in space. In addition to a fundamental role that knot theory plays in topology, its has deep interactions with many seemingly unconnected areas of mathematics and physics, and even to areas such as polymer science and biology. One of the primary aims of the project is to understand both how knots evolve over time and the surfaces that they bound in 4-dimensional space. Imagine a movie in which a piece of string freely moves in space, becoming more or less tangled over time. Further, imagine that at some frames in the movie more drastic phenomena occur such as the appearance or disappearance of a new loop of string or the gluing of two segments of the string together. Such a movie is called a "cobordism," and using this idea one can treat knotted pieces of string in much the same way that we treat numbers; namely, one can add and subtract knots in an algebraic way. Understanding this arithmetic turns out to have deep implications for the study of 4-dimensional space, one of the most difficult and least understood areas of mathematics. Primary goals in this vein include understanding knotted curves whose cobordisms arise from complex polynomials and probing the aforementioned algebraic structures using tools inspired by mathematical physics. In addition to its mathematical goals, the project will contribute to the organization of conferences and workshops, and provide critical support for students.Knot theory and its 4-dimensional aspects play a fundamental role throughout the project and can be viewed as its primary focus. Specific goals include determining the faithfulness of geometric maps between concordance and cobordism groups, topological characterizations of knots that bound complex curves in Stein domains, and the development of algebraic tools to facilitate computations of subtle invariant of knots, tangles, and 3-manifolds defined by way of Lagrangian Floer homology and non-abelian gauge theory. The project crosses boundaries between topology, analysis, algebra, and combinatorics, and uses techniques from homological algebra, gauge theory, knot theory, contact geometry, and the theory of pseudo-holomorphic curves in symplectic manifolds.