Functoriality in Gromov-Witten Theory and Beyond

本文的研究重点在于代数几何与数学物理的相互作用。具体来说，它涉及的Gromov-Witten（GW）理论和变化的霍奇结构（VHS）的一面和双有理几何，代数配边，镜像对称的其他。该提案的主要主题是在crepant变换下GW理论的函子性和在极值转换下GW和VHS的扩展函子性（在物理学文献中也称为时空拓扑变化）。 Gromov-Witten理论是数学和物理学中许多令人兴奋的研究领域的交叉点。一方面，该理论本身具有显著的结构，既有既定的结构，也有结构性的结构。调查这些结构需要一些新的见解和技术工具以及其他领域的投入。 这为数学中的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之间的深层关系和联系，特别是拓扑弦理论。

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.