Invariants, complexity and quivers

不变量、复杂性和颤动

• 批准号: 1302032
• 负责人: Harm Derksen
• 金额: $ 15.64万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1302032&HistoricalAwards=false
• 关键词: Invariants; complexity; quivers

The PI will study representations of quivers and is currently collaborating with Jerzy Weyman on an introductory book. The relationship between quivers with potentials and cluster algebras that has been explored in joint work with Weyman and Zelevinsky will be studied further. He will also study Classical Invariant Theory related to Schur-Weyl duality. This leads to a generalization of a theorem by Schrijver and a "super" version of wheeled PROPs. He also will apply Invariant Theory to (Algebraic) Complexity Theory and investigate the complexity of the Graph Isomorphism Problem. The PI will try to generalize the invariant for polymatroids that he has introduced to knots. He will also continue his collaboration with David Masser on recurrence sequences, and linear equations over multiplicative groups.A fundamental problem is to determine whether 2 objects are essentially the same. In the Graph Isomorphism, for example, one would like to determine whether two given graphs are the same after reordering of the vertices. The PI is working on an algorithm that can be proven to be efficient for various classes of graphs. One way to distinguish objects is by using invariants. An invariant is a function on objects that may have distinct values for objects that are not the same. The PI has introduced such an invariant for graphs, and would like to extend this invariant to knots. A quiver representation is just a directed graph, where the vertices represent vector spaces and the arrows represent linear maps. Quiver representations are related to quantum groups and string theory, and the PI will investigate some of the theoretical aspects.

PI将研究箭袋的表示，目前正在与Jerzy Weyman合作编写一本介绍性书籍。与韦曼和Zelevinsky的联合工作中已经探讨了与潜在的和集群代数的箭图之间的关系将进一步研究。他还将研究与Schur-Weyl对偶相关的经典不变理论。这导致了Schrijver定理的推广和轮式PROPs的“超级”版本。他还将不变量理论应用于（代数）复杂性理论，并研究图同构问题的复杂性。PI将尝试推广不变量的polymatroids，他已经介绍了结。他还将继续他的合作与大卫马瑟的递归序列，和线性方程的乘法groups.A基本问题是要确定是否2对象基本上是相同的。例如，在图同构中，人们想要确定两个给定的图在顶点重新排序后是否相同。PI正在研究一种算法，该算法可以被证明对各种类型的图都是有效的。区分对象的一种方法是使用不变量。不变量是一个对象上的函数，它可能具有不同的值，而这些值对于不同的对象来说是不同的。PI已经为图引入了这样一个不变量，并希望将此不变量扩展到节点。图的表示就是一个有向图，其中顶点表示向量空间，箭头表示线性映射。箭袋表示与量子群和弦理论有关，PI将研究一些理论方面。