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。特别是，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。合点的研究，或方程之间的关系，长期以来一直在代数中发挥着核心作用，其历史从19世纪的不变量理论到21世纪的理论物理。 这项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，研究者正在研究与嵌入射影空间的黎曼曲面的外在几何相关的代数不变量与其内在几何的关系。相关的代数不变量是坐标环的最小自由分辨率的贝蒂数，如希尔伯特所定义的，而内在几何则编码在布里-诺特理论的不变量中。研究人员将探索长期存在的基本原理，预测这些不变量之间的精确关系。该项目研究这些结构和推广他们使用几个新的技术，如相交理论，曲线的模空间，赫维茨空间，和向量束技术。

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.