Until recently there was known an essentially unique way to resolve singularities of varieties of
characteristic zero in a canonical way (though there were different descriptions and proofs of correctness).
Recently a few advances happened --
1) The algorithm was extended to varieties with log structures and even to morphisms -- This requires to consider
more general logarithmic blow ups and results in a stronger logarithmic canonicity of the algorithms even for resolution
Abstract: I'll discuss various "ratio mixing" properties of transformations preserving infinite measures e.g. "Krickeberg mixing" (based on the example in Hopf's 1936 book) & "rational weak mixing". I'll also introduce a new one connected to "tied down" renewal theory.
Contains joint works with Hitoshi Nakada, Dalia Terhesiu & Toru Sera
Abstract: Arrow's impossibility theorem states that the only voting rule satisfying certain natural requirements is a dictatorship. Gil Kalai showed that even if we relax some of these requirements so that they only hold with high probability, we are not getting genuine new voting rules. Arrow's theorem is just one of many paradoxes in social choice theory.