A^1 homotopy theory provides tools to refine geometric invariants on integers to quadratic forms. A key such invariant is the quadratic Euler characteristic. Ayoub's motivic nearby cycles are a tool to study singularities in the same world of A^1-homotopy theory. In the talk I will explain how to compute the quadratic Euler characteristic on the motivic nearby cycles spectrum for certain singularities, using an explicit semistable reduction construction. This, together with a recent work of Levine, Pepin Lehalleur and Srinivas, adds up to a quadratic conductor formula on schemes with semi-quasihomogeneous singularities, refining formulas of Milnor and Deligne. Time permitting, I will describe how, in a work in progress with Emil Jacobsen, we use a similar semistable reduction argument to compute the motivic monodromy on nearby cycles, generalising to motives the Picard-Lefschetz formula of Deligne.
