## group 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?

Mathematicians are busy tidying up the largest proof in history.

An impossible equation, two tragic heroes and the mathematical study of symmetry

In 1982 Dan Shechtman discovered a crystal that would revolutionise chemistry. He has just been awarded the 2011 Nobel Prize in Chemistry for his discovery — but has the Nobel committee missed out a chance to honour a mathematician for his role in this revolution as well?

Topologists famously think that a doughnut is the same as a coffee cup because one can be deformed into the other without tearing or cutting. In other words, topology doesn't care about exact measurements of quantities like lengths, angles and areas. Instead, it looks only at the overall shape of an object, considering two objects to be the same as long as you can morph one into the other without breaking it. But how do you work with such a slippery concept? One useful tool is what's called the fundamental group of a shape.

The Rolf Schock Prize in Mathematics 2011 has this week been awarded to Michael Aschbacher "for his

fundamental contributions to one of the largest mathematical projects ever, the classification of

finite simple groups".

*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?

**Burkard Polster**and

**Marty Ross**explain why, and explore the maths behind bell ringing.