Moduli of Stable Log Maps and Applications

稳定日志图模及其应用

• 批准号: 1700682
• 负责人: Qile Chen
• 金额: $ 15.8万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-15 至 2021-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700682&HistoricalAwards=false
• 关键词: Moduli; Stable; Log; Maps; Applications

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.

该项目位于代数几何的区域，该几何形状是研究代数品种的数学分支，即多项式方程的解决方案集。该项目的主题与物理学的字符串理论密切相关，该理论使用代数品种来描述我们宇宙的一部分。一个了不起的事实是，在另一个较大的种类中，各种类型的品种本身就是品种。对此类品种收集的研究不仅提供了研究数学几何问题的有力工具，而且还提供了提取物理学家有趣的不变性方法的有用方法。该项目的主要主题旨在开发一种数学工具来计算弦物理学的这些不变性。该项目的其他主题着重于使用上述工具来解决几何和数字理论的问题。该项目的主要主题是通过对数光滑的变性来研究Gromov-witten不变性。短期目标是开发刺穿的地图理论，该理论是用“杆子”稳定的对数图。长期目标是证明有效的退化公式，该公式表达了格罗莫夫（Gromov）编织的不变式，作为刺穿图作为基本构建块的不变性的组合。该项目的进一步主题包括（1）通过对数紧凑型研究Teichmuller动力学； （2）研究代数曲线功能场的积分点的密度问题。这两个主题都是稳定对数图理论的进一步应用。

项目成果

Virtual cycles of stable (quasi-)maps with fields

• DOI: 10.1016/j.aim.2021.107781
• 发表时间: 2019-11
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Qile Chen;F. Janda;Rachel Webb

The logarithmic gauged linear sigma model

• DOI: 10.1007/s00222-021-01044-2
• 发表时间: 2019-06
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: Qile Chen;F. Janda;Y. Ruan

Spin and hyperelliptic structures of log twisted differentials

对数扭曲微分的自旋和超椭圆结构

• DOI: 10.1007/s00029-019-0467-x
• 发表时间: 2019
• 期刊: Selecta Mathematica
• 作者: Chen, Dawei;Chen, Qile
• 通讯作者: Chen, Qile