number theory
Eron Lindenstrauss got the Fields Medal for developing tools in the area of dynamical systems and using them to crack hard problems in the seemingly unrelated area of number theory. 
Results in mathematics come in several flavours — theorems are the big important results, conjectures will be important results one day when they are proved, and lemmas are small results that are just stepping stones on the way to the big stuff. Right? Then why has the Fields medal just been awarded to Ngô Bào Châu for his proof of a lemma?

On March 14 2010 a mathematician and a magician teamed up to perform what they believed to be the world's largest live magic trick. The trick involved a thousand volunteers from the around the world who, using free choice, each came up with a number that was only known to themselves. And although the volunteer might be on the other side of the globe, the mathematician and the magician were able to read their mind and tell them which number they had chosen. 

The primes are the building blocks of our number system, but there's no general formula that will give you all of them. If you want them, you have to hunt them down one by one. Abigail Kirk investigates a method that does just that.

Neil Pieprzak tells the fascinating story of Andrew Wiles who, with intense devotion and in secret, proved a deceptively simplelooking conjecture that had defeated mathematicians for almost 400 years.

6174 is a very mysterious number. Yutaka Nishiyama explains why, and how beautiful mathematical oddities can inspire us to discover new mathematics.
