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CAREER: Algebraic Curves and Their Moduli: Degenerations and Combinatorics

职业：代数曲线及其模：简并和组合学

基本信息

批准号：
1844768
负责人：
Melody Chan
金额：
$ 39.99万
依托单位：
Brown University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1844768&HistoricalAwards=false
关键词：
CAREER Algebraic Curves Their Moduli

项目摘要

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.

代数几何是数学的中心学科，与数学、物理和工程学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。研究这些空间的一个强有力的方法，在这个项目中使用，是退化的方法，其中一个参数化的代数簇族在极限中分裂成碎片。粗略地说，这个想法是，人们研究组合学，即离散数据，为了推导出更复杂的原始空间的东西。该项目的教育部分包括一个妇女在代数几何讲习班和一个关于数学的多样性和包容性的系列研讨会。该奖项支持的研究将集中在使用现代退化技术，特别是那些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阿贝尔簇的经典模空间的拓扑。在另一个方向上，PI还将使用集值tableaux的组合学来推进我们对Brill-Noether品种的理解，并调查Brill-Noether理论中的程序所激发的tableau组合学中的问题。该奖项还将支持一个妇女在代数几何讲习班和一个系列研讨会的多样性和包容性的数学。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的知识价值和更广泛的影响审查标准。