Following Grothendieck's vision that a motive of an algebraic variety should capture many of its cohomological invariants, Voevodsky introduced a triangulated category of motives which partially realises this idea. After describing some of the properties of this category, I explain how to define the motive of certain algebraic stacks. I will then focus on defining and studying the motive of the moduli stack of vector bundles on a smooth projective curve and show that this motive can be described in terms of the motive of this curve and its symmetric powers. If there is time, I will give a conjectural formula for this motive, and explain how this follows from a conjecture on the intersection theory of certain Quot schemes. This is joint work with Simon Pepin Lehalleur.[-]

In moduli theory, it happens often that a moduli space is constructed as a quotient. This is a powerful tool, in fact from the quotient structure one can infer several interesting properties about the properties of the moduli space itself. In this talk, I will recall briefly a construction of the moduli stack of trigonal curves as a quotient stack, that I gave in a joint work with Vistoli a few years ago. Then I will move to a recent work with Loenne, where we draw from this construction some surprising results on the fundamental group of the moduli space, that reveals to be of completely different nature from the space of hyperelliptic curve.[-]

Reidemeister torsion defines an element in the determinant line of a finite CW complex. I will explain its family version which allows one to define a volume form on a mapping stack whose source has a simple homotopy type. One family of examples is given by character stacks of finite CW complexes: for surfaces one recovers the symplectic volume form while for 3-manifolds one obtains orientation data necessary to define cohomological DT invariants. Another family of examples is given by the volume form on the derived loop space related to the Todd class. This is a report on work joint with Florian Naef.[-]

We introduce a generalization of Temkin's reduction in an absolute setting. It takes the form of a category of graded log schemes, containing valuative spaces as a full subcategory, as well as more exotic objects such as the reduction mod $p^{n}$ of a p-adic rigid space. We will compare the log étale and log syntomic topologies on these objects, and we will show that the ramification filtrations of Abbes-Saito, Saito and Kato-Thatte measure precisely the lack of topological invariance of the corresponding log syntomic toposes. As a byproduct, we recover and generalize results of Deligne and Hattori on the ramification of extensions of truncated discrete valuation rings.[-]