Invariant Theory and Algebraic Combinatorics

不变理论和代数组合学

• 批准号: 0901298
• 负责人: Harm Derksen
• 金额: $ 37.91万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901298&HistoricalAwards=false
• 关键词: Invariant; Theory; Algebraic; Combinatorics

In Invariant Theory, one studies algebraic expressions which remain the same under spatial symmetries. Such algebraic expressions are called invariants. For example, the distance to the origin does not change under the rotation of the plane around the origin. The distance to the origin is an invariant. The PI will study a new approach to classical invariant theory. In collaboration with mathematicians and engineers at UIUC, the PI will study applications of theory of subspace arrangements to computer vision and image compression.The PI will work on various other topics in commutative algebra, algebraic combinatorics, number theory. Cluster algebras were introduced by Fomin and Zelevinsky. These algebras have deep connections with the theory of quiver representations. The PI will explore this further in collaboration with Weyman and Zelevinsky. To a matroid or polymatroid one can associate various quasi-symmetric functions. The PI will study universal properties of such invariants.The PI will collaborate with Masser to find an effective version of the Mordell-Lang problem in positive characteristic.

在不变量理论中，人们研究在空间对称下保持不变的代数表达式。这样的代数表达式称为不变量。例如，到原点的距离在平面围绕原点旋转的情况下不会改变。到原点的距离是一个不变量。PI将研究经典不变理论的新方法。PI将与UIUC的数学家和工程师合作，研究子空间排列理论在计算机视觉和图像压缩中的应用。PI将研究交换代数，代数组合学，数论等各种其他主题。簇代数是由Fomin和Zelevinsky引入的。这些代数与代数表示理论有着深刻的联系。PI将与Weyman和Zelevinsky合作进一步探索这一点。一个拟阵或多拟阵可以与各种拟对称函数相关联。PI将研究这类不变量的普适性质，并与Masser合作，找到一个有效的正特征形式的Mordell-Lang问题。