Articles
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. 
Quantum mechanics and general relativity are incompatible — and this has led to a decadeslong search for a theory of quantum gravity that could combine the two. But the particle physicist Richard Woodard thinks that the mismatch between the two could be nothing more than an illusion, created by the complicated maths techniques used in attempts to unite them. 
Infinity is a pain. Its paradoxes easily ensnare the unsuspecting
reasoner. So over the centuries,
mathematicians have carefully constructed
bulwarks against its predations.
But now cosmologists have developed
theories that put them squarely outside
the mathematicians' "green zone" of
safety.

John Barrow gives us an overview, from Aristotle's ideas to Cantor's neverending 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. 