# Who you gonna call? Mathematicians!

By Oliver R Southwick, on 19 March 2015

Guest blog by Oliver Southwick, PhD student, Department of Mathematics, UCL

Losing everything on your laptop may be a nightmare scenario for many of us, but what if this happened world wide? If, by some bizarre thought-experiment logic, we lost all the pieces of our scientific knowledge, painstakingly collected over hundreds of years? Everything from the sequence of the human genome to the orbits of the planets was gone, forgotten.

Well, we’d have to work it all out again, wouldn’t we? It would surely be easier the second time round. But how would you go about it?

Say you were put in charge of working out how the oceans work, how the water circulates around our planet. This is a seriously important job. We need to understand the ocean circulation to navigate ships, to predict the weather and climate and to understand the rich biology of the sea. So how would you start doing this?

You’d probably want to start measuring things. You might send some ships out and record current speeds, temperature, saltiness, wind, nutrient levels and as many other things as you thought might be important. If the guys working on space travel had really got their act together you could start using satellites to measure the height and temperature of the sea surface. You’d collect a vast dataset and start looking for patterns and explanations.

But one thing you might not think to do is to buy some pens and paper and call up the local mathematicians. Given the cost of this compared to the fleet of research ships, why not? What mathematicians bring to the table is the ability to simplify complex situations. By writing ideas as equations, we make them precise. We can then manipulate these equations to examine the ideas and deduce their consequences.

If you called the mathematicians in to look at the ocean they’d likely start from the physical facts.

One of theses facts is that if you apply a force to an object, it experiences an acceleration equal to the force divided by its own mass. This is Newton’s second law of motion. In the oceans there are many forces such as pressure, friction and the Coriolis force due to the Earth’s rotation.

By applying Newton’s second law to the forces in the oceans, you can write down equations ensuring that at every single point in the ocean and at every single point in time the forces balance out with the acceleration. Doing this gives a version of the famous Navier-Stokes equations These equations are at the root of understanding everything from how planes fly to the slow creep of glaciers. However they are much too complicated to be solved without first making some assumptions.

The successes of mathematical modeling are of course not confined to oceanography. A famous tale surrounds the Trinity Test, the first detonation of an atomic bomb. In July 1945 the scientists of the Manhattan Project detonated a plutonium bomb equivalent to 20,000 tonnes of TNT in the New Mexico desert.

Two years later, photographs of the explosion were declassified and published in the American media. The British mathematician G. I. Taylor used these pictures and some mathematical analysis, simple enough to fit on the back of an envelope, to calculate the energy released in the explosion with remarkable accuracy.

The method Taylor applied used a simple fact: the quantities on either side of a meaningful equation must be of the same type. A length must equal another length, it cannot equal a speed. Through a careful consideration of the physical system he deduced that the width of the fireball would largely be determined by three key quantities: the time since the initial detonation, the density of the air and the energy of the explosion. Using this and the principle that the two sides of an equation must be of the same type, he found the an equation giving the radius of the fireball at any point in time for a bomb with a certain energy. Working backwards he then used the width of the explosion at a point in time to calculate the energy released. The answer he found was within 10% of the official figure, an amazing feat for just one human armed with a pen and piece of paper.

So, if mathematical modeling can help us understand the world in a way we can’t by just collecting information or data, is it a problem that calling the mathematicians would have been so far down your to-do list? And what can be done to remedy the situation?

The good news is that UCL mathematicians are leading the way – telling the public about what they are doing and showing the fun, but also the importance, of the work that they do. An obvious example is Dr Hannah Fry, lecturer in the mathematics of cities from UCL’s Centre for Advanced Spatial Analysis. Hannah makes maths fun and awesome. Watch her TV programs, youtube videos, talks or read her book and you’ll see neat maths, focused around solving problems which are important, interesting and fun.

A brand new initiative comes in the form of Chalkdust, a new maths magazine with its first issue published on Tuesday 23^{rd} March. What makes this really exciting is that it’s something very new. It was formed spontaneously by a group of young UCL maths researchers not for money or any obligation but because they have ideas they want to communicate. They want to spread the exciting, the cool and the beautiful parts of their subject. I’ve not seen any of the content yet, but judging by the hype, it’s going to be great.

Researchers like Hannah Fry and the Chalkdust team are talking about a side of maths that we don’t normally think of, a subject with the tools to help us simplify and understand complex problems from ocean currents to the blood flow through our bodies. If they’re successful at making their case then maybe next time you’ve got a complex problem to tackle, you’ll know who to call… a mathematician!