Hilbert’s fourteenth problem is concerned with whether invariant rings under algebraic group actions are finitely generated. A number of examples have been constructed since the mid-20th century which demonstrate that this is not always the case. However such examples by their nature are difficult to construct, and we know little about their underlying structure. This talk aims to provide an introduction to the topic of Hilbert’s fourteenth problem, as well as the finite generation ideal - a key tool used to further understand these counterexamples. We focus particularly on the example constructed by Daigle and Freudenberg at the turn of the 21st century, and describe the work undertaken to compute the finite generation ideal of this example. 

In this talk, we will make an explicit link between self-dual Yang-Mills instantons on the Taub-NUT space, and G2-instantons on the BGGG space, by displaying the latter space as a fibration by the former. In doing so, we will discuss analysis on non-compact manifolds, circle symmetries, and a new method of constructing solutions to quadratically singular ODE systems. This talk is based on joint work with Matt Turner: https://arxiv.org/pdf/2409.03886.