Prime Characteristic Techniques in Commutative Algebra and Algebraic Geometry

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The main goal of the proposed project is to study the strong factorization conjecture for birational morphisms: every proper birational morphism between nonsingular complex varieties can be factored as a composition of smooth blowups followed by a composition of smooth blowdowns. The conjecture follows from the more fundamental problem of making a morphism toroidal by smooth blowups. There are several approaches to solving the toroidalization problem, for example, reducing it to a problem about resolution of singularities of a differential form with logarithmic poles, or using a composition of several constructions in lower dimensions to achieve toroidalization in higher dimension.The main theme of the proposed research is to study the classification of algebraic varieties. More precisely, given two varieties that are the same generically (they are the same after removing some small subsets), can one transform one variety to another by some elementary operations? The simplest elementary operations are blowups and blowdowns of subvarieties: replacing a subvariety by another subvariety of larger (resp. smaller) dimension. The blowup-blowdown conjecture states that one can always get from the first variety to the second by blowing up several times and then blowing down.