When the mathematician AK Erlang first used probability theory to model telephone networks in the early twentieth century he could hardly have imagined that the science he founded would one day help solve a most pressing global
problem: how to wean ourselves off fossil fuels and switch to renewable energy sources.

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.

It's official: the notorious Higgs boson has been discovered at the Large Hadron Collider at CERN. The Higgs is a subatomic particle whose existence was predicted by theoretical physics. Also termed the god particle, the Higgs boson is said to have given other particles their mass. But how did it do that? In this two-part article we explore the so-called Higgs mechanism, starting with the humble bar magnet and ending with a dramatic transformation of the early Universe.

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.

John Barrow gives us an overview, from Aristotle's ideas to Cantor's never-ending tower of mathematical infinities, and from shock waves to black holes.

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?