Residue Currents in Complex Analysis and Commutative Algebra

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。留数流是一个复变量留数的多元推广，它在代数和分析中有许多应用，包括Hilbert？S的Nullstellensatz、Briançon-Skoda型定理的有效版本，以及关于偏微分方程组的解的Ehrenpreis-Palamodv基本原理的显式版本。这些应用都依赖于这样一个中心思想：全纯函数的理想可以表示为留数流的零化子理想。经典的多维剩余理论涉及完全交理想。PI最近构造了代表一般理想的剩余流；这些流被用来推广以前已知的完全交的几个结果。由PI开发的使用残余流的拟议项目清单包括：构造流形上的积分公式，用残余流刻画乘子理想，以及将交循环恢复为残余流的乘积。留数演算是一种计算积分和级数的经典复数分析工具，在数学以及科学和工程中都有应用。S的工作一直处于分析、代数、分析和代数性质的andcombinatoricsandshewillcontinueworkingonproblems之间的边缘地带。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Residue currents are multivariate generalizations of one complex variable residues, which have found many applications in algebra and analysis, includingeffective versions of Hilbert?s Nullstellensatz, Briançon-Skoda type theorems,and explicit versions of the Ehrenpreis-Palamodov Fundamental Principle for solutions of systems of PDE?s. These applications all rely on the central idea that ideals of holomorphic functions can be represented as annihilator ideals of residue currents. The classical multi-dimensional residue theory concerns complete intersection ideals. The PI recently constructed residue currents representing general ideals; these currents were used to extend several results, previously known for complete intersections. The list of proposed projects using the residue currents developed by the PI includes: constructingintegral formulas on manifolds, characterizing multiplier ideals in terms of residues, and recovering intersection cycles as products of residue currents. Residue calculus is a classical complex analytic tool for computing integrals and series, with applications in mathematics as well as in science and engineering. The PI?s work has been in the borderland between analysis, algebra, andcombinatoricsandshewillcontinueworkingonproblems of both analytic and algebraic nature.