$\mathrm{SL}(2,\mathbb{C})$ Floer homology for knots and 3-manifolds

I will explain the construction of some new homology theories for knots and three-manifolds, defined using perverse sheaves on the $\mathrm{SL}(2,\mathbb{C})$ character variety. These invariants are models for an $\mathrm{SL}(2,\mathbb{C})$ version of Floer’s instanton homology. I will present a few explicit computations for Brieskorn spheres and small knots in $S^3$, and discuss the connection to the Kapustin-Witten equations, Khovanov homology, and the $A$-polynomial. The three-manifold construction is joint work with Mohammed Abouzaid, and the one for knots is joint with Laurent Cote.