Mathematical Sciences: Resolution of Singularities of an Algebraic Variety Over a Characteristic p Field

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This award supports the research in Algebraic Geometry of Professor Tzuong T. Moh of Purdue University. His project is to prove that any algebraic variety in positive characteristic may be desingularized. In the case of a curve in the plane, whose defining equation may be complicated enough that the curve crosses over itself, it is not hard to imagine lifting one intersecting branch of the curve up out of the plane so as to remove the selfintersections, or "singularities". This was done in the last century not only conceptually, but by algebraic transformations, so that the originally singular (self- intersecting) curve now sat in a higher-dimensional space but no longer intersected itself. This was done also for algebraic spaces of any dimension, but still in the geometrically intuitive domain of zero-characteristic, in 1964 by Hironaka. It is Professor Moh's aim to desingularize algebraic spaces of arbitrary dimension in the less intuitive and algebraically difficult domain of positive characteristic. This research is work in the algebraic geometry of positive characteristic. Although the field originated with notions of continuously varying geometric structures like lines and planes, in this context the discrete takes over, and methods akin to those from the theory of whole numbers are most useful. Reciprocally, the algebraic geometry of positive characteristic is now having great influence on the Theory of Numbers, and is finding application in Computer Science and Coding Theory.