(NOTE SPECIAL TIME!) Holonomy perturbations and irreducible SU(2)-representations of integral homology 3-spheres
Does any integral homology 3-sphere admit an irreducible SU(2)-representation of its fundamental group? This question is open since a long time. Various partial results have been obtained by different authors using instanton gauge theory. A few years ago we have proved this result for splicings of knots in the 3-sphere. Recently, this has been generalized to toiroidal integral homology 3-sphere in joint work with Tye Lidman, and Junanita Pinzon Caicedo. We will sketch the proofs of these results, and we also mention a few related open problems.