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

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。在这个项目中要做的研究扩展了Macaulay2中内置的计算方法。 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。Macaulay 2的实验结果帮助了许多算法的制定、开发和解决。通过对计算工具的改进，本研究将对数学和科学的许多领域产生影响。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，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，例如计算代数簇的上同调和拓扑不变量，并将开发应用于弦理论，超对称性和生物网络的离散模型。

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 in this project extends the computational methods built into Macaulay2. 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. 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.The project has two 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. Secondly, research done under this project will develop better algorithms for some key computational problems, such as computing the cohomological and topological invariants of algebraic varieties, and will develop applications to string theory, supersymmetry, and discrete models of biological networks.