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.[-]

We report on the development of localization methods useful for quadratic enumerative invariants, replacing the classical Gm-action with an action by the normalizer of the diagonal torus in SL2.
We discuss applications to quadratic counts of twisted cubics in hypersurfaces and complete intersections (joint with Sabrina Pauli) as well as work by Anneloes Vierever, and our joint work with Viergever on quadratic DT invariants for Hilbert schemes of points on P3 and on (P1)3.[-]

Le troisième groupe de cohomologie non ramifiée d'une variété lisse, à coefficients dans les racines de l'unité tordues deux fois, intervient dans plusieurs articles récents, en particulier en relation avec le groupe de Chow de codimension 2. On fera un tour d'horizon : espaces homogènes de groupes algébriques linéaires; variétés rationnellement connexes sur les complexes; images d'applications cycle sur les complexes, sur un corps fini, sur un corps de nombres.[-]

(joint work with G. Wüstholz) Roughly, $1$-dimensional periods are the complex numbers obtained by integrating a differential form on an algebraic curve over $\bar{\mathbf{Q}}$ over a suitable domain of integration. One of the alternative characterisations is as periods of Deligne $1$-motives.
We clear up the linear relations between these numbers, proving Kontsevich's version of the period conjecture for $1$-dimensional periods. In particular, a $1$-dimensional period is shown to be algebraic if and only if it is of the form $\int_\gamma (\phi+df)$ with $\int_\gamma\phi=0$. We also get formulas for the spaces of periods of a given $1$-motive, generalising Baker's theorem on logarithms of algebraic numbers.
The proof is based on a version of Wüstholz's analytic subgroup theorem for $1$-motives.[-]