Collaborative Research: A Software System for Research in Algebraic Geometry, Commutative Algebra, and their Applications

随着快速计算技术的广泛应用，计算实验在科学和数学的各个分支中变得越来越重要。Macaulay2是一个软件系统，在多学科研究中促进了符号和符号-数值计算的使用。它可以执行的计算在数学、计算机科学和自然科学的许多领域都有应用。本项目的研究扩展了Macaulay2内置的计算方法。该项目将促进研究界的Macaulay2开发人员与应用科学家和纯数学家之间的合作。本项目为学生提供研究训练机会。Macaulay2是一个开源的计算机代数系统，专门用于求解多变量多项式方程组，以及计算和处理描述解集底层几何结构的数据。本方案的三个主要方面是：对核心数学课题的研究和数学以外的应用；维护和发展Macaulay2作为研究基础设施的一部分；通过培训、会议和合作发展人力资源，特别是让年轻的研究人员参与进来。数学之外的影响包括研究物理学、生物学、计算机视觉和其他非线性代数是关键成分的领域的问题。该项目将维持和发展Macaulay2作为研究的主要工具，特别是通过改进Groebner碱基和自由分辨率的核心算法，以及改进其结合符号和数值计算的设施。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

With the advent of widely available fast computing, computational experiments have become more important in every branch of science and mathematics. Macaulay2 is a software system that leads in facilitating the use of symbolic and symbolic-numeric computation in multidisciplinary investigations. The computations it can perform have uses in many fields of mathematics, computer science, and natural sciences. The research under this project extends the computational methods built into Macaulay2. The project will promote collaborations between Macaulay2 developers from the research community, on the one hand, and applied scientists and pure mathematicians, on the other. This project provides research training opportunities for students.Macaulay2 is an open-source computer algebra system dedicated to solving systems of polynomial equations in many variables as well as computing and processing data that describe the underlying geometry of the solution sets. The three major aspects of this proposal are: research on core mathematical topics and applications outside mathematics; the maintenance and development of Macaulay2 as part of the infrastructure of research; and the development of manpower through training, conferences and collaborations, especially involving young researchers. The impact outside mathematics includes investigating problems from physics, biology, computer vision, and other areas where nonlinear algebra is a key ingredient. The project will maintain and develop Macaulay2 as a major tool for research, especially by improving the core algorithms for Groebner bases and free resolutions and by improving its facilities that combine symbolic and numerical computation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.