p-adic preliminaries

January 29, 2010

Since my last post I’ve started attending a course onp-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 ofp-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}, whereCis the ring of Cauchy sequences in\mathbb{Q}andMis the maximal ideal ofCconsisting 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 example0.9999...=1.000..., because the sequences:0,0.9, 0.99, 0.999,...and1, 1.0, 1.00 ,1.000,...both converge to1.

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Random Matrices and the Riemann Hypothesis

January 15, 2010

I made it to my second post!  This is the greatest achievement of my blogging life so far.

Having never even heard of random matrix theory before last week, I have recently been hearing talk of it bandied about all over the place.  When I asked my supervisor why this might be, he replied that they were a “hot topic”, and went on to explain the recent connections that have been made between the distribution of zeros of the Riemann zeta function, and that of eigenvalues of large random matrices.  So I’m going to write a bit about it here.

Random matrices are intuitively exactly what you might first suspect they are: matrices with “random” entries.  Of course, it is technically meaningless to say that an element of a matrix is random (hence the inverted commas), but what we can do is to use probability theory to rigorously define an analogous concept.  Without going into in too much depth, we can think of a random matrix as being a matrix with elements that are random numbers from some probability distribution (that is, random variables).  So an n\times n random matrix is actually a collection of all the possible n\times n matrices, along with probability density functions telling us how likely each of these is to occur.

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…hopefully not another first and final blog post.

January 12, 2010

Hello everyone!  Or no-one, as the case more accurately may be.   This is my first post, and hopefully not my last…although for now I will keep the existence of this blog to myself in case it does turn out to be a passing fad.  So why am I doing it at all?

In my daily struggle with mathematics, I have become frustrated by the constant seep of information from my brain.  I am pretty sure that this is a natural seepage, possibly exacerbated by abuse and neglect of my mental faculties.  It’s possible that my brain is just not designed to hold large quantities of mathematics…I don’t know, and probably never will.   What I do know is that I am spending far too much of my time relearning the same things over and over again.  Whilst mulling on this yesterday, I realised that I have never forgotten something that I have taught to someone else in the past.  And this made me suspect that the things that I keep forgetting are things that I don’t fully understand.  In order to teach something properly, one needs to have a complete understanding of it; in fact I think that one of the best ways to determine how well a person understands something is to observe how effective they are at teaching it.  And one of the best ways to learn something properly is to study it with the aim of communicating it to others.

Hence this blog.  I intend to write short essays on topics that come up in my mathematical life, whether it be through my research, the seminars I attend, or the subjects I teach.  It could be simply that I have a conversation with someone and realise that I don’t understand something as well as I’d like.  Or it might be that on one of my far-too-frequent rambling and distracting excursions through wikipedia and other mathematical blogs I happen on something which catches my interest, and which I want to know more about.

As to the level of my posting, I envisage that it will be readable by fresh undergraduates, and highly interested lay people.  I would like to develop my expository skills, as I make a living from teaching (and if this whole mathematician thing doesn’t work out then my next choice of occupation would probably be “science-writer”).   I would also like to be able to appeal to a large audience.   But at the same time I will be mainly writing about things that I will be learning myself, and these things unfortunately seem to get more difficult with each day.  So I will tread a fine line between dumbing down and incomprehensibility.

Perhaps one day someone will come across something I’ve written, and decide that mathematics isn’t all bad, and is even kind of interesting: that would be enough to make it all worthwhile.

Adam.

PS – If any mathematicians do happen upon this, then I would be grateful for even the most scathing and fastidious corrections in your comments: there is no point trying to be a good expositor of nonsense.