He was an extremely versatile mathematician, which is evident in the multitude of discoveries he has to his name. He was a unique figure in the scientific world: in an interview with the YouTube channel Numberphile, he explained how he promised himself after his first major discovery never again to worry about the scientific or societal merit of his research (in his words, whether it was “serious” or not), and to instead focus on whatever he himself found interesting. In this mathematical In Memoriam I would like to remember him through this first discovery which lifted the weight of academic pressure off his shoulders.

Conway’s Group Theory

The discovery which, in his own words, “launched him into international prominence” took place in the field of group theory, specifically so-called “simple groups”. To be able to discuss this, and to fully understand the importance of Conway’s contributions, we must first explain the idea behind a mathematical group.

Groups are related to the idea of symmetry, since the rules which groups must abide by also define the essential aspects of symmetry. They can therefore be seen as sets of symmetries of a geometrical object. As such, there is a group of symmetries for the equilateral triangle, consisting of 6 elements: 2 rotational, 3 axial, and 1 “do nothing” symmetry necessary to formally satisfy the definition of a group. Such sets of symmetries naturally exist for many shapes, including higher dimensional shapes. The more elements are in a group, the more “symmetrical” the shape corresponding to said group.

Another interesting trait of groups is that they themselves can contain other groups. For instance, the group for the equilateral triangle contains subgroups of just the rotational or axial symmetries. An elementary concept in group theory is that of “simple” groups, which cannot be split into any subgroup other than the group itself, or “trivial” groups, which consist of a single trivial element. This is comparable to prime numbers, which are divisible only by themselves (similar to a group being divided by itself) and 1 (similar to the trivial group). The fascinating thing about simple groups is that they can combine to create any possible finite group, like atoms in a molecule. The comparison between simple groups and atoms is surprisingly fitting, because mathematicians have spent 200 years working on a proof to categorize all simple groups, similarly to the periodic table of elements. This proof was formidable: it required tens of thousands of pages of work from the combined efforts of hundreds of mathematicians, one of whom was John Conway. One of his contributions was the discovery of Conway groups in 1968, to which he owes his fame among mathematicians. These are four groups, the largest of which contains 8,315,553,613,086,720,000 elements, corresponding to all symmetries of a shape in 24 dimensions. He also constructed the first of the groups now known as the “sporadic groups”. The classification of simple groups is seen as one of the most impressive results in the history of mathematics, and Conway’s contributions were enormous.

The Monster and String Theory

One of the most surprising results of group theory is the proof of the existence of a “largest possible group”. This group contains all the symmetries of what John Conway called “the Monster”, a shape which lives in 196,883 dimensions. The number of elements in the “Monster group”, or the Monster’s number of symmetries, is a whopping 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 (~8∙108). The fact that a largest possible Monster group indeed exists, means that there is no shape with more symmetries than the Monster. This gives a certain significance to the two numbers 8∙108 and 196,883 due to the implication that there is a limit to symmetry in nature.

Conway did not discover this Monster group, but he did develop “moonshine theory” with Simon P. Norton (the name shows how Conway looked at mathematics differently from the conventional scientist). This theory connects the existence of the Monster group with the “j function”, a so-called “modular form”, which is a function that lives in the world of complex numbers. John McKay, a friend of Conway’s, was the first to find a connection between the Monster group and the j function. He wrote out the Fourier expansion of the j function,

J(τ) = 1/q + 196884q + 21493760q2 + 864299970q3 + ...,

and noticed that the second coefficient (196,884) was close to the dimension in which the Monster lived (196,883). McKay showed that the coefficients form a series that can be directly linked to the Monster group, beginning with 1 and 1 + 196,883. The other coefficients can be used to form other representations of the Monster group. These observations were noticed by Conway and Norton, who developed a formal mathematical conjecture in 1979 that captured this connection. They called this “monstrous moonshine”, because the link between modular forms and the Monster group was so astonishing to them.

Still, the conjecture was proven in 1998 by a former student of Conway’s named Richard Borcherds, for which he won the Fields medal, the equivalent of the Nobel prize in mathematics. The proof was just as fascinating as the conjecture itself, for its usage of the mathematics behind string theory, a formalism in physics which supposes that the existence of particles such as protons and electrons are the result of oscillations of miniscule objects called strings. Borcherds’s proof showed that certain 26-dimensional strings have the Monster group as their symmetry group. It is an unimaginable and unpredictable result, that connected much of mathematics and even physics, and on which John Conway undeniably left his mark.

With his discovery of simple groups bearing his name, Conway was “done”, as it were. He had contributed to the study of mathematics and had made a name for himself, and everything he would do thereafter would be out of his own curiosity. This approach led to some work which would incidentally turn out to be groundbreaking, in the fields of geometry, number theory, algebra and analysis, among others. Outside of pure mathematics it also gave way to research in Conway’s other fields of interest. He invented countless mathematical games, wrote popular science books, through which he would also develop new insights which would be picked up by the scientific community. Dr. Simon Kochen, a colleague of Conway’s, described him in an interview with the New York Times as a “magical genius”, a statement I agree with wholeheartedly.