Affine quantum groups: from geometry to categorifications

Representation Theory is an area of pure mathematics at the intersection of algebra, geometry and combinatorics. Its main philosophy is to study intricate mathematical objects in an indirect way by investigating their action on very simple, concrete and well- understood objects. Metaphorically, one looks at the shades of these objects in order to guess their essential properties rather than trying to directly observe them. Representation theory has been an extremely active research are in the past decades and has very deep connections and applications to other areas such as mathematical physics. The main motivation for the projects of the Principal Investigators proposal stem from the rich representation theory of objects called quantum groups, that were introduced in order to tackle questions arising from quantum mechanics and high energy physics. The crucial features at the heart of the Principal Investigators projects are certain remarkable invariants that can be naturally constructed out of the representation theory of quantum groups. The present proposals main purpose consists in providing new interpretations of these invariants using different perspectives, such as geometry or category theory. The former may yield to highly non-trivial geometric constructions capturing a large part of the representation-theoretic data. The latter suggests deep connections with theories that have recently emerged thanks to recent breakthroughs relying on categorical tools. Many fundamental aspects of these theories remain unexplored or vastly open and therefore establishing connections with the representation theory of quantum groups could potentially have significant impact.