Logarithmic Moduli Spaces for Symplectic Geometry: Construction, Applications, and Beyond

辛流形是经典力学中相空间概念的推广。在过去的四十年里，辛几何领域得到了迅速的发展，导致了与其他重要研究领域的新联系，如代数几何、低维拓扑和高能物理。该奖项支持对被称为全纯曲线的基本对象及其对应的不变量和代数结构的研究。特别是，研究人员将解决基本问题，例如在存在称为除数的对象的情况下构造行为良好的全纯曲线族。这项研究包含了研究生和博士后可以开展的具体项目。研究人员将组织一年一度的小型研讨会，向本科生介绍几何和拓扑学的研究机会，以及他们在其他领域的应用。他还将在公共图书馆发起一个针对高中生的数学俱乐部。这一建议的主要目标是为任意对辛流形和法交辛因子构造全纯曲线的模空间，并满足特定性质。这种模空间的构造需要紧致化，需要形变理论的分析框架，需要解决横截性问题，需要证明粘合定理。这些模空间在计数几何、镜像对称、Fukaya范畴的构造以及辛几何、代数几何和弦理论的其他活跃研究领域都有直接的应用。PI与M.McLean和A.Zinger合作，引入了正交辛因子和簇的拓扑概念。他们已经构建了工具，如正则化和对数切线丛来处理这类对象。最近，PI发展了一种新的基于对数切丛的紧化和变形理论。他将使用此设置来处理施工的其余步骤。主要项目是定义相对于任意法线交叉因子的Gromov-Witten不变量。其他项目包括证明Gromov-Witten不变量的退化公式，用代数方法找出它们之间的关系，以及探索镜面对称性的应用。该项目由几何分析和既定的激励竞争研究计划(EPSCoR)共同资助。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Symplectic manifolds are geometric objects that generalize the concept of phase-space in classical mechanics. During the past four decades, the field of symplectic geometry has evolved rapidly, leading to new connections with other significant areas of research, such as algebraic geometry, low dimensional topology, and high energy physics. This award supports research on fundamental objects known as holomorphic curves and their corresponding invariants and algebraic structures. In particular, the investigator will address foundational questions, such as the construction of well-behaved families of holomorphic curves in the presence of objects known as divisors. This research contains specific projects that can be carried out by graduate students and postdocs. The investigator will organize annual mini-symposia for introducing undergraduate students to research opportunities in geometry and topology, and their applications to other fields. He will also initiate a math club at the public library aimed at high school students.The main objective of this proposal is to construct moduli spaces of holomorphic curves for arbitrary pairs of symplectic manifolds and normal crossing symplectic divisors, satisfying particular properties. Construction of such moduli spaces requires a compactification, an analytical framework for deformation theory, addressing the transversality issue, and proving a gluing theorem. These moduli spaces have immediate applications in enumerative geometry, Mirror Symmetry, construction of Fukaya categories, and other active areas of research in symplectic geometry, algebraic geometry, and string theory. In collaboration with M. McLean and A. Zinger, the PI introduced topological notions of normal crossing symplectic divisor and variety. They have constructed tools such as regularizations and logarithmic tangent bundle for working with such objects. Recently, the PI developed a novel compactification and a deformation theory based on the logarithmic tangent bundle. He will use this setup to work on the remaining steps of the construction. The main project is to define Gromov-Witten invariants relative to an arbitrary normal crossing divisor. Other projects include proving a degeneration formula for Gromov-Witten invariants, finding the relations with the algebraic approach, and exploiting the applications to Mirror Symmetry. This project is jointly funded by Geometric Analysis and the Established Program to Stimulate Competitive Research (EPSCoR).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.