Vector Bundles of Conformal Blocks on Moduli Spaces

这个项目研究代数几何学中的一个中心主题--代数曲线族与数学物理中出现的相关概念之间的关系。在代数几何中，利用模空间来研究曲线族，模空间将曲线族中的不同曲线参数化。除了提供关于曲线的深刻见解外，模空间还展示了重要的行为，这些行为有助于形成我们对几何对象如何被参数化的理解，并在其中充当更大的数学世界的使者，为高维理论的发展提供有价值的见解。最近的工作表明，曲线的模空间的某些方面反映了潜在的几何结构。这个研究项目调查了与这些发现相关的悬而未决的问题。研究生参与了这个项目，研究人员共同组织了数学系的MathCamp高中项目。项目围绕四个一般主题进行组织。对于光滑曲线，用射影簇上的某些充分线丛的整体截面来识别共形块。项目一旨在研究这种描述在多大程度上适用于非光滑曲线。方案二提出了模空间的双调模型，即由齐次簇中嵌入的曲线上支承的加权点的构形，推广了射影空间中指定有理正规曲线所给出的模型。为了回答表示论中关于乘法特征多面体和某些第一类共形块向量丛的集合所跨越的锥体的公开问题，项目三的目标是利用从两者的子集之间的映射收集的信息。项目四旨在理解由共形块向量丛给出的高余维正循环。

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.