Baumslag 9626231 The investigator and his colleagues develop software to study finitely presented groups. Many problems that arise in diverse mathematical disciplines can be distilled into related problems about finitely presented groups. One of the major developments of the decade beginning in the mid 1950s has been the proof that almost all questions about such finitely presented groups are algorithmically undecidable. Nevertheless, the very importance of these groups makes it essential to try to obtain at least partial information about them. In particular, although problems about finitely presented groups are unsolvable in general, it may be possible in a number of specific instances to solve such problems. In order to gain some insight into these problems, it is often necessary to try to carry out some experiments and computations with finitely presented groups. Such experiments and computations are very time-consuming and usually impossible to carry out by hand. This project aims to make such experiments and computations possible by using very fast computers. enabling one to gather enough information together about many important problems to facilitate their solution. The project creates and implements a package of group-theoretical algorithms and procedures, which can be run by means of a graphical user interface and which is easily accessible by a widevariety of users. Groups are mathematical entities that encapsulate, among other things, the intrinsic meaning of symmetry. This explains,in part, why they turn out to be a very important tool in physics, chemistry, biology and other mathematical disciplines. The understanding of these "groups" provide insight into these disciplines, as well as a means for studying them. For example, groups play an essential role in quantum mechanics and particle physics and they also provide a means for studying the space people live in. Many of these groups have seemingly simple descriptions, even though t hey themselves can be very complicated. This complexity helps to explain why, for example, certain important parts of chemistry and biology are so difficult. They also give rise to new problems and provide for solutions to problems in theoretical computer science, cryptography and other fields. Their ubiquitousness and usefulness make the creation of a software package that will lead to a deeper understanding of many of them an extremely important enterprise.
