My Tryst with the Monstrous Moonshine
I have been reading the book, "The Prime Number Conspiracy" by the folks at Quanta. I would highly recommend this to anyone who has even the slightest of interest in abstract mathematics. We journey through the vast and varied landscapes of mathematics ranging from algebraic geometry to complex analysis and much much more.
To give an idea of one of the topics that I found particularly enrapturing, I'll try to distill and present the so called monstrous moonshine concept. It is an enigmatic connection between group theory and number theory.
Let us get a few of the pre-requisities out of the way. To get started, we need to know what a group is, and we need to know what modular functions are. To know what a group is, we first need to know about symmetries.
The symmetries of any given shape can be viewed through an arithmetic lens. For example, starting from a square ABCD, if we rotate it 90 degrees, we get DABC and then if we flip it horizontally we get CBAD. However, this is the same as flipping the original square ABCD across a diagonal. therefore, we see that a 90-degree rotation + horizontal flip = diagonal flip.
During the 19th century, mathematicians realized that they could distill this type of arithmetic into an algebraic entity called a group. The same abstract group can represent the symmetries of many different shapes, giving mathematicians a tidy way to understand the commonalities in different shapes.
Moving on to modular functions, we see that these are a special class of functions related to elliptic curves whose graphs have repeating patterns similar to M. C. Escher’s tessellation of a disk with angels and devils (shown on the left) which shrink ever smaller as they approach the outer boundary. These functions are the heroes of number theory as they satisfy or explain many beautiful and surprising numerical identities (about partitions and sums of square among others). They also played a crucial role in Andrew Wiles’ 1994 proof of Fermat’s Last Theorem.
We will work with a special group called the monster group. It has a remarkable number of 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 symmetries in total. This number is mind-bogglingly large (more that there are atoms in the sun!). This is the largest number of symmetries that any finite sporadic simple group can have. That is why it is called the Monster. Moreover, this symmetrical behemoth of an object exhibits special characteristics in spaces of certain dimensions, of which1 and 196,883 are the first two. Interestingly, if we consider the fourier expansion of a modular function called the j-function, then its first coefficient comes out to be 196,884 which is equal to 1 + 196,883. The next special space in which this monster lurks is of the dimension 21,296,876 while the second coefficient of the j-function is 21,493,760 which equals 1 + 196,883 + 21,296,876. The next higher-dimensional space for the monster has 842,609,326 dimensions while the third coefficient in the expansion is 864,299,970 which equals 1 + 196,883 + 21,296,876 + 842,609,326. The pattern is glaringly clear. The sum of the dimensions of the monster group somehow get reflected in the coefficients of the j-function.
The above pattern might seem like a trite co-incidence or a mathematical gimmick at first glance. However, this connection is as strange as strange comes. In way of analogy, it is as if archeologists excavating the Harappan site of the Indus Valley Civilization were to chance upon stone carvings which happen to resemble the scripture of the Mayan Empire. They would have no choice but to infer some contact between these two civilizations, despite the fact that they existed in the opposite ends of Earth, that too around 2000 BC. John Conway’s excavations had revealed a similar link between two mathematical sites: the modular function from number theory and this Monstrous symmetry from group theory. The two things didn’t appear to have anything to do with each other (Hence the name 'moonshine'). Yet the secret of which dimensional space this Monstrous creature lived in seemed to be coded into the modular function.
A proof of this link was finally given by Richard Borcherds in 1992. Interestingly, the proof works by exploiting a particular string theory model. String theory is based on the idea that the universe has tiny hidden dimensions, too small to measure, in which 1-dimensional strings vibrate to produce the physical effects we experience at the macroscopic scale. In the proof, the j-function and the monster group both play crucial roles- The coefficients of the j-function count the ways strings can oscillate at each energy level while the monster group captures the model’s symmetry at those energy levels.