CAREER: Algebraic Curves and Their Moduli: Degenerations and Combinatorics

代数几何是数学中的一门中心学科，与数学、物理和工程中的许多领域都有联系和应用。代数几何学家研究由多项式方程定义的称为代数簇的空间。在这个项目中，研究这些空间的一个强有力的方法是退化方法，在这种方法中，一族参数化代数簇在极限中分解为碎片。粗略地说，这个想法是研究碎片的组合学，即离散数据，以推导出关于更复杂的原始空间的东西。该项目的教育部分包括代数几何中的女性工作坊和关于数学中的多样性和包容性的研讨会系列。该奖项支持的研究将集中于使用现代退化技术，特别是来自热带几何领域的技术，从代数几何研究经典空间。在一个方向上，PI将使用这些技术来研究曲线和交换簇的经典模空间的拓扑。在另一个方向上，PI还将促进我们利用集值图论的组合学来理解Brill-Noether簇，并研究由Brill-Noether理论中的程序所激发的图论中的问题。该奖项还将支持代数几何中的女性研讨会和关于数学多样性和包容性的研讨会系列。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.