The principal investigator and his colleagues continue developing the software package Magnus, which has been designed to carry out computations in and experiments with, in the main, finitely presented groups. This software has been set up so that it can be used remotely, via the internet. This means that researchers world-wide can use the software by logging on to the Magnus server. The project uses Internet-based facilities to maintain a list of outstanding open problems and to set up several discussion groups focusing on related mathematical areas. These support collaboratory interactions among users, including researchers in Europe, Australia, New Zealand, Hong Kong, and China. Software development includes the introduction of genetic algorithms as well as measures to allow users to add their own routines to Magnus, extending the scope of the tool. The investigators also use Magnus to study problems in pure group theory and in commutative algebra. The project includes the training of students. In mathematics, group theory aims to capture and study the essence of symmetry, particularly in patterns. Such patterns occur frequently in nature, for instance as crystals. Its connection with symmetry makes group theory important in physics, in biology, and in the study of our physical world, as well as in geometry and other mathematical fields such as the theory of computation and cryptography. The investigators' goal in this project is to build an easy-to-use computational tool that makes it possible for a user to carry out experiments and solve problems about the most ubiquitous of the mathematical objects involved here, the so-called finitely presented groups. These are groups that can be described completely by a finite set of rules; examples include the group of rigid geometric motions of an object and the groups of quantum mechanics. The computational tool, Magnus, has a simple but limited interface. Nevertheless, Magnus has already been used to explore and solve a number of difficult problems in group theory. The present project focuses on algorithmic developments and techniques to allow users to add their own routines to Magnus, extending the scope of the tool. The project also provides Internet-accessible computation and information resources in a collaboratory setting, fostering Internet-based interchanges and collaborations among researchers. Finally, the project helps train students in both computation and mathematics.
