mark.spivakovsky@math.univ-toulouse.fr
The problem of resolution of singularities asks whether, given an algebraic variety $X$ over a field, there exists a non-singular algebraic variety $X'$ and a proper map $X'\rightarrow X$ which is one-to-one over the non-singular locus of $X$. The local version of the problem, Local Uniformization, is stated in terms of valuations. The Local Uniformization Theorem was proved by O. Zariski in 1940 in the case when $char k=0$ and is a major open problem in the field when $char k=p>0$. We will first recall Zariski's valuative approach and Hironaka's (non-valuative) proof of resolution in characteristic zero. We will explain the difficulties arising in characteristic $p>0$ and the ideas for overcoming them using differential operators. Finally, we will discuss two main technical tools of our program for proving local uniformization: key polynomials and universal Puiseux expansion in generalized power series with non-well-ordered support.