Abstract: I will talk about joint work with Baohua Fu, Paul Levy and Eric Sommers about the geometry of nilpotent cones of complex semisimple Lie algebras, particularly in exceptional types (Kraft and Procesi studied classical types in the 1980's). One very particular singularity in E8 led us to consider a family of 4-dimensional isolated symplectic singularities; in another paper with two more authors (Gwyn Bellamy and Cédric Bonnafé), we show that they admit many different descriptions and that they have trivial local fundamental group (so they are not quotient singularities), thus answering a question that Beauville raised in 2000.