\begin{center}\textbf{\LARGE Désingularisation en dimension~3, caractéristique mixte} \end{center} \begin{center} \textbf{\Large Vincent Cossart} \end{center} \begin{center} {\large 13 septembre 2013}\ \end{center} \vskip 5mm {\it \hskip 5mm Conférence d'ediée à Shreeram Shankar Abhyankar, 1930-2012.} \vskip 10mm Travail en commun avec Olivier Piltant. \noindent\textbf{Theorem 1. (Cossart-Piltant)} \emph{Let $C$ be an integral Noetherian curve which is excellent and ${\cal X}/C$ be a reduced and separated scheme of finite type and dimension at most three. There exists a proper birational morphism $\pi : \ {\cal X}' \rightarrow {\cal X}$ with the following properties: \begin{itemize} \item [(i)] ${\cal X}'$ is everywhere regular; \item [(ii)] $\pi$ induces an isomorphism $\pi^{-1}(\mathrm{Reg}({\cal X})) \simeq \mathrm{Reg}({\cal X})$; \item [(iii)] $\pi^{-1}(\mathrm{Sing}({\cal X}))$ is a normal crossings divisor on ${\cal X}'$. \end{itemize} If furthermore ${\cal X}\backslash \mathrm{Sing}{\cal X}$ is quasi-projective, one may furthermore take ${\cal X}'$ projective.}\ Par une réduction "à la Abhyankar" \cite{CoP1}, le théorème ci-dessus est une conséquence du théorème suivant~:\ \noindent\textbf{Theorem 2. (Cossart-Piltant)} \emph{Let $(S,m_S,k)$ be an excellent regular local ring of dimension three, quotient field $K:=QF(S)$ and residue characteristic $\mathrm{char}k=p>0$. Let \begin{equation} h:=X^p+f_1X^{p-1}+ \cdots +f_p \in S[X], \ f_1, \ldots , f_p \in S \end{equation} be a reduced polynomial, ${\cal X} :=\mathrm{Spec}(S[X]/(h))$ and $L:=\mathrm{Tot}(S[X]/(h))$ be its total quotient ring. Assume that $h$ satisfies one of the following assumptions: \begin{description} \item[(i)] ${\cal X}$ is $G$-invariant, where $\mathrm{Aut}_K(L)=\mathbf{Z}/p =:G$, or \item[(ii)] $\mathrm{char}K=p$ and $f_1= \cdots =f_{p-1}=0$. \end{description} Let $\mu$ be a valuation of $L$ which is centered in $m_S$. There exists a composition of local Hironaka-permissible blowing ups: \begin{equation}\label{eq102} ({\cal X}=:{\cal X}_0,x_0) \leftarrow ({\cal X}_1,x_1) \leftarrow \cdots \leftarrow ({\cal X}_r,x_r), \end{equation} where $x_i \in {\cal X}_i$ is the center of $\mu$, such that $({\cal X}_r,x_r)$ is regular.}\ Le cas (ii) est d'ejà résolu \cite{CoP2}. Dans cet exposé, nous allons expliciter le cas (i) à l'aide de la théorie des polyèdres d'Hironaka et des gradués associés~: si le discriminant de $h$ est monomial, les formes initiales de $h$ pour les valuations correspondant aux faces du polyèdre sont alors d'Artin-Schreier ou purement inséparables. C'est le point clef de notre preuve. La preuve complète du théorème de d'esingularisation sera exposée du 1 au 11 octobre à Ratisbonne. http://tinyurl.com/CPschool13 \bigskip \begin{thebibliography}{99} \bibitem{CoP1} {\sc Cossart V., Piltant O.}, Resolution of singularities of threefolds in positive characteristic I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings, {\it J. Algebra} {\bf 320} (2008), no. 3, 1051-1082. \bibitem{CoP2} {\sc Cossart V., Piltant O.}, Resolution of singularities of threefolds in positive characteristic II, {\it J. Algebra} {\bf 321} (2009), no. 7, 1836-1976. \bibitem{CoP3} {\sc Cossart V., Piltant O.}, Characteristic polyhedra of singularities without completion, {\it preprint} arXiv:1203.2484 (2012), 1-6. \bibitem{CoP4} {\sc Cossart V., Piltant O.}, Resolution of Singularities of Threefolds in Mixed Characteristics. Case of small multiplicity, to appear in {\it RACSAM} (2013), 1-39. \end{thebibliography}