## Mathematics in Music

February 12, 2010

Leaving aside the topic of$p$-adic numbers (I feel as though I should learn more about it myself before I make any mistakes), I’m going to get back to a subject I hinted at a couple of posts back: the role of mathematics in music.  When I tell people that I studied music before switching to mathematics, they often say something along the lines of that the subjects are very similar/interconnected/both use “the left side of the brain”.   This isn’t really quite as true as seems to be commonly thought: you can certainly find a lot of mathematical patterns and structure in music; but so can you in any art, and indeed – arguably – in anything if you look hard enough!  And while I am personally averse to “side of the brain” arguments, if we are stooping to that level then I would argue that there is a creative right-side element integral to the creation and appreciation of music which is largely absent from mathematics.

However, it is true that many mathematicians are also involved in music – especially classical music – to some degree .  And it is true that the mathematical and logical structure of music is much more apparent and easier to appreciate than with other art-forms.   One of the best ways to see this is to look at tuning systems.

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$1.$

## 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.