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.
