Homological and quantum invariants in low dimensional topology
8th to February 12th 2021
The aim of this workshop is to continue the study of low-dimensional topology invariants built in parallel from quantum groups or homologies of configuration spaces, and to observe their relations. Recent results indeed relate the Verma modules of Uq(𝖘𝖑2) very precisely to configuration space homologies on punctured disks.
On the one hand, recipes for constructing quantum invariants are known but their topological content is often lost, giving rise to numerous conjectures (volume conjecture, conjecture AMU etc.). On the other hand, the methods for constructing invariants from configuration spaces are less developed while the homological information remains apparent. The rich algebraic structures coming from both homological tools (Poincaré duality, cup and cap product, etc.) and representation theory (tensor product, decomposition into weight spaces, etc.) should shed light on each other. Also, the generalization of this homological approach to invariants coming from other quantum groups is still little known. Finally, it is reasonable to think that this interpretation in terms of configuration spaces will bring new point of views on quantum knot polynomials and their categorifications.