The present project is in the broad area of geometric representation theory. The basic goal of the subject is to use relations between geometry and algebra to get better understanding of each of them. This particular research project would deal on the geometric side with moduli of representations of quivers, instantons and sheaves on curves and surfaces. These are related on the algebraic side with Kac-Moody and vertex algebras, their representations and more general tensor categories.
Representation theory uses symmetries in the study of various objects. In particular this project deals with sheaves on curves (which are closely related to number theory, i.e. the study of integers) and quiver varieties (whose origins go back to Platonic solids and so predate number theory). The amazing fact is that solids and numbers have closely related symmetries. Various attempts at understanding (and exploiting) this connection constitute a large part of modern mathematics.