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Interactions between Newton-Okounkov Bodies, Cluster Algebras, and Orbit Closures

牛顿-奥孔科夫体、簇代数和轨道闭包之间的相互作用

基本信息

批准号：
1855598
负责人：
Laura Escobar Vega
金额：
$ 17.28万
依托单位：
Washington University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2023-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855598&HistoricalAwards=false
关键词：
Interactions between Newton Okounkov Bodies

项目摘要

The interplay between combinatorics, which studies with discrete structures, and algebraic geometry, which is concerned with solutions of polynomial equations, has immensely enriched both areas. Combinatorics provides discrete objects that can be used to encode information about an algebraic variety. Conversely, the rich structure of a combinatorial object is often better understood when seen as a discrete shadow of an algebro-geometric phenomenon. The research in this project develops the interplay between these areas utilizing tools which include degenerations of varieties and high-dimensional analogues of polygons.This project aims to understand various aspects of the interplay between combinatorics and algebraic geometry for Newton-Okounkov bodies, symmetric orbit closures, and subword complexes. Generalizing the relation between polytopes and toric varieties, Newton-Okounkov bodies provide tools from convex geometry to study algebraic varieties. The PI will develop the theory of Newton-Okounkov bodies by constructing combinatorial parameter spaces for these bodies together with a wall-crossing formula. The study of symmetric orbit closures is relevant to the theory of Kazhdan-Lusztig-Vogan polynomials and Harish-Chandra modules for a certain real Lie group, mirroring the connection of Schubert varieties to Kazhdan-Lusztig polynomials and representation theory. The PI will investigate the projections of symmetric orbit closures to Grassmannians. Knutson and Miller introduced subword complexes to study the cohomology of Schubert varieties. The PI will relate these complexes with toric degenerations of Schubert varieties and their desingularizations.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.

研究离散结构的组合学和研究多项式方程解的代数几何之间的相互作用极大地丰富了这两个领域。组合学提供了离散对象，可用于对代数变量的信息进行编码。相反，当将组合对象的丰富结构视为代数-几何现象的离散阴影时，通常会更好地理解。本项目的研究利用包括品种退化和高维多边形类似物在内的工具开发了这些领域之间的相互作用。该项目旨在了解牛顿-奥昆科夫体、对称轨道闭包和子词复合体的组合学和代数几何之间相互作用的各个方面。牛顿-奥肯科夫体推广了多面体与环变之间的关系，为凸几何研究代数变提供了工具。PI将通过构造这些体的组合参数空间以及一个过壁公式来发展牛顿-奥孔科夫体的理论。对称轨道闭包的研究涉及到某实李群的Kazhdan-Lusztig- vogan多项式和Harish-Chandra模的理论，反映了Schubert变分与Kazhdan-Lusztig多项式和表示理论的联系。PI将研究对称轨道闭包对格拉斯曼的投影。Knutson和Miller引入子词复合体来研究Schubert变异的上同源性。PI将把这些复合体与舒伯特变种的环向退化及其去广域化联系起来。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。