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CAREER: Connections Between Tropical Geometry, Arithmetic Geometry, and Combinatorics

职业：热带几何、算术几何和组合数学之间的联系

基本信息

批准号：
2044564
负责人：
Farbod Shokrieh
金额：
$ 40万
依托单位：
University of Washington
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2026-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2044564&HistoricalAwards=false
关键词：
CAREER Connections Between Tropical Geometry

项目摘要

This award is at the intersection of algebraic geometry, arithmetic geometry, and combinatorics. In algebraic geometry, one studies the geometry of solutions to a collection of polynomial equations. In arithmetic geometry, one is concerned with the study of integers (whole numbers) and rational numbers (fractions). Combinatorics is an area of mathematics concerned with the study of finite structures. An example of such finite structures is the notion of graphs or networks. The aim is to make substantial progress on various foundational questions in these areas. The interplay between these disparate fields will continue to have significant consequences and will create and stimulate entirely new research problems.The PI will study fundamental questions at the intersection of non-archimedean and tropical geometry, arithmetic and Arakelov geometry, and combinatorics and convex geometry. The three main aspects of the research are: further study of the interplay between tropical methods and arithmetic geometry of abelian varieties, the study of various combinatorial and convex geometric questions arising from this interplay, and the study of invariants on degenerating families of curves and abelian varieties. The educational and outreach activities will complement the research. The centerpiece is a yearly extended research experience for graduate students on projects at the intersection of combinatorics, tropical geometry, and arithmetic and Arakelov geometry.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.

该奖项是代数几何、算术几何和组合学的交叉学科。在代数几何中，人们研究一组多项式方程的解的几何。在算术几何中，研究整数(整数)和有理数(分数)。组合学是一个研究有限结构的数学领域。这种有限结构的一个例子是图或网络的概念。其目的是在这些领域的各种基本问题上取得实质性进展。这些不同领域之间的相互作用将继续产生重大影响，并将创造和激发全新的研究问题。PI将研究非阿基米德几何和热带几何、算术和阿拉克洛夫几何以及组合学和凸几何的交叉点上的基本问题。研究的三个主要方面是：进一步研究热带方法和阿贝尔变种的算术几何之间的相互作用，研究由这种相互作用产生的各种组合和凸几何问题，以及研究退化曲线和阿贝尔变种上的不变量。教育和外联活动将补充这项研究。该奖项的核心是每年为研究生在组合学、热带几何、算术和阿拉克洛夫几何交叉点的项目提供延长的研究经验。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。