Residue Currents in Complex Analysis and Commutative Algebra

点击翻译按钮获取中文摘要

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.