Why does this blog exist? I am required to maintain a blog of my research activities as mandated by my contract with POEMA. Secondly this blog will serve as record for evaluating my progress (both for me and my superiors) as well as a platform to showcase my interests (to possible collaborators). Thirdly, it serve as a source of information to anyone interested in me or the topics I study. I must warn the reader that I am a novice to the practice of blogging. Hence, constructive criticism is welcomed. What is the content of this blog? In this blog I will record my mathematical endeavours, indexed by the week in which they take place. The mathematical endeavours are directed towards the topic of : Approximation hierarchies for (non-)commutative polynomial optimisation. This topic falls under general polynomial optimisation techniques and is know to use tools from (but not limited to): real algebra moment theory operator theory semidefinite optimisation. In addition to f