Valuation Rings and Local Factorization

Cutkosky 9610132 主要研究者将继续研究映射的局部因式分解。 研究者最近证明了双有理映射沿着一个赋值的因子分解是可能的，通过构造一个具有非奇异中心的monoidal变换的三角形，在三维和特征零。 一般来说，两个环都必须爆破。 主要研究者正在追求的一些新方向是：（1）找到一个更“规范”的算法，它将通过monoidal变换修补到全局因式分解;（2）将结果扩展到正特征;（3）找到一个在所有维度上都有效的局部结果。 他还继续研究等价性和有限的决定性的映射，并对科恩，麦考利协调环。 这是代数几何领域的研究。 代数几何是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。 在其起源，它处理的数字，可以定义在平面上的最简单的方程，即多项式。 如今，该领域不仅使用代数方法，还使用分析和拓扑学方法，相反，在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。

Cutkosky 9610132 The principal investigator will continue a study of local factorization of mappings. The investigator has recently proven that factorization of birational maps along a valuation is possible in dimension three and characteristic zero by constructing a triangle of monoidal transforms with nonsingular centers. In general, it is necessary that both rings be blownup. Some new directions which the principal investigator is pursuing are: (1) Find a more "canonical" algorithm, which will patch to a global factorization by monoidal transforms; (2) Extend the result to positive characteristic; (3) Find a local result valid in all dimensions. He is also continuing research on equivalence and finite determinancy of mappings, and on Cohen-Macaulay coordinate rings. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.