Computational Group Theory

计算群论

• 批准号: 9973233
• 负责人: Gilbert Baumslag
• 金额: $ 45万
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-04-15 至 2003-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9973233&HistoricalAwards=false
• 关键词: Computational; Group; Theory

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.

Baumslag9973233 首席研究员和他的同事们继续开发软件包Magnus，该软件包被设计用于在主要的研究小组中进行计算和实验。 这个软件已经设置好了，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到Magnus服务器来使用该软件。 该项目使用基于互联网的设施来保持一个未解决问题的列表，并建立几个讨论组，集中在相关的数学领域。这些支持用户之间的合作实验室互动，包括欧洲，澳大利亚，新西兰，香港和中国的研究人员。 软件开发包括引入遗传算法以及允许用户将自己的例程添加到Magnus的措施，从而扩展了工具的范围。 调查人员还使用马格努斯研究纯群论和交换代数中的问题。 该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性的本质，特别是在模式中。 这种图案在自然界中经常出现，例如晶体。 它与对称性的联系使得群论在物理学、生物学、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）中非常重要。 在这个项目中，研究人员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这里涉及的最普遍的数学对象的问题，即所谓的随机呈现群。 这些群可以完全用一组有限的规则来描述;例子包括物体的刚性几何运动群和量子力学群。 计算工具Magnus有一个简单但有限的界面。 尽管如此，马格努斯已经被用来探索和解决一些困难的问题，在群论。 本项目的重点是算法开发和技术，允许用户将自己的例程添加到Magnus中，扩展该工具的范围。该项目还在合作实验室环境中提供可通过互联网访问的计算和信息资源，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在计算和数学方面的能力。