In the 1980's Bernstein-Zelevinski calculated the centre of the abelian category of smooth representations on $GL_{n}K$ when $K$ is a local field. Soon after Deligne generalised this to all reductive algebraic groups $G$ over $K$. The centre of a category consists of all families $z_{A} \in End(A)$ as $A$ varies through all objects such that for any morphism in the category $f:A \longrightarrow B$ we have $fz_{A} = z_{B}f$. Deligne's answer comes in terms of distributions on $G$.

Over the last decade or so, I developed the notion of monomial resolutions for such representations. This amounts to an embedding of the representation category into a derived category of monomial objects. Using Bruhat's thesis I shall explain how to interpret the monomial morphisms in terms of spaces of distributions and thereby to re-derive Deligne's result.

I know to my cost how technical this stuff can get - so I shall try to navigate by means of conceptual insights. For example, for us topologists, I shall explain how sheaves of distributions behave in a manner precisely analogous to a famous result of Swan and Serre about sections of topological vector bundles.

Let $p>2$ be a prime number and $K$ be a number field. Let $S$ be a finite set of primes of $K$ and let $K_S$ be the maximal pro-$p$ extension of $K$ unramified outside $S$; put $G_S=$ Gal $({K_S}{K})$. If $S$ contains the primes above $p$, we know that $cd(G_S)$ less than or equal to $2$, but what is going on if this is not the case?

Thanks to a criteria of Labute, Mináč and Schmidt, we can exhibit mild pro-$p$ groups $G_S$ (and then of cohomological dimension $2$). In this talk I will explain the question of the propagation of the mildness property along some quadratic extensions of $\mathbb{Q}$. In particular, I will give some statistics and some theoretical results.