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的对象，可用于对有关代数变体的信息进行编码。相反，组合对象的丰富结构通常在被视为代数几何现象的离散阴影时更容易被理解。这个项目的研究利用包括变种的退化和多边形的高维类似的工具来开发这些领域之间的相互作用。本项目旨在了解组合学和代数几何之间相互作用的各个方面，如牛顿-奥库科夫体、对称轨道闭合和子词复合。推广了多面体和环簇之间的关系，牛顿-奥库科夫体提供了从凸几何到研究代数簇的工具。PI将通过构造牛顿-奥孔科夫体的组合参数空间和一个跨越墙的公式来发展牛顿-奥库科夫体理论。对称轨道闭包的研究与某个实李群的Kazhdan-Lusztig-Vogan多项式理论和Harish-Chandra模有关，反映了Schubert簇与Kazhdan-Lusztig多项式和表示理论的联系。PI将调查对称轨道关闭对格拉斯曼尼亚人的预测。Knutson和Miller引入子词复合体来研究Schubert簇的上同调。PI将把这些复合体与舒伯特变种及其设计的环形退化联系在一起。这一奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.