Algebraic geometry of moduli spaces

Gromov-Witten理论是一个迅速发展的领域，与当前数学和物理研究的许多核心领域有着基本的联系。这里提出的项目是基于过去几年的技术和发现对Gromov- Witten理论进行广泛的研究。主要内容包括：带边界曲线模空间上积分的定义和精确求值，普遍Virasoro约束的证明，Gromov-Witten/ Donaldson-Thomas/Pairs对应的建立，以及重言类的研究。这些主题指向几个不同的方向：拓扑弦理论、可积层次和经典代数几何。每个主题都是该领域进展的核心，每个主题都将以新的观点进行讨论。代数变量是由多项式方程的零点定义的，是古典数学和现代数学的基本对象。代数几何是对代数变种的研究。辛几何和弦理论物理的思想开辟了代数几何的新领域：通过曲线的Gromov-Witten理论和轴的Donaldson-Thomas理论来研究代数变异。由于该课题在多个方向上有基本的联系，其进展将直接影响到邻近的领域。拓扑弦理论是最明显的联系，这两个领域经常接触。但是，例如，从深谷类别到随机三维分区的主题也会受到影响。

Gromov-Witten theory is a rapidly expanding field with basic connections to many central areas of current research in mathematics and physics. The project proposed here is a wide ranging study of Gromov- Witten theory based on the techniques and discoveries of the last few years. The main topics covered are: the definition and exact evaluations of integrals on the moduli space of curves with boundaries, the proof of the universal Virasoro constraints, the establishment of the Gromov-Witten/ Donaldson-Thomas/Pairs correspondences, and the study of tautological classes. These topics point in several differentdirections: topological string theory, integrable hierarchies, and classical algebraic geometry. Each topic is central to progress in the field, and each will be addressed with a new point of view.Algebraic varieties, defined by the zeros of polynomial equations, are basic objects in both classical and modern mathematics. Algebraic geometry is the study of algebraic varieties. Ideas from symplectic geometry and string theoretic physics have opened new fields in algebraic geometry: the study of algebraic varieties via the Gromov-Witten theory of their curves and the Donaldson-Thomas theory of their sheaves. Since the topic has basic connections in several directions, progress will have a direct impact on the neighboring fields. Topological string theory is the most obvious connection and the two fields are in frequent contact. But also, for example, topics varying from the Fukaya category to random 3-dimensional partitions will be affected.