Motivic nearby cycles and their monodromy at a singular point

In this survey, we explain how to compute both the quadratic Euler characteristic of nearby cycles, and the motivic monodromy, at a quasi-homogeneous singularity. This gives, for such singularity, a quadratic refinement to the Deligne–Milnor formula in characteristic zero, and an enhancement of the Picard–Lefschetz formula to Voevodsky motives with rational coefficients.