In this talk I shall illustrate, by means of two examples, an on-going program at Utah State University to create Maple software and Java applets for implementing the solution to various classification problems in mathematics.
The first classification problem to be discussed is Petrov's remarkable classification of all 4 dimensional local group actions which admit an Lorentz invariant metric. (This is different from Petrov's classification of spacetimes according to the algebraic properties of the Weyl tensor.) Petrov's classification contains over 100 distinct local group ations and associated invariant metrics and thus it is a non-trivial task to identify a given local group action or a given invariant metric in Petrov's list. I shall demonstrate a web-based Java applet which allows one to identify any metric with symmetry in Petrov's list and also to search for group actions or invariant metrics in Petrov's list with prescribed properties. The results of some preliminary efforts to automate the verification of Petrov's classification will be shown.
The second classification problem will focus on the classification of low dimensional Lie algebras. There are many such tables of Lie algebras in the literature but once again the problem remains of explicitly identifying a given Lie algebra in these tables, a task which is further complicated by the existence of families of Lie algebras containing arbitrary parameters.
The programs which support the implementation of these classification problems are part of Vessiot, an integrated suite of Maple packages for computations in differential geometry, Lie algebras and the variational calculus on jet spaces. The talk will conclude with a quick overview and demonstration of these programs.