Deep connections

Katie Steckles

As part of Monday’s session of Interviews with Recent Laureates, maths writer Alex Bellos interviewed this year’s Abel Prize winners, Hillel Furstenburg and Gregory Margulis. The conversation covered many aspects of their work and the award, including what they’re going to spend the money on (Margulis has not given this much thought yet and “can do that later”), but it also touched on something interesting and fundamental about mathematics.

The 2020 prize winners are recognised for ‘pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics’. Their work connects all of these branches of mathematics, and Alex Bellos, as the interviewer, asked how this work has broken down walls. Furstenburg responded that he thinks of it more as a case of bringing things together – uncovering the “greater harmony to which these things belong”.

Despite what they teach you at school – that maths is one subject, that you do in one room once a week, and mainly involves adding things up – there are a huge variety of different kinds of maths: algebra, topology, number theory, probability, dynamics, geometry, analysis, logic and set theory. Even though these topics are studied separately by different groups of mathematicians, there are often unexpected connections between them.

The story of Monstrous Moonshine is a great example of this – a connection that was found between the abstract algebraic structures of group theory, and the complex analysis involved in studying modular forms – such a surprising connection it’s inspired several books, including the very accessible Finding Moonshine by Marcus du Sautoy. In the interview, Furstenburg mentioned another mathematical idea which spans across fields – he describes as ‘the most striking example of this unity’, Dirichlet’s theorem on prime numbers in arithmetic progressions.

Prime numbers – numbers only divisible by themselves and 1 – are hugely important in number theory, as is the concept of progressions – sequences of numbers which follow a pattern. An arithmetic progression is a sequence of numbers with an equal gap between them – for example, 1, 2, 3, 4, 5… is an arithmetic progression with a gap of 1 (starting at 1), and 6, 11, 16, 21, 26… is an arithmetic progression with a gap of 5, starting at 6. 

In general, we can say that the numbers in the progression are of the form a + nd, where a is the number it starts from, d is the gap each time, and n takes each of the whole number values 1, 2, 3, 4, 5… to give a list of numbers.

Dirichlet’s theorem on primes in arithmetic progressions (and it’s important to specify, since Dirichlet was sufficiently prolific that there are at least three theorems and several other concepts that could be referred to as Dirichlet’s) states that for any two positive integers a and d (with the additional condition that a and d are coprime, so they don’t have a common divisor other than 1), there are infinitely many prime numbers of the form a + nd.

This wonderful theorem extends a classical theorem of Euclid, which states that there are infinitely many prime numbers. Euclid’s statement can be proved quite neatly, if you realise that while the definition of a prime requires it simply not to be divisible by anything except itself and 1, it’s sufficient to check that it’s not divisible by any other prime number (since if a number is divisible by a non-prime, that non-prime itself must have a prime divisor).

Let’s say there are not infinitely many primes – a trick known as ‘proof by contradiction’. Then, Euclid argues, simply multiply together all the prime numbers (which is significantly easier to do if there are finitely many of them), and add 1. This new number will not be divisible by any of the numbers you already considered to be prime – if you divide by any of them, you’ll always get a remainder of 1. Hence this number must be a prime, which wasn’t on your list already! This is a contradiction, since our list was meant to contain all primes. This means our starting assumption – the only real assumption we’ve made, that there are finitely many primes – must be false. QED (where the E stands for Euclid).

Proving Dirichlet’s theorem, however, is not as trivial. The proof uses analytic number theory – a branch of number theory which makes use of methods from mathematical analysis – like calculus, and taking infinite limits – to study the integers (whole numbers). As Furstenburg says, “[the proof] used a tool called the theory of functions of a complex variable, which on the face of it should have nothing to do with numbers and prime numbers in particular, but it turns out to be very significant.”

It’s fascinating that these two separate areas of maths could be seemingly studying such different things, but techniques from one find a useful application in another – but it’s something mathematicians often find to be the case. In the words of Gregory Margulis, “Mathematics is divided into several parts; sometimes people feel connections between these parts, and sometimes not; my understanding of unity is trying to find these deep connections between various parts of mathematics.” Furstenburg explains, “That aspect of the beauty of mathematics is what attracted me to the subject, and seems to be guiding my work”.

The post Deep connections originally appeared on the HLFF SciLogs blog.