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?

We talk to delegates of the International Conference of Women Mathematicians, taking place in Hyderabad, India, about the challenges faced by female mathematicians around the world.

Over 3000 mathematicians have gathered to sounds of the tabla and bansuri playing traditional Indian music in the Hyderabad International Convention Centre, and they are all baiting their breath. They are all waiting for the announcement of the Fields medals, the highest honour in mathematics, and the prestigious Gauss, Nevanlinna and Chern Prizes. The hall is a colourful scene filled with mathematicians from around the world, including many local participants in beautiful saris and suits. You can hear a pin drop as the entire hall stands for the entrance of Pratibha Patil, the President of India, who will award the medals to the prize winners...

Researchers have unveiled the first prototypes of robots that can develop emotions and express them too. If you treat these robots well, they'll form an attachment to you, looking for hugs when they feel sad and responding to reassuring strokes when they are distressed. But how do you get emotions into machines that only understand the language of maths?

Two computer geeks claim to have calculated the number pi to 5 trillion digits — on a single desktop and in record time. That's 2.3 trillion digits more than the previous world record held by the Frenchman Fabrice Bellard.

It's not just evil villians who can blow smoke rings, it seems peat moss has been doing it for millennia.

How do you judge the risks and benefits of new medical treatments, or of lifestyle choices? With a finite health care budget, how do you decide which treatments should be made freely available on the NHS? Historically, decisions like these have been made on the basis of doctors' individual experiences with how these treatments perform, but over recent decades the approach to answering these questions has become increasingly rational.

Infectious diseases hardly ever disappear from the headlines — swine flu is only the last in a long list containing SARS, bird flu, HIV, and childhood diseases like mumps, measles and rubella. If it's not the disease itself that hits the news, then it's the vaccines with their potential side effects. It can be hard to tell the difference between scare mongering and responsible reporting, because media coverage rarely provides a look behind the scenes. So how do scientists reach the conclusions they do?

One of the greatest advances in the biomedical sciences has been the unravelling of our genetic code. This new understanding sheds light on what makes organisms function and how they are related to each other, helps to combat diseases, and to convict criminals. But it also poses great mathematical challenges: the genetic revolution is an information explosion which can only be tamed using mathematical methods.

Two lines in a plane always intersect in a single point ... unless the lines are parallel. This annoying exception is constantly inserting itself into otherwise simple mathematical statements. Burkard Polster and Marty Ross explain how to get around the problem.