Geometric aspects of quiver representations

PI提出研究颤振表示与代数的其他分支之间的相互作用。提出了几个问题:研究锥环的重量的城墙就颤动的累积量体重空间的多样性,描述有限类型和驯服的累积量的颤动,颤动表征之间的连接和集群代数和预期轨道之间的连接关闭Dynkin颤动的余维数3和4,余维数三的完美理想的结构和戈伦斯坦余维数四的理想。涉及到簇代数的部分包括用Igusa-Orr图理论研究连接和研究具有超势的颤振的突变。这个提议是关于颤栗的表示，交换代数和表示理论之间的相互作用-代数的不同分支。颤振的表示为线性代数问题（即涉及向量和矩阵的问题）的编码提供了一个很好的组合工具。交换代数和代数几何是研究由多项式方程定义的集合的数学分支。PI计划研究的一些相互作用是最近发现的，并为所有这些领域提供了新的见解。在目前的提议中，颤振与完美和戈伦斯坦理想的结构之间可能的新的意想不到的联系将被探索。如果这部分成为现实，它可能会对交换代数产生根本性的影响。

DMS-0600229Jerzy M. WeymanThe PI proposes to study interactions between the quiver representations and other branches of algebra. Several problems are proposed: study of the walls of cones of weights of rings of semi-invariants of quivers and the multiplicities of their weight spaces, characterizing finite type and tame quivers in terms of semi-invariants, connections between quiver representations and cluster algebras and an expected connection between the orbit closures for Dynkin quivers of codimension three and four and the structure of perfect ideals of codimension three and Gorenstein ideals of codimension four. The part involving cluster algebras includes studying connections with the theory of pictures of Igusa-Orr and studying the mutations of quivers with superpotential.This proposal is concerned with interactions between the representations of quivers, commutative algebra and representation theory - different branches of algebra. The representations of quivers give a nice combinatorial tool for coding the linear algebra problems, i.e. problems involving vectors and matrices. Commutative algebra and algebraic geometry are branches of mathematics studying the sets defined by polynomial equations. Some of the interactions the PI proposes to study were discovered recently and provided new insights to all these areas. In the present proposal the possible new unanticipated connection between quivers and the structure of perfect and Gorenstein ideals will be explored. If this part materializes, it could have fundamental consequences in commutative algebra.