RUI: Combinatorial Algebraic Geometry: Curves and Their Moduli

代数几何是数学的一个中心领域，它研究由多项式方程组定义的几何对象。模量理论旨在通过考虑它们在一个这样的品种家族中的行为来理解特定的品种。特别地，模空间由特定类型的所有几何对象组成。该研究项目包括代数几何与其他数学领域的新发展之间富有成效的相互作用所产生的问题，如组合学，它涉及组织和分析离散结构。该项目将资助本科生研究，PI将继续努力扩大数学科学中代表性不足群体的成员的参与。这些项目旨在更好地理解在自然界中组合丰富的模空间及其上同调理论。研究的基本对象包括退化轨迹，齐次空间，如格拉斯曼和旗品种，仿射格拉斯曼，和模空间的曲线。更具体地说，项目包括研究退化轨迹及其motivic类，并应用于Brill-Noether理论;齐性空间的量子上同调和量子K理论以及与仿射格拉斯曼的联系; Gromov-Witten理论和表示论产生的曲线模空间上的基点自由类;以及代数几何和代数组合学中的其他问题，该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准。

Algebraic geometry is a central area of mathematics that studies varieties, which are geometric objects defined by systems of polynomial equations. Moduli theory aims to understand specific varieties by considering how they behave in a family of such varieties. In particular, a moduli space consists of all geometric objects of a particular type. This research project consists of problems that arise from the fruitful interactions between algebraic geometry and new developments in other fields of mathematics such as combinatorics, which is concerned with organizing and analyzing discrete structures. This project will fund undergraduate research and the PI will continue efforts towards broadening participation of members of underrepresented groups in the mathematical sciences. The projects aim to better understand moduli spaces that are combinatorially rich in nature and their cohomology theories. Fundamental objects of investigation include degeneracy loci, homogeneous spaces such as Grassmannians and flag varieties, the affine Grassmannian, and the moduli space of curves. More specifically, projects include the study of degeneracy loci and their motivic classes, with applications to Brill-Noether theory; quantum cohomology and quantum K-theory of homogeneous spaces and connections to the affine Grassmannian; base point free classes on the moduli space of curves arising from Gromov-Witten theory and from representation theory; and other problems in algebraic geometry and algebraic combinatorics, including several for undergraduate students.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.