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? 
In this excellent talk the mathematician Vicky Neale gives a fascinating and easytofollow introduction to the prime numbers. 

Marcus du Sautoy talks about football, cryptography, and numbers. 
There's been progress on one of the biggest open problems in maths: the twin prime conjecture. 
Here's an easy game that leads you straight to an unsolved question in maths. 
The Fields medallist reveals the beauty of numbers, how he drew inspiration from Sanskrit maths and a Rubik's cube, and how his maths fits with his tabla playing. 
We enjoyed Manjul Bhargava's Fields medal lecture so much we wanted to share it with you! 
Manjul Bhargava is being honoured as a number theorist of "extraordinary creativity," with "a taste for simple problems of timeless beauty." 