A brief Introduction to Invariant Theory

Invariant Theory is a rich branch of algebra that originated from a work by Cayley in 1845. It has lead to outstanding results such as Hilbert’s celebrated papers in the 1890’s or Mumford’s development of Geometric Invariant Theory (GIT) for reductive groups. In recent years there has been great interest in computational aspects of Invariant Theory and its applications, and at the theoretical frontier non-reductive GIT is being developed.

The goal of this talk is, first, to advertise Invariant Theory and, second, to explain why reductive GIT „works well“ - thereby giving some insight which challenges non-reductive GIT needed to overcome. For this, we treat the following topics: reductive groups, stability notions and their applications, the Hilbert-Mumford criterion and the Kempf-Ness theorem. At the end, we catch a glimpse of non-reductive GIT.