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

研究离散结构的组合学和研究多项式方程解的代数几何学之间的相互作用极大地丰富了这两个领域。组合数学提供了离散对象，可用于编码有关代数簇的信息。相反，组合对象的丰富结构往往被视为代数几何现象的离散阴影时更好地理解。本项目的研究利用包括变种退化和多边形高维类似物在内的工具来发展这些领域之间的相互作用。本项目旨在了解牛顿-奥肯科夫体、对称轨道闭合和子字复体的组合学和代数几何之间相互作用的各个方面。牛顿-奥昆科夫体是凸几何中研究代数簇的工具，它推广了多面体和复曲面簇之间的关系。PI将通过为这些机构构建组合参数空间以及跨壁公式来发展牛顿-奥肯科夫机构的理论。对称轨道闭包的研究与某个真实的李群的Kazhdan-Lusztig-Vogan多项式和Harish-Chandra模的理论有关，反映了Schubert簇与Kazhdan-Lusztig多项式和表示理论的联系。PI将研究对称轨道闭合对格拉斯曼的投影。Knutson和米勒引入子字复形来研究舒伯特簇的上同调。PI将这些复杂的复曲面退化舒伯特品种和他们的desingularizations.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.