Syzygies, Moduli Spaces, and Brill-Noether Theory

这个项目涉及代数几何的研究，多项式方程的研究。研究者致力于研究方程的代数性质和它们所定义的空间的几何性质之间的联系。特别地，这个研究项目使用syzygies来研究关于黎曼曲面的基本问题，黎曼曲面是最重要的几何对象之一。从19世纪的不变量理论到21世纪的理论物理学，对合子或方程之间关系的研究长期以来一直在代数中发挥着核心作用。本研究的应用包括模空间、矩阵分解、弦理论、枚举几何和镜像对称。更详细地说，研究者正在研究将嵌入射影空间的黎曼曲面的外在几何与其内在几何相关联的代数不变量。相关的代数不变量是Hilbert定义的坐标环的最小自由分辨率的Betti数，而内在几何是用Brill-Noether理论的不变量编码的。研究者将探索长期的基本猜想，预测这些不变量之间的精确关系。本课题利用交叉理论、曲线模空间、Hurwitz空间、向量束等新技术对这些猜想进行了研究和推广。

This project concerns research in algebraic geometry, the study of polynomial equations. The investigator works on the connections between the algebraic properties of equations and the geometric properties of the spaces they define. In particular, this research project uses syzygies to study fundamental questions regarding Riemann surfaces, one of the most important classes of geometric objects. The investigation of syzygies, or the relations amongst equations, has long played a central role in algebra, with a history ranging from 19th-century invariant theory to 21st-century theoretical physics. Applications of this research include moduli spaces, matrix factorizations, string theory, enumerative geometry, and mirror symmetry.In more detail, the investigator is working on relating the algebraic invariants associated to the extrinsic geometry of a Riemann surface embedded in projective space to its intrinsic geometry. The relevant algebraic invariants are the Betti numbers of the minimal free resolution of the coordinate ring, as defined by Hilbert, whereas the intrinsic geometry is encoded in the invariants of Brill-Noether theory. The investigator will explore longstanding fundamental conjectures predicting precise relationships between these invariants. The project investigates these conjectures and generalizations of them using several new techniques such as intersection theory, the moduli space of curves, Hurwitz space, and vector bundle techniques.