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

该奖项支持在代数几何的研究， Tzuong T.教授普渡大学的Moh。他的计划是 证明任何具有正特征的代数簇都可以 去奇异化在平面中的曲线的情况下，其 定义方程可能足够复杂， 穿越自身，不难想象， 使曲线的交叉分支向上离开平面，以便 消除自相交或“奇异点”。这样做是 在上个世纪，不仅在概念上， 转换，因此最初的奇异（自 交叉）曲线现在坐在一个高维空间，但没有 更长时间的自我反省。这也是为了代数 任何维度的空间，但仍然是几何直观的 1964年，Hironaka提出了零特征域。是 莫教授的目标是去奇异化代数空间， 在不太直观和代数的任意维 困难领域的积极特征。 本文的研究工作是在代数几何的正 特色虽然这个领域起源于 连续变化的几何结构，如直线和平面， 在这种情况下，离散接管，和方法类似， 来自整数理论的那些是最有用的。 正特征的代数几何 现在对数论有很大的影响， 在计算机科学和编码理论中找到应用。

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.