a change of area

The last two weeks I’ve discussed fairly simple (to understand) mathematical problems which were both relating to prime numbers. I don’t want to give the impression that number theory is all there is to do in maths so today I thought I’d talk about something completely different…

The brain!

Yes, the brain. Specifically how it transmits information. About this time last year I was finishing off my honours year. One of the coolest things about honours is that everyone does different projects. We were all working in the same room, and discussing our work, but there was pretty much no crossover between research areas. So anyway, one of my friends was studying the neuron firings of the brain.

How do you study that? What is there to do? Well, in the world at the moment we have a lot of data. We record a lot of things without knowing what they mean. The world of mathematical modelling is about finding the mathematical equations that govern the things we have measured.

Mathematical modelling often ties into other subject areas like biology and physics.

Equations are about plugging some facts in and receiving another fact out (that is, describing the relationship between two or more things, if you’re being fussy). To find out your average speed on a trip you plug in the variables of distance and time taken. With the neuron firing of the brain you plug in physical factors like the current in the brain and the action of other neurons.

As I was writing that last sentence I felt like a fraud. I have no idea what those things really mean. That’s the crazy thing about it all. I could probably understand the equations and even work with them (at a stretch!) but I have no idea about the biology behind them.

But this stuff is really cool. Why are we still researching it? Because we don’t have a perfect equation, like we do for average speed, yet. We have approximate models that are improving – that is they are getting closer to the measured data – but none accurately represent what’s going on.

And I think it’s pretty cool that there are people researching the mathematical equations behind how our brain transmits information…

NB: This stuff is too complicated to really get into the details of it but if you’re interested in mathematical modelling then click here, and if you’re particularly interested in modelling the neuron firing of the brain then you probably know more about it than I do but here is a link anyhow

another prime problem

Last weekend I wrote about Goldbach’s conjecture which involved the sums of prime numbers. Keeping with the theme of easy-to-state-number-theory-problems, I decided to mention the Twin Prime Conjecture this week.

We call a prime number a twin prime if you can reach another prime by either adding or subtracting 2. For example: 3 + 2 = 5, 13 – 2 = 11.

The conjecture is that there are an infinite number of twin primes. (We know there are an infinite number of primes.) Computers have been used to calculate twin primes which are very large (5,000+ digits) but it will take more than pure calculation to solve this conjecture…

Research in mathematics?

I am asked, regularly, how there can still be things to research in maths. How can there be new maths? Two is still two, right? What could there possibly be to discover?

So, for those of you who wonder how the career “mathematician” still exists, or what I could possibly spend three years researching, I have decided to create a weekly feature of famous unsolved mathematical problems/current research in mathematics.

The first thing that jumps to my mind is Goldbach’s Conjecture. Goldbach conjectured*¬† in the 1700s that any positive even number (>2) is equal to the sum of two primes. This is easy to see for the first few examples: 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5 and so on. Using the amazing powers of computers we know that this conjecture is true up to ridiculously high numbers. But no one has ever proved that it must always be true.

This is such a great problem because it can be understood by anyone who knows what a prime is but it’s obviously incredibly difficult to solve.

Personal note: I was set this question as ‘homework’ by the head of the Adelaide Uni Maths Society a few years ago. He knew that it was a famous unsolved conjecture but I did not. I spent several hours trying to solve it but was sadly unsuccessful. My dad laughed when I told him what I was working on. He knew what it was too. Now you all know and can set the problem for your mathematically inclined friends.

*Actually he conjectured some very similar things, one of which turns out to be equivalent. Go look it up on wikipedia.