Collaborative Research: A Software System for Algebraic Geometry Research

Macaulay2是一个免费的计算机代数系统，致力于定性研究多变量多项式方程组。它是由丹尼尔·格雷森和迈克尔·斯蒂尔曼在NSF的资助下开发的。格雷森和斯蒂尔曼将继续开发Macaulay 2。他们将升级现有算法，开发和发布新算法，并实施新算法。特别是，他们将开发具有Macaulay2的S优势的符号计算与代数几何中的新浮点算法的交互作用，这些算法现在由Andrew Sommese、Jan Verschelde、Anton Leykin、Frank Schreyer和其他人开发。预计这些将使一类全新的问题可以进行实验，在许多情况下还可以解决。艾森巴德将为与其他系统的扩展集成组织联系，并将邀请其他数学家参与需要完成的开发工作。该项目的核心是继续扩大合作，这一直是Macaulay2开发的标志。为此目的，在为期两年的赠款期间，将举行两次Macaulay2工作组会议。一个要攻击的特殊研究问题是：使用计算系统(概率地)反驳或建议证明关于仿射空间的多项式自同构的Jacobian猜想(这将需要使用新的浮点算法)。新算法可以产生影响的其他领域包括研究数字系统、具有特定类型分母的分数、理想因式分解、变量乘法不满足交换律的系统、几何优化、对基因表达水平随时间的观察结果的分析，以及生物信息学。Macaulay2是支持涉及多变量多项式方程系统的数学研究的基础设施的一部分。对这种多项式方程组的研究是纯数学和应用数学以及物理学的核心，最近在密码学、机器人学和弦论等领域产生了新的影响。计算机能力的提高和Macaulay2等程序的出现使实验达到了一个新的水平。在Macaulay2上发现的实验结果有助于数学和物理学中易于处理的猜想的形成和发展。衡量Macaulay对S影响的一个指标是，至少有270篇研究论文引用了Macaulay 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将进一步开发该软件，并将从研究社区招募开发人员。他们将通过伯克利的工作组和任命研究生为研究生助理，向研究生和数学家介绍在研究数学中使用计算机以及编程和开发算法所需的技能。

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.