Collaborative Research: A Software System for Algebraic Geometry Research

该项目有三个主要方面。首先，它解决了符号计算的基础设施作为Macaulay 2支持的研究工具。这包括新功能的实现、错误修复、用户支持、进一步的文档以及新版本的多平台发布，这些新版本将维护和开发Macaulay 2作为广泛领域研究的主要工具。第二，通过培训年轻人使用Macaulay 2，并通过参与和扩大作为Macaulay 2发展标志的合作，解决人力需求。通过这种方式，它将使更多的研究社区发展这种实验数学的能力。 第三，在这个项目下所做的研究将为一些关键的计算问题开发更好的算法，例如计算归一化。研究的其他目标是改进核心Groebner基算法的实现，发现研究生物网络的新方法，并在新兴的数值代数几何领域开发更好和更可靠的方法。Macaulay 2是一个免费的计算机代数软件系统，致力于多变量多项式方程组的定性研究。它可以执行的计算在数学和科学的许多领域都有应用，从代数几何到基因组学。在这项资助下进行的研究扩展了Macaulay 2中内置的计算方法。此外，赠款将提供许多机会，培训年轻的数学家和其他科学家使用这些工具，并带来其他专家？通过会议、学校和以这一领域的重要问题为中心的“密切合作研讨会”，向他们传授知识。该项目将使来自研究社区的Macaulay 2软件开发人员与物理学和生物学科学家以及纯数学家之间的新合作成为可能。它将介绍研究生，博士后，初级和高级数学家在研究数学中使用计算机，并帮助他们获得强大的编程和算法开发技能，这将提高他们自己的研究。Macaulay 2的实验结果帮助了许多算法的制定、开发和解决。通过计算工具的改进，所提出的研究将影响数学和科学的许多领域。

The project has three main aspects. First, it addresses the infrastructure of symbolic computation as a research tool supported by Macaulay2. This includes the implementation of new features, bug fixes, user support, further documentation, and multiplatform releases of new versions that will maintain and develop Macaulay2 as a major tool used for research in a broad range of fields. Second, it addresses manpower needs, by training young people in the use of Macaulay2, and by engaging and expanding the collaborations that have been a hallmark of Macaulay2 development. In this way it will enable more of the research community to develop competence in this kind of experimental mathematics. Third, research done under this project will develop better algorithms for some key computational problems, such as computing normalization. Other goals of the research are to improve the implementation of the core Groebner basis algorithms, uncover new methods for the study of biological networks, and develop better and more reliable methods in the emerging field of numerical algebraic geometry.Macaulay2 is a free computer algebra software system dedicated to the qualitative investigation of systems of polynomial equations in many variables. The computations it can perform have uses in many fields of mathematics and science, from algebraic geometry to genomics. The research to be done under this grant extends the computational methods built into Macaulay2. In addition, the grant will provide many opportunities to train young mathematicians and other scientists in the use of these tools, and to bring other experts? knowledge to bear on them through conferences, schools, and "Intense Collaboration Workshops" centered on the important issues in this field. The project will enable new collaborations between Macaulay2 software developers from the research community and scientists from physics and biology as well as pure mathematicians. It will introduce graduate students, postdocs, and junior and senior mathematicians to the use of computers in research mathematics and help them acquire powerful skills in programming and development of algorithms that will enhance their own research. Experimental results found with Macaulay2 have helped in the formulation, development, and solution of many conjectures. Through the improvement of computational tools, the proposed research will impact many fields of mathematics and science.