A Software System for Algebraic Geometry Research

代数几何研究的软件系统

• 批准号: 9970085
• 负责人: Daniel Grayson
• 金额: $ 15.94万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-04-15 至 2003-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970085&HistoricalAwards=false
• 关键词: Software; System; Algebraic; Geometry; Research

Grayson9970085 Grayson and his colleague Stillman collaborate in developingthe computer software system Macaulay 2, a symbolic computeralgebra program that supports large scale research computingprojects dealing with systems of polynomial equations in manyvariables. The program's many functions depend mainly onvariants of the Groebner basis algorithm for computing standardbases and syzygies of ideals and modules over polynomial rings.The results yield quantitative and qualitative information aboutsolutions of such systems of equations. In this project theinvestigators develop new algorithms, improve existing algorithmsand implementations, improve the documentation, add new features,improve the program's efficiency, and make it more useable. Theyalso provide a web-accessible computation server to facilitatecontributions from and collaborations between the Internet-basedcommunity of users. These interchanges improve the algorithmsand scope of Macaulay 2. Stillman also continues his research incomputational commutative algebra and algebraic geometry, as wellas into problems involving generalized Green's conjectures,Castelnuovo-Mumford regularity, toric varieties and binomialideals. Grayson also continues his research into algebraicK-theory concerning the connection between algebraic K-theory andmotivic cohomology as mediated by filtrations of the K-theoryspace. The goal of this project is to turn Macaulay 2 into thepremiere collaborative research tool for mathematicians andothers around the world whose work involves systems of polynomialequations. Such research is important because systems ofpolynomial equations arise whenever mathematics is used as a toolin science, for example, in engineering, in physics, and inbiology. The project extends the algorithms, efficiency, andscope of Macaulay 2, provides an Internet server for users, andprovides an Internet-based forum for users to contribute Macaulay2 packages and to collaborate on the solution of mathematics andapplications problems. The project also involves the training ofstudents in both mathematics and symbolic computation.

格雷森9970085 格雷森和他的同事斯蒂尔曼合作开发的计算机软件系统麦考利2，一个符号计算机代数程序，支持大规模的研究计算项目处理系统的多项式方程在manyvariables。 该程序的许多功能主要依赖于Groebner基算法的变体，该算法用于计算多项式环上理想和模的标准基和合轨，其结果产生关于此类方程组解的定量和定性信息。 在这个项目中，研究人员开发了新的算法，改进了现有的算法和实现，改进了文档，增加了新的功能，提高了程序的效率，使其更加可用。 他们还提供了一个网络可访问的计算服务器，以促进基于互联网的用户社区之间的合作和贡献。 这些交换改进了Macaulay 2的算法和范围。 Stillman还继续他的研究在计算交换代数和代数几何，以及到问题涉及广义绿色的代数，Castelnuovo-Mumford正则性，环面品种和binomialideals。 格雷森还继续他的研究到algebraicK-理论有关之间的连接代数K-理论和动机上同调介导的过滤的K-theoryspace。 该项目的目标是将Macaulay 2变成世界各地数学家和其他工作涉及多项式方程组的人的首要合作研究工具。 这样的研究是重要的，因为多项式方程组出现时，数学被用作科学的工具，例如，在工程学，物理学和生物学。 该项目扩展了Macaulay 2的算法、效率和范围，为用户提供了一个互联网服务器，并为用户提供了一个基于互联网的论坛，供用户贡献Macaulay 2软件包，并就数学和应用问题的解决方案进行协作。 该项目还涉及培训学生在数学和符号计算。