Moduli of Stable Log Maps and Applications

本课题属于代数几何领域，代数几何是研究代数变量的数学分支，即多项式方程的解集。这个项目的主题与物理学中的弦理论密切相关，其中代数变量被用来描述我们宇宙的一部分。一个令人惊奇的事实是，某种类型的品种在另一个更大的品种内的集合本身就是品种。对这些变量集合的研究不仅为研究几何和数学问题提供了有力的工具，而且为物理学家提取感兴趣的不变量提供了有用的方法。本项目的主要课题旨在开发一种数学工具来计算弦物理中的这些不变量。本项目的其他主题侧重于使用上述开发的工具来解决几何和数论中的问题。本课题的主要课题是通过对数平滑退化研究Gromov-Witten不变量。短期目标是发展穿孔图理论，粗略地说，穿孔图是具有“极点”的稳定对数图。长期目标是证明一个有效的退化公式，该公式将Gromov-Witten不变量表示为来自穿刺映射的不变量的组合，作为基本构建块。本项目的进一步课题包括：(1)通过对数紧化研究Teichmuller动力学；(2)研究代数曲线函数场上积分点的密度问题。这两个主题都是稳定对数映射理论的进一步应用。

This project lies in the area of algebraic geometry, a branch of mathematics studying algebraic varieties, namely, the set of solutions of polynomial equations. The topic of this project is closely related to string theory from physics, where algebraic varieties are used to describe a piece of our universe. An amazing fact is that collections of varieties of certain type inside another bigger one are themselves varieties. The study of such collections of varieties provides not only powerfully tools to study problems in geometry in mathematics, but also useful methods to extract invariants interesting to physicists. The main topic of this project is aimed at developing a mathematical tool to calculate these invariants from string physics. Other topics of this project focus on using the above developed tool to solve problems from geometry and number theory. The major topic of this project is to study Gromov-Witten invariants via logarithmic smooth degenerations. The short term goal is to develop the theory of punctured maps which are, roughly speaking, stable logarithmic maps with "poles". The long term goal is to prove an effective degeneration formula which expresses Gromov-Witten invariants as a combination of invariants from punctured maps as the basic building blocks. Further topics of this project include (1) studying Teichmuller dynamics via the logarithmic compactification; and (2) studying density problem of integral points over function fields of algebraic curves. Both topics are further applications of the theory of stable logarithmic maps.