Collaborative Research: A Software System for Algebraic Geometry Research

该项目主要有三个方面。首先，它讨论了作为Macaulay2支持的研究工具的符号计算的基础结构。这包括实施新功能、错误修复、用户支持、进一步的文档和新版本的多平台发布，这些新版本将维护和开发Macaulay2，使其成为用于广泛领域研究的主要工具。其次，它通过培训年轻人使用Macaulay2，并通过参与和扩大作为Macaulay2开发标志的合作来满足人力需求。通过这种方式，它将使更多的研究团体发展这种实验数学的能力。第三，在该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，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，发现生物网络研究的新方法，并在新兴的数值代数几何领域开发更好和更可靠的方法。Macaulay2是一个免费的计算机代数软件系统，致力于定性研究多元多项式方程组。它可以执行的计算在数学和科学的许多领域都有用途，从代数几何到基因组学。根据这项拨款进行的研究扩展了Macaulay2中内置的计算方法。此外，这笔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、学校和围绕这一领域的重要问题举办的“密集协作研讨会”，向他们传授知识。该项目将促进来自研究界的Macaulay2软件开发人员与来自物理和生物学的科学家以及纯数学家之间的新合作。它将向研究生、博士后以及初级和高级数学家介绍计算机在研究数学中的使用，并帮助他们在编程和开发算法方面获得强大的技能，这将加强他们自己的研究。用Macaulay2发现的实验结果有助于许多猜想的形成、发展和解决。通过计算工具的改进，拟议的研究将对数学和科学的许多领域产生影响。

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.