Functoriality in Gromov-Witten Theory and Beyond

本文的研究方向是代数几何与数学物理之间的相互作用。具体地说，它一方面涉及Gromov-Witten(GW)理论和Hodge结构的变分(VHS)，另一方面涉及二次几何、代数余弦论和镜像对称性。该提议的主要主题是GW理论在Crepant变换下的功能性，以及GW和VHS在极端跃迁(在物理学文献中也被称为时空拓扑变化)下的扩展功能性。格罗莫夫-维腾理论是数学和物理学中许多激动人心的研究领域的交汇点。一方面，这一理论本身具有非凡的结构，既有既定的，也有猜测的。调查这些结构需要一些新的见解和技术工具，以及来自其他领域的投入。这为数学经典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，特别是拓扑弦理论之间的深刻联系和联系。

The proposed research lies on the interaction between algebraic geometry and mathematical physics. Specifically, it concerns the Gromov-Witten (GW) theory and variation of Hodge structures (VHS) on one side and birational geometry, algebraic cobordism, and mirror symmetry on the other. The main themes of the proposal are the functoriality of GW theory under crepant transformations and the extended functoriality of GW and VHS under extremal transitions (also known as space-time topology change in the physics literature). Gromov-Witten theory lies in the intersection of many exciting research areas in mathematics and physics. On the one hand, the theory itself has remarkable structures, both established and conjectural. Investigating these structures requires some new insights and technical tools and input 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, especially the topological string theory.