ABHYANKAR 82nd BIRTHDAY CONFERENCE SPEAKER: Vincent Cossart TITLE: Desingularization in dimension~3, mixed characteristic ABSTRACT: Joint work with Olivier Piltant. The main Theorem is the following. MAIN THEOREM. Let $C$ be an regular excellent curve with function field $F$. Let $S/F$ be a reduced algebraic projective surface and $\X$ be a flat projective $C$-scheme with generic fiber $\X_F=S$. Then there exists a projective birational morphism $\pi : \X'\fleche \X$ with the following properties: $\bullet$ $\X'$ is everywhere regular. $\bullet$ $\pi$ induces an isomorphism $\pi^{-1}(\Reg(\X)) \iso \Reg(\X)$. $\bullet$ $\pi^{-1}(\Sing(\X))$ is a normal crossings divisor on $\X'$. In this conference we would like to focus on the uniformization problem. We want to explain the main step in reducing the local uniformization problem to the -already solved- uniformization problem in characteristic $p>0$ for Artin-Schreier or purely inseparable extensions of degree $p$ over a regular ring of dimension~3.