Which comes first?

For an unknown reason, I haven't been sleeping too well of late, but despite this — or perhaps in spite of it — my work has been progressing well. Tiredness is a state that provides your brain with more capability to concentrate for shorter periods of time and it is in those periods that I have been at my most fruitful, moving to the point where I am equipped (mathematically speaking) to lunge into the major challenge of my PhD. It may turn out that this preparation will prove to be half of the battle, but it will only be after the first half that I will know that for sure, so we'll reserve judgement on that one.

My supervisor is currently in the process of writing a paper introducing the area we are working on (the fine details of which can be found in the research section. He is writing it such that, by the time it is accepted for publication and has been published, we should have the answers to the open questions he is posing in it. Thus, the place of our work in the field of symmetry will be given a firm basis and, from a PhD point of view, I will (hopefully) have very definite answers to very definite questions.

The main question — as it stands at the moment — is a bit chicken and egg: we would like to solve an equation using a condition we have come up with, but are only able to check the condition using equations we know the solution of. Once we've done this enough times, though, then patterns should start emerging and we'll be able to put two and two together, thus placing the egg firmly back in the chicken.