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学习代数K-理论，重点是确定群环的结构。另一位主要研究人员将致力于将上同调的概念扩展到任意权，这是代数几何中的一个重要问题。此外，还将开展p-进动场理论方面的工作。我们将考虑将Buchberger理论和算法推广到具有互补赋值环的环上的问题。此外，这个项目还包括投射簇与合子之间的关系的研究，Castelnuovo-Mumford正则性的研究以及它与标准Groebner基和合子的复杂性的关系。这个项目涉及对代数、代数计算和数论中的各种问题的研究。虽然这项研究涵盖了广泛的主题，但在这个项目上，研究人员之间存在着很大的重叠和互动。这代表着随着计算的使用和研究，数学中与许多不同领域相互作用的趋势越来越大。

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.