K. Lakshmanan Memorial Distinguished Lecture
3.30-4.30 p.m., Seminar Hall
Resolution of Singularities and Local Uniformization
University of Missouri, USA.
An algebraic variety X over an algebraically closed field k is a space which is locally defined by the vanishing of a set of polynomials. The nonsingular points of X are the points where the tangent space to X has the same dimension as X. A resolution of singularities of X is a proper algebraic transformation from a nonsingular variety X' to X which is an isomorphism over a dense open subset of X. Resolutions of singularities exist over fields of characteristic zero (such as the complex numbers) as shown by Hironaka, and resolutions exist for varieties of dimension less than or equal to 3 over fields of positive characteristic as shown by Abhyankar.
We discuss ideas in the construction of resolution of singularities in characteristic zero and why the proof does not extend to positive characteristic. We also discuss the related problem of local uniformization, which is resolution at the center of a given valuation.