In this lecture, we shall mention some principal points in the integration theory of Hrushovski-Kazhdan over algebraically closed valued fields of equal characteristic zero. In particular, we consider a certain Grothendieck ring of definable subsets and give a sketch of proof of the structure theorem of this ring. Furthermore, we recall works by Hrushovski-Loeser concerning the structure theorem and motivic Milnor fiber. In fact, they are able to construct a ring homomorphism between the previous ring and the Grothendieck ring of algebraic varieties, from which the motivic Milnor fiber can be described in terms of analytic Milnor fiber viewed as a efinable subset. We shall close the lecture by a proof of Kontsevich-Soibelman’s integral identity conjecture using the works by Hrushovski-Loeser as well as elements of Hrushovski-Kazhdan’s integration.
salle 6