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

该奖项支持普渡大学莫宗棠教授在代数几何方面的研究。他的计划是证明任何正特征的代数变化都可以去具体化。对于平面上的曲线，其定义方程可能足够复杂，以至于曲线与自身交叉，不难想象将曲线的一个相交分支从平面上抬起，以消除自交，或“奇点”。这在上个世纪不仅在概念上完成了，而且通过代数变换完成了，因此原来的奇异（自相交）曲线现在位于高维空间中，但不再与自身相交。这也适用于任何维的代数空间，但仍然是在零特征的几何直观域，由Hironaka在1964年完成。Moh教授的目标是在不太直观和代数上困难的正特征域中对任意维的代数空间进行非具体化。本研究是在正特征的代数几何领域进行的。虽然该领域起源于连续变化的几何结构，如线和平面，但在这种情况下，离散接管了，类似于整数理论的方法是最有用的。反过来，正特征代数几何也对数论产生了重大影响，并在计算机科学和编码理论中得到了应用。

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.