Invariants, complexity and quivers

PI将研究颤抖的表现，目前正在与杰西·韦曼合作写一本介绍性的书。与Weyman和Zelevinsky共同研究的带势颤振与簇代数之间的关系将得到进一步的研究。他还将学习与Schur-Weyl对偶有关的经典不变理论。这导致了Schrijver定理的推广和轮式PROPs的“超级”版本。他也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将尝试推广他引入到结点的多拟阵的不变量。他还将继续与David Masser合作研究递归序列，以及乘法群上的线性方程。一个基本问题是确定两个对象是否本质上相同。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。不变式是对象上的函数，对于不相同的对象可能具有不同的值。PI已经为图引入了这样一个不变量，并且想把这个不变量扩展到结点。抖动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。颤振表示与量子群和弦理论有关，PI将研究一些理论方面。

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.