The 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.
