Quantum Algebras, Quiver Varieties and Applications

Representation theory is the study of symmetries in mathematics, while algebraic geometry is the study of spaces that can be described by algebraic equations. The interface between these two fields is a rich subject, which has had recent applications to various branches of mathematics, theoretical physics, and combinatorics. In this project the principal investigator will study a particular class of spaces called quiver varieties, which are determined by certain graphs, from which they inherit many interesting symmetries. Abstracting the properties of quiver varieties allows mathematicians to discover many fascinating formulas, whose applications range between such distant fields as string theory and the study of knots.

The approach of the principal investigator is two-fold: first, to study the general properties of quiver varieties (cohomology, K-theory, derived categories) by using a technical tool called the shuffle algebra, and second, to apply these techniques in important particular cases in order to solve concrete problems. For example, the study of flag Hilbert schemes leads to a geometric realization of knot invariants. Similarly, the study of moduli spaces of higher rank sheaves leads to a mathematical understanding of the relations between gauge theory and conformal field theory. These and other applications will be pursued by the principal investigator and his coauthors.