Symmetry: Mark Ronan's website

The use of symmetry in mathematics made a huge step forward when Joseph Louis Lagrange introduced it in his work on algebraic equations in 1770. Other mathematicians later took his ideas further, in particular …variste Galois, who died in 1832 at the age of twenty. He introduced the concept of a group of symmetries, now known simply as a 'group' in modern mathematics. Galois used groups in creating a new theory of algebraic equations, and they have since led to important advances in many areas of mathematics, sometimes in cases where there is no obvious symmetry.

In Symmetry and the Monster I describe the quest to find the basic building blocks for all finite groups (i.e. groups of symmetries). These are the so-called 'simple groups', which are not simple in the usual sense, but can be very complex and interesting. Most of them fall into one of several different families, and are relatively well understood. But there are 26 exceptions — called sporadic groups — that do not fit into the general pattern. Being outsiders makes them particularly interesting.

Among these exceptions the largest is called the Monster, and contains all but six of the others. Investigating the Monster led to some very surprising connections with number theory and with the mathematical physics of string theory. This started with a strange coincidence between two numbers: 196,883 and 196,884, the first appearing in a natural way from the Monster, and the second from an important sequence in number theory.

This surprising fact was originally dubbed moonshine, and it led to further coincidences between the Monster and other branches of mathematics, which in some cases have been explained in connection with string theory. However there are coincidences as yet unexplained, one of which concerns the number 163.