Vector Bundles of Conformal Blocks on Moduli Spaces

本专题研究代数曲线族之间的关系，这是代数几何的中心课题，也是数学物理中出现的相关概念。在代数几何中，曲线族是通过模空间来研究的，模空间将族中的不同曲线参数化。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，这些行为有助于我们理解几何对象如何被参数化，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了有价值的见解。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。本研究项目调查与这些发现相关的开放问题。研究生参与了这个项目，研究者参与了数学系的数学夏令营高中项目。对于光滑曲线，共形块被确定为某些充分线丛在射影簇上的整体截面。 项目一的目的是研究在何种程度上这种描述可能或可能不适用于非光滑曲线。项目二提出了模空间的双有理模型，作为嵌入齐次变量的曲线上支持的加权点的配置，推广了射影空间中的点有理正规曲线。 为了回答关于表示论中的乘法本征多面体的开放问题，以及由保形块的向量丛的第一陈类的某些集合所张成的锥，项目三的目标是利用从两者的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向量丛给出的高余维的正循环。

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.