Ran Azouri: Motivic nearby cycles and quadratic conductor formulas

Various tools may be used to investigate degenerations in a motivic setting: The nearby cycles functor of Ayoub in motivic homotopy theory; nearby cycles in the context of motivic integration; comparing the Euler characteristics of the singular and generic fibers. I will report on a quadratic conductor formula for hypersurfaces in a local setting (recent work by Levine, Pepin Lehalleur and Srinivas) with the motivic, compactly supported Euler characteristic, which takes values at the Grothendieck-Witt ring of the base field. Then I will show how reinterpreting it in terms of motivic nearby cycles and computing it along certain coverings (defined by Denef and Loeser) of pieces of the singular fiber, allows to extend the formula to a more general degeneration with a few (quasi-)homogeneous singularities.