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

随着快速计算的广泛应用，计算实验在科学和数学的各个分支中变得越来越重要。Macaulay 2是一个软件系统，它有助于在多学科研究中使用符号和符号数字计算。 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。 该项目下的研究扩展了Macaulay 2中内置的计算方法。 该项目将促进Macaulay 2开发人员与应用科学家和纯数学家之间的合作。Macaulay 2是一个开源的计算机代数系统，致力于求解多变量多项式方程组，以及计算和处理描述解集的基础几何的数据。 这项建议的三个主要方面是：研究数学以外的核心数学课题和应用;维护和发展Macaulay 2作为研究基础设施的一部分;以及通过培训、会议和合作，特别是年轻研究人员的参与，发展人力资源。 数学之外的影响包括研究物理学，生物学，计算机视觉和其他领域的问题，其中非线性代数是一个关键因素。 该项目将保持和发展Macaulay 2作为一个主要的研究工具，特别是通过改进Groebner基地和自由分辨率的核心算法，并通过改进其设施，结合联合收割机符号和数值计算。该奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.