This research proposal lies at the interaction between representation theory, low-dimensional topology, symplectic topology and algebraic geometry. Our aim is to develop a deep intradisciplinary connection between the world of quantum invariants, whose initial description comes from representation theory, the world of invariants coming directly from 3-dimensional topology and algebraic geometry.
Conjectures from physics predict that quantum invariants encode topological information of knot complements. The main result of the project leader from 2023 shows that Uq(sl(2))-quantum invariants can be seen directly from a graded intersection pairing of two Lagrangian submanifolds in a symmetric power of a surface. The explicit form of these Lagrangians opens up a new geometric framework for studying these quantum invariants and their categorifications. We have three main objectives which interplay between the 4 research areas mentioned above and the first two have as starting point my result from 2023.
Objective I: Categorifications for quantum invariants from topological models (Projects 1, 2 and 3)
Objective II: The 3-dimensional geometry of quantum invariants via configuration spaces (Projects 4, 5 and 6)
Objective III: Compactification of moduli of Hermite-Einstein metrics (Projects 7 and 8)