Vector Bundles of Conformal Blocks on Moduli Spaces

模空间上共角块的向量丛

• 批准号: 1820718
• 负责人: Angela Gibney
• 金额: $ 9.18万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1820718&HistoricalAwards=false
• 关键词: Vector; Bundles; Conformal; Blocks; Moduli

This project studies the relationship between the families of algebraic curves, a central topic in algebraic geometry, and related notions arising in mathematical physics. In algebraic geometry, families of curves are studied by the use of moduli spaces, which parametrize the different curves in the family. Besides providing deep insights about curves, moduli spaces exhibit important behaviors which help shape our understanding of how geometric objects may be parametrized, and therein serve as envoys of a larger mathematical world, giving valuable insights into the development of higher dimensional theory. Recent work has revealed that certain aspects of moduli spaces of curves reflect underlying geometric structures. This research project investigates open questions related to these discoveries. Graduate students are involved in the project, and the investigator co-organizes the Mathematics Department's MathCamp high-school program.The projects are organized along four general themes. For smooth curves, conformal blocks are identified with global sections of certain ample line bundles on a projective variety. Project One aims to study to what extent this description may or may not hold for non-smooth curves. Project Two proposes birational models of the moduli space as configurations of weighted points supported on curves embedded in homogeneous varieties, generalizing those given by pointed rational normal curves in projective space. To answer open problems about the multiplicative eigenpolyhedra from representation theory, and cones spanned by certain sets of first Chern classes of vector bundles of conformal blocks, the goal of Project Three is to leverage information gathered from maps between subsets of the two. Project Four aims to understand positive cycles of higher codimension given by vector bundles of conformal blocks.

该项目研究代数曲线族（代数几何的中心主题）与数学物理中出现的相关概念之间的关系。在代数几何中，曲线族是通过使用模空间来研究的，模空间对族中的不同曲线进行参数化。除了提供关于曲线的深刻见解之外，模空间还表现出重要的行为，有助于塑造我们对几何对象如何参数化的理解，并在其中充当更大数学世界的使者，为高维理论的发展提供有价值的见解。最近的工作表明，曲线模空间的某些方面反映了底层的几何结构。该研究项目调查与这些发现相关的悬而未决的问题。研究生参与了该项目，研究人员共同组织了数学系的 MathCamp 高中项目。这些项目按照四个一般主题组织。对于平滑曲线，共形块由投影序列上某些充足线束的全局部分来识别。 项目一旨在研究这种描述在多大程度上适用于非平滑曲线。项目二提出了模空间的双有理模型，作为嵌入同质簇的曲线上支持的加权点的配置，概括了射影空间中由尖有理正态曲线给出的模型。 为了回答有关表示论中的乘法本征多面体以及由共形块的第一陈类向量束的某些集合跨越的锥体的开放问题，项目三的目标是利用从两者子集之间的映射收集的信息。项目四旨在了解由共形块的向量束给出的更高余维的正循环。