Working Group on Algebraic Number Theory

We review the construction of the triangulated categories of motives over a base scheme (following the method of Morel and Voevodsky). We then explain quickly the construction of various operations between these categories as well as some realization functors (Betti and $\ell$-adic). If time permits, we also discuss the rigid analytic variant and develop an application to the theory of nearby and vanishing cycles.