Algebraic geometry of moduli spaces

格罗莫夫-维腾理论是一个迅速扩展的领域，与当前数学和物理研究的许多中心领域有基本联系。这里提出的项目是基于过去几年的技术和发现，对格罗莫夫-维滕理论进行的广泛研究。主要内容包括：具有边界的曲线的模空间上积分的定义和精确计算，泛Virasoro约束的证明，Gromov-Witten/Donaldson-Thomas/Pair对应的建立，以及重言式类的研究。这些主题指向几个不同的方向：拓扑弦论、可积层次和经典代数几何。每个主题都是该领域进步的中心，每个主题都将以新的观点加以阐述。由多项式方程的零点定义的代数族是古典和现代数学中的基本对象。代数几何是研究代数簇的学科。辛几何和弦理论物理的思想在代数几何中开辟了新的领域：通过Gromov-Witten曲线理论和Donaldson-Thomas理论研究代数簇。由于这个话题在几个方向上有基本的联系，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，这两个领域是经常接触的。但是，例如，从Fukaya类别到随机的3维分区的主题也将受到影响。

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.