Invariant Theory and Algebraic Combinatorics

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，平面绕原点旋转时，到原点的距离不会改变。到原点的距离是一个不变量。PI将研究经典不变量理论的一种新方法。与UIUC的数学家和工程师合作，PI将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将致力于交换代数、代数组合学和数论的其他各种主题。簇代数是由Fomin和Zlevinsky引入的。这些代数与箭图表示理论有很深的联系。PI将与Weyman和Zlevinsky合作进一步探讨这一问题。对于一个拟阵或多个拟阵，人们可以把各种准对称函数联系起来。PI将研究这些不变量的普遍性质。PI将与Masser合作，寻找具有正特征的Mordell-Lang问题的有效版本。

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.