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

Macaulay2是一个免费的计算机代数软件系统，致力于定性研究多变量多项式方程组。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。该项目中将要进行的研究扩展了Macaulay2中内置的计算方法。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。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，提出的研究将影响数学和科学的许多领域。该项目主要有两个方面。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，该项目的研究将为一些关键的计算问题开发更好的算法，如计算代数簇的上同调不变量和拓扑不变量，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.