Categorical symmetries in representation theory

The concept of symmetry is an important organising principle in science. Representation theory is the field ofmathematics concerned with studying symmetries. This project aims to develop categorical symmetries ofcentral objects in mathematics such as braid groups, the Hilbert scheme of points, and the Virasoro algebra. Theproblems proposed have connections to many different areas including algebra, geometry, topology, andmathematical physics. Progress on this project would be a majoradvance in pure mathematics with potentially dramatic consequences in all the related fields.