Mathematical Sciences: Tight Closure and Absolute Integral Closures, Briancon-Skoda Theorems and Graded Rings

9413870阿伯巴赫该奖项支持交换代数的研究。主要研究人员将探索优秀局部环的绝对积分闭包的结构。他还将研究分次环的Cohen-Macaulay性，以寻找更高解析偏差的理想。这项研究涉及到交换代数和代数几何中的一些问题。代数几何研究多项式方程族的解。人们既可以研究解集的几何，也可以通过研究解集上的某些函数来代数地处理问题，这些函数构成了所谓的交换环。这种双重视角在交换代数和代数几何之间建立了紧密的联系。

9413870 Aberbach This award supports research in commutative algebra. The principal investigator will explore the structure of the absolute integral closure of excellent local rings. He will also study the Cohen-Macaulayness of graded rings for ideals of higher analytic deviation. This research is concerned with a number of questions in commutative algebra and algebraic geometry. Algebraic geometry studies solutions of families of polynomial equations. One can either study the geometry of the solution set or approach problems algebraically by investigating certain functions on the solution set that form what is called a commutative ring. This dual perspective creates a close connection between commutative algebra and algebraic geometry.