Gromov-Witten theory under extremal transitions and birational transformations

Gromov-Witten （GW）理论是数学和物理学中许多令人兴奋的研究领域的交叉。粗略地说，它计算一个几何物体中满足规定条件的曲线的数量。一方面，该理论本身具有显著的结构，既有既定的结构，也有推测的结构。研究这些结构需要一些来自其他领域的新见解和技术工具。这为古典数学学科提供了许多有趣的问题。另一方面，这些新的见解和技术工具有助于发现现有数学和理论物理之间的深层关系和联系。这个项目将促进对这一活跃研究领域的理解。该项目包括培养研究生和博士后，以及国际合作。这个研究项目是代数几何和数学物理的交叉。具体来说，它一方面涉及Gromov-Witten理论和Hodge结构（VHS）的变化，另一方面涉及两种几何和镜像对称。项目的主要主题是研究GW束和爆炸理论，以及GW和VHS在极端跃迁（在物理文献中也称为时空拓扑变化）下的扩展功能。本文还将研究GW理论中的蠕变猜想与Landau-Ginzburg / Calabi-Yau对应关系。

Gromov-Witten (GW) theory lies in the intersection of many exciting research areas in mathematics and physics. Roughly speaking it counts the number of curves meeting prescribed conditions in a geometric object called a variety. On the one hand, the theory itself has remarkable structures, both established and conjectural. Investigating these structures requires some new insights and technical tools from other areas. This provides a lot of interesting problems for classical subjects in mathematics. On the other hand, these new insights and technical tools help to discover deep relations and connections between existing mathematics and theoretical physics. This project will advance understanding in this active area of research. The project includes training of graduate students and postdocs, as well as international collaborations.This research project lies on the intersection of algebraic geometry and mathematical physics. Specifically, it concerns the Gromov-Witten theory and variation of Hodge structures (VHS) on one side and birational geometry and mirror symmetry on the other. The main themes of the project are the study of GW theory of bundles and blowing-ups and the extended functoriality of GW and VHS under extremal transitions (also known as space-time topology change in the physics literature). The relations between the crepant transformation conjecture in GW theory and the Landau-Ginzburg / Calabi-Yau correspondence will also be studied.