## A year’s work, lessons learnt

October 11, 2010

I’m back!  And rather surprisingly, I seem to have gained a lot of readers in my absence.   Having not even logged on to WordPress for a few months, I have returned to see that my google reader subscription rate has doubled, and the number of people visiting the blog has increased by more than at any point since I started writing it.  I’m not really sure what lesson to take from this.  Probably it is just the natural result of a gradual snowballing effect: over time more people click on your site, the google rankings go up, causing more people to click on your site…

Then again it’s  possible that people just prefer it when I don’t write anything!  Well I’m sorry those people, but I intend to start again.  Although possibly even more erratically than before.

Anyway, I will explain the terrible sequence of events which led me to abandon blog shortly, but first, a shameless Rupert Murdoch-style using of one of my products to promote another! (I would do this at the end, but am rather doubtful as to how many people actually make it to the end of my posts).  Having been introduced to Vietnamese coffee by my father-in-law a while ago, I have utterly fallen in love with it, and realised that it is very difficult to find here in the UK.  So I have set up a (very) small business selling it.  The website is here.  Try it!  You won’t be disappointed.

Sorry about that.  Now, this is what happened.  Having struggled with the proof of a knotty mathematical problem for the better part of a year, I was advised by my supervisor to publish what I had.  So I put the paper on the arXiv (an online preprint archive), not really expecting to achieve anything, but generally wanting to share the knowledge out of a spirit of altruism (and self-promotion).  Within a few hours of it appearing, a certain Peter Mueller had read it, and proved the last part of the conjecture!  (So there you go doubters: people do read your preprints).  This was wonderful news; we invited him to be a co-author, and set about writing a final draft. I also wrote a whole long blog post about how great this was, and what it all meant.  But then a couple of days later Prof. Mueller sent me some rather less good news: he had found a  mistake in my work, which completely invalidated the whole thing…

## Quadratic Equations! (or: what do mathematicians actually do?)

April 4, 2010

When I tell people that I study mathematics they tend to have one of two reactions:

1. They make some impressed-sounding noise, or mention that they were terrible at maths at school, and then quickly make it clear that they wish to change the subject.

2. They are genuinely interested, and want me to tell them exactly what it is that I study.

The second reaction is the one that I fear most!  And at this point it is usually me who tries to change the subject.  I have an ongoing competition with myself to increase the length of time which I can spend explaining my research to someone before their eyes glaze over and their body language starts to say “I want to be somewhere else now”.  I am currently up to about 15 seconds.  And the subject I am currently working on (somewhere between graph theory and galois theory) is fairly accessible compared to some of the more exotic branches of mathematics!

The main problem in explaining pure mathematics to a non-mathematician is the level of abstraction involved in the subject.  Most people’s view of mathematics is that it deals with numbers, and it is hard for people to imagine what exactly it is that mathematicians do…add and subtract really big numbers?  Many seem to find it difficult to imagine how it could be that all the mathematics that could be done hasn’t been done already.* People rarely encounter abstract mathematics before university; and for good reason, as the transition from dealing with concrete quantities in a familiar setting, to treating those quantities and that setting as merely one very special case in a vast world of abstraction, can be rather bewildering.

January 29, 2010

Since my last post I’ve started attending a course on$p$-adic numbers.  Initially my only real motivation for doing so was that a closely related concept had come up in my research; I had previously been of the opinion that the study of$p$-adic numbers was something of a niche pursuit that bore little relevance to other areas of mathematics.  However, having attended 2 lectures, I am finding the subject quite fascinating, and pleasing in the way it relates concepts from algebra, number theory and analysis.  So today I’m going to write highly non-rigorously about some of the interesting bits…perhaps I will even do a short series of posts on the subject.

So what are the p-adic numbers?  I think the best way to explain this is to start by talking about something a bit more familiar: the real numbers.  A space is complete if, intuitively, it “has no gaps”; this is a very desirable property from the analyst’s point of view (in fact analysis can only be done in a complete space, as the notion of a limit does not make sense if there are gaps in the space).  The formal definition of a complete space is one in which every Cauchy sequence – that is one in which the gaps between elements eventually get infinitesimally small – converges to a point in the space.  The rational numbers are not complete because, for example, we can construct a sequence$(a_i)$that converges to$\sqrt{2}$by defining:

$a_1=1$

$a_2=1.4$

$a_3=1.41$

$a_4=1.414$

…and so on.  The real numbers$\mathbb{R}$can be obtained by completing the rational numbers$\mathbb{Q}$, that is, by “filling in the gaps”.  The way we do this is to take every Cauchy sequence in$\mathbb{Q}$and let$\mathbb{R}$be the set of points that these sequences converge to (for the more technically-minded,$\mathbb{R}$ is the quotient ring$\frac{C}{M}$, where$C$is the ring of Cauchy sequences in$\mathbb{Q}$and$M$is the maximal ideal of$C$consisting of all sequences converging to zero).  A helpful way to think of this is by envisaging the decimal expansion of every number as being a convergent sequence, in the same way as we saw above for$\sqrt{2}$.  Sequences are considered to be equivalent if they converge to the same point, and so for example$0.9999...=1.000...$, because the sequences:$0,0.9, 0.99, 0.999,...$and$1, 1.0, 1.00 ,1.000,...$both converge to$1.$