Entanglement as a concept is simultaneously plainly obvious and quite abstract. Anyone can look at an object or a drawing and have an immediate intuitive understanding of whether it is tangled or not. However, if a person was asked why they claimed a certain object was entangled, they can quite easily struggle to satisfactorily define what it is about the object that makes us feel like it is entangled. What we would like to have is a specific set of criteria that could immediately determine whether or not a given object is entangled and be usable in the large variety of contexts that entanglement is found in.
This was my main interest in the topic of entanglement when my supervisor, Dr Vanessa Robins, proposed the idea to me. How can we rigorously define when a certain object is experiencing the quite broad and amorphous quality of entanglement? How can mathematical tools help in this endeavour to understand entangled structures?
While this was my main source of interest, there are certainly more practical reasons to be interested in the topic of entanglement as it arises quite often in chemistry, biology and materials science where the entanglement of a crystal structure or macromolecule can help determine its energy and stability amongst many other physical and chemical properties. It also presents a challenge for many people working in these fields to see if they can physically make structures that have been shown to be mathematically interesting. A common feature of the readings cited in my literature review is a complex description of a very abstract kind of entanglement followed by the citation of an inorganic chemist who had managed to make this abstract form of entanglement in a laboratory setting.
As it turns out there are many different mathematical tools which have been employed in order to understand entanglement both using universally applicable methods and methods that rely on particular properties of a specific type of entanglement that help to create a deeper understanding of the role of entanglement in a certain context. These approaches come from many different areas within mathematics including, topology, group theory, spatial graph theory, numerical analysis, knot theory and linear algebra, and one of the really cool things about studying this topic was seeing how these approaches worked together to create a more holistic characterisation of the concept of entanglement.
This led me to the nice realisation that understanding the entanglement of physical objects really exposes the inextricable links between different areas of mathematics that are themselves tangled together in complex ways.
Australian National University