CAREER: Algebraic Curves and Their Moduli: Degenerations and Combinatorics

代数几何是数学的中心学科，在数学、物理和工程的许多领域都有联系和应用。代数几何学者研究由多项式方程定义的称为代数变异的空间。研究这些空间的一个强大的方法，正如在这个项目中使用的，是退化的方法，其中一个参数化的代数变量族在极限中分解成碎片。粗略地说，其思想是研究组合学，即碎片的离散数据，以便推断出更复杂的原始空间的事情。该项目的教育部分包括一个“代数几何中的妇女”讲习班和一个关于数学多样性和包容性的系列研讨会。该奖项支持的研究将集中于利用现代退化技术，特别是热带几何领域的退化技术，从代数几何中研究经典空间。在一个方向上，PI将使用这些技术来研究曲线和阿贝尔变体的经典模空间的拓扑结构。在另一个方向上，PI也将推进我们使用集值表的组合学对Brill-Noether变量的理解，并研究由Brill-Noether理论中的程序激发的表组合学中的问题。该奖项还将支持一个“代数几何中的女性”研讨会和一个关于数学多样性和包容性的系列研讨会。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Algebraic geometry is a central subject in mathematics and has connections and applications to many areas in mathematics, physics, and engineering. Algebraic geometers study spaces called algebraic varieties that are defined by polynomial equations. One powerful method of studying these spaces, as used in this project, is the method of degenerations, where a parametrized family of algebraic varieties breaks into pieces in the limit. Roughly speaking, the idea is that one studies the combinatorics, i.e. the discrete data, of the pieces, in order to deduce things about the more complicated original space. The educational component of the project includes a Women in Algebraic Geometry Workshop and a seminar series on diversity and inclusion in mathematics.The research supported by this award will center on using modern degeneration techniques, especially those from the field of tropical geometry, to study classical spaces from algebraic geometry. In one direction, the PI will use these techniques to study the topology of classical moduli spaces of curves and abelian varieties. In another direction, the PI will also advance our understanding of Brill-Noether varieties using the combinatorics of set-valued tableaux, and investigate questions in tableau combinatorics that were motivated by the program in Brill-Noether theory. This award will also support a Women in Algebraic Geometry workshop and a seminar series on diversity and inclusion in mathematics.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.