Many people's impression of mathematics is that it is an ancient edifice built on centuries of research. However, modern quantitative finance, an area of mathematics with such a great impact on all our lives, is just a few decades old. The Isaac Newton Institute quickly recognised its importance and has already run two seminal programmes, in 1995 and 2005, supporting research in the field of mathematical finance.
In the first part of this article we explored Landau's theory of phase transitions in materials such as magnets. We now go on to see how this theory formed the basis of the Higgs mechanism, which postulates the existence of the mysterious Higgs boson and explains how the particles that make up our Universe came to have mass.
Remember Frank Lampard's disallowed goal in the 2010 World Cup match against Germany? The ball hit the crossbar, landed well behind the line but then bounced out again. And it all happened too quickly for the ref to spot it was a goal. How these kind of (non)-goals happen and what can we do about them?
Horses, like all animals, have a number of different gaits. But how can they perform these complicated leg movements without having to stop and think? And why do they switch to a new gait when they want to go faster? Mathematics can shed some light on these questions.
The holy grail for 21st century physics is to produce a unified theory of everything that can describe the world at every level, from the tiniest particles to the largest galaxies. Currently the strongest contender for such a theory is something called M-theory. So what is this supposed mother of all theories all about?
Some things are so familiar to us that they are simply expected, and we may forget to wonder why they should be that way in the first place. Sex ratios are a good example of this: the number of men and women in the world is roughly equal, but why should this be the case? A simple mathematical argument provides an answer.