Resolution of singularities and the defect
In 1964 Hironaka proved resolution of singularities for algebraic varieties of arbitrary dimension over fields of characteristic 0. For this result, which has applications in many areas of pure and applied mathematics, he received the Fields Medal. In positive characteristic, resolution has only been proven for dimensions up to 3 by Abhyankar and recently by Cossart and Piltant. The general case has remained open although several working groups of algebraic geometers have attacked it.
Following ideas of Zariski, one can also consider a local form of resolution, called local uniformization. Already in 1940 he proved it to hold for algebraic varieties of all dimensions in characteristic 0. But again, the case of positive characteristic has remained open. By its definition, local uniformization is a problem of valuation theoretical nature. In my talk, I will give a quick introduction to valuations and sketch the main idea of local uniformization.
In positive characteristic, finite extensions of valued fields can show a nasty phenomenon, the defect. It has been identified as one of the main obstacles to local uniformization. I will present examples of defects and some results, as well as some main open problems.