Computational Group Theory

首席研究员和他的同事们继续开发软件包Magnus，该软件包主要用于在有限呈现的群体中进行计算和实验。这个软件已经安装好了，可以通过互联网远程使用。这意味着世界各地的研究人员可以通过登录magnus服务器来使用该软件。该项目使用基于互联网的设施来维护一个突出的开放问题列表，并建立几个讨论小组，重点关注相关的数学领域。这些支持用户之间的协作交互，包括欧洲、澳大利亚、新西兰、香港和中国的研究人员。软件开发包括遗传算法的引入，以及允许用户向Magnus添加自己的例程的措施，扩展了工具的范围。研究者也使用Magnus来研究纯群论和交换代数中的问题。这个项目包括对学生的培训。在数学中，群论旨在捕捉和研究对称的本质，特别是在模式中。这种模式在自然界中经常出现，例如晶体。群论与对称性的联系使得群论在物理学、生物学、对我们的物理世界的研究，以及几何学和其他数学领域（如计算理论和密码学）中都很重要。研究人员在这个项目中的目标是建立一个易于使用的计算工具，使用户能够进行实验并解决有关这里涉及的最普遍的数学对象的问题，即所谓的有限呈现群。这些群体可以用一组有限的规则来完全描述；例子包括物体的刚性几何运动组和量子力学组。计算工具Magnus有一个简单但有限的界面。尽管如此，Magnus已经被用来探索和解决群论中的许多难题。目前的项目侧重于算法的发展和技术，允许用户在Magnus中添加自己的例程，扩展工具的范围。该项目还在合作实验室环境中提供可访问互联网的计算和信息资源，促进研究人员之间基于互联网的交流和合作。最后，该项目有助于培养学生的计算和数学能力。

Baumslag9973233 The principal investigator and his colleagues continuedeveloping the software package Magnus, which has been designedto carry out computations in and experiments with, in the main,finitely presented groups. This software has been set up so thatit can be used remotely, via the internet. This means thatresearchers world-wide can use the software by logging on to theMagnus server. The project uses Internet-based facilities tomaintain a list of outstanding open problems and to set upseveral discussion groups focusing on related mathematical areas.These support collaboratory interactions among users, includingresearchers in Europe, Australia, New Zealand, Hong Kong, andChina. Software development includes the introduction of geneticalgorithms as well as measures to allow users to add their ownroutines to Magnus, extending the scope of the tool. Theinvestigators also use Magnus to study problems in pure grouptheory and in commutative algebra. The project includes thetraining of students. In mathematics, group theory aims to capture and study theessence of symmetry, particularly in patterns. Such patternsoccur frequently in nature, for instance as crystals. Itsconnection with symmetry makes group theory important in physics,in biology, and in the study of our physical world, as well as ingeometry and other mathematical fields such as the theory ofcomputation and cryptography. The investigators' goal in thisproject is to build an easy-to-use computational tool that makesit possible for a user to carry out experiments and solveproblems about the most ubiquitous of the mathematical objectsinvolved here, the so-called finitely presented groups. These aregroups that can be described completely by a finite set of rules;examples include the group of rigid geometric motions of anobject and the groups of quantum mechanics. The computationaltool, Magnus, has a simple but limited interface. Nevertheless,Magnus has already been used to explore and solve a number ofdifficult problems in group theory. The present project focuseson algorithmic developments and techniques to allow users to addtheir own routines to Magnus, extending the scope of the tool.The project also provides Internet-accessible computation andinformation resources in a collaboratory setting, fosteringInternet-based interchanges and collaborations among researchers.Finally, the project helps train students in both computation andmathematics.