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）资助的。残馀电流是一个复杂变量残馀的多元推广，在代数和分析中有许多应用，包括Hilbert？s Nullstellensatz， brianon - skoda型定理，以及关于PDE系统解的Ehrenpreis-Palamodov基本原理的显式版本。这些应用都依赖于一个中心思想，即全纯函数的理想可以表示为残馀电流的湮灭子理想。经典的多维剩余理论涉及完全交理想。PI最近构建了代表一般理想的剩余电流；这些电流被用来扩展先前已知的完全相交的几个结果。由PI开发的使用剩余电流的建议项目列表包括：在流形上构造积分公式，用剩余来表征乘法器理想，以及恢复作为剩余电流乘积的相交循环。残数微积分是计算积分和级数的经典复解析工具，在数学和科学工程中都有广泛的应用。π吗?她的工作一直处于分析、代数和组合之间的边界，她将继续研究分析和代数性质的问题。

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.