The proposal consists of several interrelated parts. The first part is to study the rings of semi-invariants of quivers and of quivers with relations. The investigator proposes to continue study the walls of cones of weights of rings of semi-invariants of quivers and multiplicities of weight spaces for these rings. In particular case this includes the cones defined by Klyachko inequalities. For the semi-invariants of quivers with relations the main problem is to characterize the finite type and tame quivers with relations in terms of semi-invariants. Another aspect is the connection of quiver representations and cluster algebras. The principal investigator plans to study quivers with potential and the mutations of related Jacobian algebras. He also plans to study the connection of certain sphere triangulations and the Igusa-Orr theory of pictures related to nilpotent groups. The second part is devoted to studying defining ideals of equivariant varieties. Several types of varieties are proposed: orbit closures for representations with finitely many orbits, tangential and secant varieties of orbits of highest weight vectors, and orbit closures for Dynkin quivers corresponding to Schubert varieties in the Grassmannian. The third part involves problems related to Boij-Soderberg conjectures for Betti tables of graded modules. The principal investigator proposes to study cohomology tables of vector bundles on homogeneous spaces and equivariant refinements of Boij-Soderberg conjectures.
This proposal is related to several branches of mathematics: representations of quivers and commutative algebra. A representation of a quiver is a way to associate vector space data to the vertices of some oriented graph. The edges can be viewed as relations between these data. Abstract algebra allows to study such objects systematically and the results of research might lead to better algorithms dealing with linear algebra problems. Commutative algebra studies polynomial functions of geometric objects. The second part of the proposal is devoted to studying polynomial equations defining objects characterized geometrically such as rank conditions on matrices. Various conditions of this type on tensors are of interest for engineers and computer scientists. The third part of the proposal studies Betti tables: certain family of numerical invariants associated to modules over a polynomial ring.
