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? 
The natural numbers, 1, 2, 3, 4, ..., are nice. So what could be nicer than discovering interesting patterns within them? 
This year has seen a flurry of results as mathematicians hunt down the elusive proof of the twin prime conjecture. Will they get their wish for Christmas this year? 
Number theory is famous for problems that everyone can understand and that are easy to express, but that are fiendishly difficult to prove. Here are some of our favourites. 
Agreeing to pay £50,000 for something worth £2 wouldn't win you any haggling competitions. In mathematics, however, a similar result can bring you international acclaim. This is the case with recent progress towards the famous twin prime conjecture. 
An "electric atomosphere" is not what you expect at a maths lecture. But it is what prevailed when Andrew Wiles announced his proof of a 350yearoldold problem, Fermat's last theorem, exactly 20 years ago. 
This year's Abel Prize has been awarded to the Belgian mathematician Pierre Deligne for "seminal contributions to algebraic geometry and for their transformative impact on number theory, representation theory, and related fields". 
This year's Abel Prize goes to Endre Szemerédi for his
"fundamental contributions to discrete mathematics and theoretical
computer science."

The number 1 can be written as a sum of unit fractions, that is fractions with 1 in the numerator. But how long can we make such a sum? 