The Holy Grail of Maths: Langlands. (specialization vs generalization).

This year’s TOK question for Mathematics is the following:

“How can we reconcile the opposing demands for specialization and generalization in the production of knowledge? Discuss with reference to mathematics and one other area of knowledge”

This is a nice chance to discuss the Langlands program which was recently covered in a really excellent Numberphile video. 

image by Bjorn Oberg for Quanta Magazine

Langlands program

The Langlands program involves exceptionally complicated mathematics but at its heart it is motivated by a belief that there are deep connections between different branches of mathematics such that a solution to a problem in one branch can provide a solution to a problem in a completely different field. 

Bridges between fields of mathematics signal at the potential of an underlying mathematical framework we don’t yet know and finding this has been described as the “holy grail of mathematics” – similar to a unified theory in Physics.

Fermat’s Last Theorem

Probably the most famous problem in number theory (which deals in integer solutions to equations) is Fermat’s last theorem:

This theorem states that the above equation has no integer solutions for any value of n greater than 2.  In reality this deserves to be called Fermat’s last conjecture – as there is no evidence that he actually proved this beyond a scribbled note in a book margin.  Still his conjecture remained unproved despite the best efforts of mathematicians for over 350 years.

Linking to the TOK essay idea, Fermat was very much an archetypal mathematician from history, an individualist who explored specialisms in discrete fields.  His conjecture would however only be proved through a number of collaborative building blocks in the 20th century and through a drive to discover general connections between branches of mathematics.

Langland program

Illustration of Robert Langland (illustrator:  Jonathan Dyck for The Walrus)

Robert Langland is a Canadian mathematician who in the 1960s proposed the potential links between different branches of mathematics, i.e that previous results in different mathematical fields could be generalised and brought together under a larger framework.  Some of his contemporaries (Taniyama, Shimura, Weil) had earlier conjectured that there was an association between elliptical curves in modulo form and Dirichlet series and their generating functions – two apprarently completely branches of mathematics.  

In the 1980s, another mathematician, Ken Ribet was able to prove that if the Taniyama-Shimura-Weil conjecture was true, then Fermat’s Last Theorem was also true.  So now mathematicians had a completely new way to attempt to prove Fermat’s Last theorem – without even needing to consider Fermat’s equation at all.

In the 1990s, Andrew Wiles was the mathematician who famously managed to prove the Taniyama-Shimura-Weil conjecture for sufficient cases such that this implied that Fermat’s Last Theorem was also true. 

So, a problem in number theory which had remained unsolved for over 350 years, withstanding the best efforts of exceptional mathematicians working in the specialist field of number theory had finally been proved via a collaborative approach which sought to generalise from a problem in number theory to a wider relationship bridging different branches of the subject.

An example of the maths

The mathematics behind this is exceptionally difficult – but below I will go through hopefully enough to get a flavour for this idea.  To understand this topic properly is probably beyond even most post-doctoral mathematicians.

Elliptical curves

We start with an elliptical curve:

This graph is shown below:

We can see that this has a number of integer solutions (points on the graph where both coordinates are integers).  For example (0,0), (1,0), (0,-1) and (1,-1) are all shown on the graph.

However we can also look at the solutions to this equation mod some number.  For example in mod 2, we only need to consider the number 0 and 1.  For any number larger than 1 we take the remainder when divided by 2.  So, 3 ≡ 1 (mod 2).

Elliptical curves mod p

If we then look at the integer solutions mod 2, then only possible coordinates to check are (0,0), (0,1), (1,0), (1,1).  All of these are solutions and so this elliptical curve has 4 solutions mod 2.

Let’s look at integer solutions mod 3.  This time we need to check (0,0), (0,1), (0,2), (1,0), (2,0), (1,1), (1,2), (2,1), (2,2).  Of these 9 possible solutions we have (0,0), (0,2), (1,2), (1,0) so this elliptical curve has 4 solutions mod 3.

We can carry this method on for different prime powers to get the following table:

The question then arises, is there any relationship between p and a(p)?  There doesn’t appear to be any obvious pattern.  We can calculate more solutions and then plot all the values of a(p) for p for the primes less than 100:

We can see that there is some oscillatory behaviour between positive and negative and that the oscilations appear to be growing larger – but there is no clear pattern to help us predict a(p) for a given p.

Generating functions 

Here is the magic of the idea in the Langlands Program – we can actually find an expression which contains within it all the information needed to unlock the pattern between p and a(p).  However this expression is from a completely different branch of mathematics involving infinite summations.  We can start with the following infinite product:

Which we can write as:

We can rearrange this as (we have to be careful about rearranging these sorts of infinite products – but this is allowed here!):

We can then expand out the first few brackets:

We can then find the coefficients of the prime powers of q.  So for q squared we will have:

and for q cubed:

and for q to the power 5:

We can then put these into a table:

and when we calculate these coefficients we see something almost magical – we get the same answers for a(p) we calculated before for the elliptical curve.

Magical connections

This is worth reflecting on!  The following short expression:

has within it all the information needed to tell us about all the solutions to our elliptical curve for every modulo prime.  This expression contains within it all the number of solutions to an infinite number of elliptical curves in different modulo, without needing any reference to elliptical curves, graphs or different modulo.

We can see that there appears to be therefore some connection between these completely different branches of mathematics – i.e information gained in one branch can then be used to provide information in an (apparently) completely different field.

The ability to generalise between fields has now opened up powerful new methods of proving conjectures that previously were unable to be solved in one area (such as Fermat’s Last Theorem), but now can be proved by different branches of mathematics.  How deep this connection is, and why this connection exists is still to be discovered!