Vector Bundles of Conformal Blocks on Moduli Spaces

本课题研究代数曲线族之间的关系，这是代数几何中的一个中心话题，以及数学物理中出现的相关概念。在代数几何中，利用模空间对曲线族进行了研究，模空间对曲线族中的不同曲线进行了参数化。除了提供关于曲线的深刻见解之外，模空间表现出重要的行为，有助于塑造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。最近的研究表明，曲线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开放性问题。研究生们也参与了这个项目，研究者还共同组织了数学系的MathCamp高中项目。这些项目是根据四个主题组织的。对于光滑曲线，保形块被确定为在投影变化上的某些充足线束的整体截面。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。方案二提出模空间的二元模型作为嵌入齐次变量曲线上的加权点的构型，推广了射影空间中由点有理法向曲线给出的那些。为了回答关于表示理论中的乘法特征多面体的开放问题，以及由共形块的向量束的某些第一Chern类集合所跨越的锥体，项目三的目标是利用从两者子集之间的映射中收集的信息。项目四旨在理解由共形块的矢量束给出的高协维的正循环。

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.