Computational Invariant Theory: A survey and some new developments
Abstract: The 19th century Classical Invariant Theory always had a strong computational aspect. Even Hilbert's famous papers "Ueber die vollen Invariantensysteme" from 1893 was written in order to give a constructive approach to the finiteness theorem. This aspect was lost, as was the whole subject until Mumford's "Geometric Invariant Theory" which was the starting point of a completely new development.
But the computational aspects remained in the background. Only recently, some interesting new results in this direction were discovered. We slowly start to understand what was classically called "the symbolic method". But even now, with all our knowledge in representation theory and using our heavy computational tools, we haven't been able to push the computations much further than the classics. However, we got a much clearer insight into the structure and discovered some nice features which are clearly beyond the classical methods.