Local uniformization can be understood as a local form of resolution of singularities. In 1940, Zariski proved the Local Uniformization Theorem for places of algebraic function fields over base fields of characteristic 0. In our talk, we will sketch the proof of the following theorem:
Let F|K be a function field of arbitrary characteristic, and P an Abhyankar place of F|K such that FP|K is separable. Further, take any elements \zeta_1,\ldots,\zeta_m in the valuation ring of P on F. Then there exists a model of F|K on which P is centered at a smooth point whose local ring contains \zeta_1,...,\zeta_m.
Abhyankar places of F|K are places of F which are trivial on K and for which equality holds in the Abhyankar inequality. They are very "representative" as they lie dense in the Zariski space of all places, with respect to topologies much finer than the Zariski topology. (This fact can be proved using the model theory of valued fields.)
We also present the arithmetic version of the above theorem, where the Abhyankar places are allowed to be extensions of p-adic places on Q. Moreover, we discuss the possibility of simultaneous uniformization of finitely many Abhyankar places (work in progress); this would amount to the simultaneous local resolution of finitely many singularities.
The proofs use the theory of henselian elements which was developed in joint work with Peter Roquette. An element z in an algebraic extension of a field K equipped with a place P is called henselian if there is a polynomial h with coefficients in the valuation ring of K and such that h(z)=0 and the residue zP is a simple root of the reduction of h modulo P.
Finally, we will discuss local uniformization for arbitrary places in arbitrary characteristic after a finite Galois extension of the function field. Alternatively, one can also seek to minimize the extensions of value group and residue field induced by the extension of the function field. We will give a corresponding result.