These lectures will revolve around applications of Hrushovski and Kazhdan's theory of motivic integration. It associates motivic invariants to semi-algebraic sets in algebraically closed valued fields. Following the work of Hrushovski and Loeser, and in collaboration with Yin, we shall see that when applied to the non-archimedean Milnor fiber, the motivic volumes recover some classical invariants of the Milnor fiber. Finally, we will see how these methods can be applied to a singularity arising as the quotient of a smooth variety by a linear group. When the group is finite, the orbifold formula of Batyrev and Denef–Loeser provides a motivic version of the McKay correspondence. In collaboration with Loeser and Wyss, we establish a similar formula for a general linear group.[-]

These lectures will revolve around applications of Hrushovski and Kazhdan's theory of motivic integration. It associates motivic invariants to semi-algebraic sets in algebraically closed valued fields. Following the work of Hrushovski and Loeser, and in collaboration with Yin, we shall see that when applied to the non-archimedean Milnor fiber, the motivic volumes recover some classical invariants of the Milnor fiber. Finally, we will see how these methods can be applied to a singularity arising as the quotient of a smooth variety by a linear group. When the group is finite, the orbifold formula of Batyrev and Denef–Loeser provides a motivic version of the McKay correspondence. In collaboration with Loeser and Wyss, we establish a similar formula for a general linear group.[-]

These lectures will revolve around applications of Hrushovski and Kazhdan's theory of motivic integration. It associates motivic invariants to semi-algebraic sets in algebraically closed valued fields. Following the work of Hrushovski and Loeser, and in collaboration with Yin, we shall see that when applied to the non-archimedean Milnor fiber, the motivic volumes recover some classical invariants of the Milnor fiber. Finally, we will see how these methods can be applied to a singularity arising as the quotient of a smooth variety by a linear group. When the group is finite, the orbifold formula of Batyrev and Denef–Loeser provides a motivic version of the McKay correspondence. In collaboration with Loeser and Wyss, we establish a similar formula for a general linear group.[-]

I will present for each integer d > 3 a K-trivial log canonical variety over the complex numbers of dimension d that does not admit a Beauville-Bogomolov decomposition. That is, for the universal cover X of the variety, there is no decomposition of X as a product of an affine space and of three types of projective varieties: strict Calabi-Yau, symplectic and rationally connected varieties. Note: the counterexample is sharp in the sense that for Kawamata log terminal varieties the decomposition does hold.[-]

The $p$-adic Igusa zeta function, topological and motivic zeta function are (related) invariants of a polynomial $f$, reflecting the singularities of the hypersurface $f = 0$. The first one has a number theoretical flavor and is related to counting numbers of solutions of $f = 0$ over finite rings; the other two are more geometric in nature. The monodromy conjecture relates in a mysterious way these invariants to another singularity invariant of $f$, its local monodromy. We will discuss in this survey talk rationality issues for these zeta functions and the origins of the conjecture.[-]