Collaborative Research: A Software System for Algebraic Geometry Research

Macaulay2是一个免费的计算机代数系统，致力于定性 多元多项式方程组的研究。 它 是由丹尼尔格雷森和迈克尔斯蒂尔曼在国家科学基金会的资助下开发的。 Grayson和Stillman将继续开发Macaulay2。 他们 将升级现有算法，开发和发布新算法， 实现新的算法。特别是，他们将发展互动 象征性的计算是Macaulay 2的力量， 代数几何中的浮点算法， 作者：Andrew Sommese，Jan Verschelde，Anton Leykin，Frank Schreyer 等人 预计这些将使一个全新的类 这些问题可以通过实验来解决，在许多情况下，也可以解决。 Eisenbud将组织联系人，以便与其他 系统，并将从事其他数学家的发展工作， 必须完成 该项目的核心是继续扩大 合作是Macaulay2开发的标志。 为此，将在以下地点举行两次Macaulay 2工作组会议： 两年的补助金。 一个特别的研究问题是 攻击的是：使用计算系统（概率） 反驳，或建议证明，雅可比猜想多项式 仿射空间的自同构（这将需要使用新的浮动 点算法）。 新算法可以产生影响的其他领域 包括研究数字系统，具有特定类型的分数， 分解器，理想分解，系统，其中乘法的 变量不满足交换律，几何优化， 分析随时间推移的基因表达水平的观察结果，以及 生物信息学 Macaulay2是支持数学的基础设施的一部分， 涉及多变量多项式方程组的研究。 的 研究这样的多项式方程组是纯和 应用数学和物理学，最近在这些领域的新影响 密码学，机器人学和弦理论 提高计算机能力 像Macaulay2这样的程序的可用性正在使 实验可能。 Macaulay 2的实验结果表明， 正在帮助制定和发展易于处理的战略， 数学和物理都是如此。衡量麦考利影响力的一个指标是， 至少有270篇研究论文引用了麦考利2，几位数学家 贡献了代码，书籍和课程材料现在都在使用它。 PI将进一步开发软件，并将招募开发人员 来自研究界。 他们将介绍研究生， 数学家在数学研究中使用计算机， 在编程和算法开发的必要技能，通过 工作组在伯克利和通过任命的研究生 作为研究生助理

Macaulay2 is a free computer algebra system dedicated to the qualitative investigation of systems of polynomial equations in many variables. It was developed by Daniel Grayson and Michael Stillman with NSF funding. Grayson and Stillman will continue the development of Macaulay2. They will upgrade existing algorithms, develop and publish new algorithms, and implement new algorithms. In particular, they will develop the interaction of the symbolic computations that are Macaulay2's strength with the new floating point algorithms in algebraic geometry that are now being developed by Andrew Sommese, Jan Verschelde, Anton Leykin, Frank Schreyer and others. It is anticipated that these will make a whole new class of problems accessible to experimentation and, in many cases, solution. Eisenbud will organize contacts for the extended integration with other systems and will engage other mathematicians in the development work that needs to be done. Central to the project are the continued expansion of the collaborations that have been the hallmark of Macaulay2 development. For this purpose two Macaulay2 Workgroup Meetings will be held in the course of the two-year grant. One particular research problem to be attacked is: the use of computational systems to (probabilistically) disprove, or suggest a proof of, the Jacobian Conjecture on polynomial automorphisms of affine spaces (this will require the use the new floating point algorithms). Other areas where new algorithms can make an impact include the study of numerical systems, fractions with specified types of denominators, ideal factorization, systems where the multiplication of the variables doesn't satisfy the commutative law, geometric optimization, the analysis of observations of gene expression levels over time, and bioinformatics. Macaulay2 is part of the infrastructure that supports mathematical research involving systems of polynomial equations in many variables. The study of such systems of polynomial equations is central in pure and applied mathematics and in physics, with recent new impacts in such fields as cryptography, robotics and string theory. Increasing computer power and the availability of programs like Macaulay2 are making a new level of experimentation possible. The experimental results found with Macaulay2 are helping in the formulation and development of tractable conjectures in mathematics as well as in physics. A measure of Macaulay2's impact is that at least 270 research papers have cited Macaulay2, several mathematicians have contributed code, and books and course materials are now using it. The PI's will develop the software further and will recruit developers from the research community. They will introduce graduate students and mathematicians to the use of computers in research mathematics and the requisite skills in programming and development of algorithms, through workgroups at Berkeley and through the appointments of graduate students as graduate assistants.