From 1963 to 1966 I studied Mathematics at University College, London – and I have a B.Sc. (Special) in that subject. The more I have learned about maths the more I know how much I don’t know. It is a truly infinite field.
My specialist topics were Mathematical Logic and Astrophysics, though the only (mathematical) paper I ever wrote was on Topology. My topological paper produced the interesting result that “nobody knows the answer” – of course, the answer may have been found since then, but (to be honest) I haven’t looked to see whether there has been any recent research in that particular sub-field.
Starting from what I was told about astrophysics I was disturbed that there were – no, there are – two successful branches of mathematical physics which have not yet been hammered together. We do not yet know the TOE (Theory Of Everything), which would allow all of Relativity, and Quantum physics and the analytic unification of the four fundamental forces. Gravity, as they say, is a bitch.
Part of the current (2017) analysis suggests that there is a very (very!) large majority of the physical universe which we just cannot see. I do not mean “it’s too far away” but matter and energy which is definitely in our field of view – but just invisible. I am now looking at some of the hypotheses which have given rise to this state – my research into that is described more in the section on Heretical Science. I do know that I have had to blow the dust off my knowledge of integral calculus to be able to read the papers giving the background theories. A lot of dust collects in half a century!
And my topological paper? It was an attempt to discover how many doodles you can draw. Well, that’s clearly an infinite number, so to make a more sensible question I asked “what is the number of topologically distinct plane closed curves, consisting of a single line, that crosses itself n times?” and trying to get a formula related to n. My co-researcher (Pete Nalder) and I used many ways of partitioning the sets of curves, n by n (so to speak), but we found that the number grows faster than (n+1)! – which gets very large very quickly. We were able to examine all the cases up to n=4, and most (we could not be certain it was all) the cases with n=5. n=6 was a silly thing to look at.
If P(n) is the number of curves with n crossings then we found that P(0) = 0, P(1) = 2, P(2) = 5, P(3) >12, and so on. We found that P(4)>120 and that P(5)>>720. The “P(n)≳(n+1)!” rapidly becomes a useless approximation.
If any one out there has a better answer, do let me know!