A Software System for Algebraic Geometry Research

代数几何研究的软件系统

• 批准号: 9970348
• 负责人: Michael Stillman
• 金额: $ 20.74万
• 依托单位: Cornell University
• 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=9970348&HistoricalAwards=false
• 关键词: Software; System; Algebraic; Geometry; Research

Stillman9970348 Stillman and his colleague Grayson 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.

Stillman和他的同事Grayson合作开发了计算机软件系统Macaulay 2，这是一个符号计算机代数程序，支持处理多变量多项式方程系统的大规模研究计算项目。该程序的许多功能主要依赖于格罗布纳基算法的变体，用于计算多项式环上的标准基和理想和模块的协同。结果给出了这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，并使其更可用。它们还提供了一个可访问网络的计算服务器，以促进基于互联网的用户社区之间的贡献和协作。这些交换改进了Macaulay 2的算法和范围。Stillman还继续他在计算交换代数和代数几何方面的研究，以及涉及广义格林猜想、Castelnuovo-Mumford正则、环变和二项式的问题。格雷森还继续他对代数k -理论的研究，以k -理论空间的过滤为媒介，探讨代数k -理论与动机上同的关系。这个项目的目标是将Macaulay 2变成数学家和世界上其他涉及多项式方程系统的人的首要合作研究工具。这样的研究很重要，因为每当数学被用作科学的工具时，多项式方程系统就会出现，例如在工程学、物理学和生物学中。该项目扩展了Macaulay2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来贡献Macaulay2软件包，并在数学和应用问题的解决上进行协作。该项目还包括对学生进行数学和符号计算方面的培训。