Mathematical Sciences: Research in Algebra and Algebraic Number Theory

该项目支持代数、数论和代数几何的基础和计算研究。其中一名主要研究人员将研究代数k理论，特别强调确定群环的结构。另一位首席研究员将致力于将动机上同的概念扩展到任意权，这是代数几何中的一个重要问题。我们还将研究p进场的理论。将Buchberger理论和算法推广到具有互补估值环的环的问题。此外，本项目还包括射影品种与协同关系的研究，Castelnuovo-Mumford正则性及其与标准Groebner基和协同复杂性的关系的研究。该项目涉及代数、代数计算和数论方面的广泛问题的研究。虽然研究涵盖了广泛的主题，但在这个项目的研究人员之间存在很大的重叠和互动。这代表了数学与许多不同领域的相互作用以及计算的使用和研究的增长趋势。

This project supports fundamental and computational research in algebra, number theory and algebraic geometry. One of the principal investigators will study algebraic K-theory with a particular emphasis on determining the structure of group rings. Another principal investigator will work on extending the notion of motivic cohomology to arbitrary weights which is a significant problem in algebraic geometry. Work will also be done on the theory of p-adic fields. Problems concerning the extension of Buchberger theory and algorithms to rings which have complementary valuation rings will be considered. In addition, this project includes research on the relationship between projective varieties and syzygies, a study of Castelnuovo-Mumford regularity and its relationship to the complexity of standard Groebner bases and syzygies. This project involves research on a wide variety of problems in algebra, algebraic computation and number theory. Although the research covers a wide breath of topics, there are large overlaps and interactions among the investigators on this project. This typifies a growing trend in mathematics of interaction with many diverse areas along with the use and study of computation.