In algebraic geometry, local uniformization is a weak form of resolution of singularities, stating that a variety can be desingularized near any valuation, or in other words that the Zariski–Riemann space of the array is in some sense non-singular.
Zariski (1940) proved local uniformization of varieties in any dimension over fields of characteristic 0, and used this to prove resolution of singularities for varieties in characteristic 0 of dimension at most 3.
Cossart and Piltant (2008, 2009) extended Abhyankar's proof of local uniformization of 3-folds to the remaining characteristics 2, 3, and 5.
Temkin (2013) showed that it is possible to find a local uniformization of any valuation after taking a purely inseparable extension of the function field.
Local uniformization in positive characteristic for varieties of dimension at least 4 is (as of 2019) an open problem.