Proving Without Knowing - The 2021 Abel Prizes in Mathematics

Andy Yin

A major prize was awarded on March 17 — the Abel Prize. It’s sometimes called the ‘Nobel Prize of maths’, because, cruelly, there is no Nobel for maths — Alfred Nobel was, apparently, more interested in rewarding practical discoveries than theory.

What makes a mathematician’s work worthy of elevation by an award like the Abel? Many are elevated for finding connections between different areas of maths, exposing similarities between the seemingly different. Mathematicians prize these connections because they can turn difficult problems in one mathematical ‘language’ into approachable problems in another.

This year’s laureates — László Lovász of Hungary, and Avi Wigderson of Israel — were chosen for their lifetime achievements in linking two major theoretical disciplines: theoretical computer science and discrete mathematics.

Discrete maths is concerned with ‘discrete’ objects — consisting of distinct parts — as opposed to ‘continuous’ objects like curves and functions. One of the major objects of study in discrete maths are graphs — collections of points connected by edges.

It’s easy to see how these might be relevant to computer science, as models of networks or even the components of a computer, but graph theory has much older origins. One of the oldest and most famous problems in graph theory, originally posed in 1852, is the four-colouring of a map.

Can every map be coloured with up to four colours such that no two adjacent regions share a colour?

In other words — how many colours of ink does a cartographer need to have in stock? It seems like a daunting task.

But the key, as is common in maths, is to forget about the extraneous details. To determine if a region can be coloured red, all that matters is whether the regions it shares borders with are coloured red or not. The size or shape of any region is irrelevant.

Therefore, a map can be replaced by an abstract model — a graph. Each region is represented by a single dot. To indicate that two regions are adjacent, a line is drawn between their dots.

This graph tells you all you need to know about a map, without excess.

Now, this is just the beginning — rephrasing the question in an abstract way. The answer to the question turned out to be yes, but the correct proof (first found in 1976) was so monstrous it needed the aid of a computer. That’s often the case with problems in graph theory and discrete mathematics — easy to understand in lay terms, extraordinarily challenging to solve.

Lovász: Easier Said Than Done For many people, that’s what makes discrete maths so appealing. That was the case for László Lovász. As a high school student at Budapest’s elite Fazekas, his class would be frequently visited by one Paul Erdős, a Hungarian mathematician who would bring the students deceptively tricky puzzles.

Erdős deserves a paragraph. His reputation is singular among modern mathematicians. Unlike many academics who look for stable employment, Erdős preferred to wander, moving from university to university, living at other mathematicians’ homes and collaborating on a few papers before moving on. The number of papers he published (more than 1500) far exceeds that of any other mathematician in history.

Erdős challenged Lovász with unsolved problems in graph theory, and, in Lovász’s words: “so started my lifelong commitment to graph theory.” In fact, Lovász gained more than that from Erdős. He picked up his mentor’s love for travel and collaboration. Moreover, he picked up an appreciation for one of Erdős’s favourite maths techniques: randomness. Among his many accomplishments, Paul Erdős pioneered the probabilistic method. The idea is that many questions in graph theory, like the fourcolouring problem, boil down to “is there a way to _”. Explicitly finding a way is not always easy. If I had to colour the states of Australia with four colours, I wouldn’t break a sweat, but if I had to colour the countries of Africa or Europe, I’d have more trouble. How do you prove a theorem like the four-colour theorem without manually working out a way to colour every possible map? After all, there are infinitely many.

Erdős’s answer might be: do it randomly. You can certainly generate a random colouring of a graph by randomly picking one of four colours for each point. If you do this, you’ll almost certainly paint two neighbouring regions the same colour and hence fail the four-colouring problem. But you don’t need this technique to succeed every time — you just need to know that it’s possible for it to succeed.

One of the accomplishments that Lovász was awarded his Abel Prize for is the Lovász local lemma — a lemma (small theorem) that gives a way to estimate the success probability of a technique like this.

This method wasn’t actually used to prove the four-colour theorem, but it has been used to prove many things. And it should be surprising! Unlike a theory of nature, which can be overturned by new evidence, a mathematical proof is definite, a logical certainty. So it seems unusual that random generation can be a part of it.

But to Lovász, randomness is a natural part of a mathematician’s toolbox — as natural as calculus or basic algebra.

Wigderson: Knowing Nothing Except That I Am Right If the probabilistic method is surprising, then Avi Wigderson’s — Lovász’s co-laureate — revelations about the nature of proof are even more so.

One of Wigderson’s areas of expertise is the concept of zero-knowledge proofs.

The name is spooky, but it’s actually something that’s pretty realistic. Let’s say you are registering a patent for an ingenious new map-colouring machine. You want to be able to demonstrate to the patent examiner that your invention does indeed do what you say it does. But you don’t want them to know how it works — maybe they’re crooked and will use your designs to put a machine of their own on the market.

How can you convince someone that something works without showing them how? Say you have a map with you, and you claim that your machine can colour it with just three colours (which isn’t possible for every map). But, to protect your work, you can’t convince them by showing them the coloured map.

You can do something subtle: colour the map with your machine, but hide the colours by putting an opaque sticker over each region. The examiner won’t believe at first that you’ve succeeded, so you invite them to peel off the stickers on two bordering regions. The examiner will see that they have different colours. “Okay,” they say, “But how do I know that you haven’t messed up somewhere else in the map and coloured two adjacent countries the same?”

They ask, “Can I peel off another two stickers?” Every sticker they peel off increases their confidence that you have coloured the map correctly, without using the same colour for any two bordering countries. But, the more stickers they peel, the more of the map they see — peel off enough, and they might be able to steal your work.

The trick is this: each time they peel off two stickers, you take the map away and recolour it. A map that’s coloured with red, blue, and yellow can always be recoloured in blue, yellow, and red, or red, yellow, and blue, and so on. Then you put the stickers back on and invite the examiner to peel off another two.

The examiner gets to see, every time, that you have not coloured two bordering countries the same. Moreover, they don’t learn anything about your method, because you’re not showing them a single colouring — you’re showing them bits of different colourings. This method won’t convince the examiner 100% like showing them the full colouring of the map would. But, the more stickers they peel, the more confident they are that you are correct — all the while they are gaining no knowledge whatsoever about how you actually solved the problem.

What is remarkable is that a huge class of mathematical problems — called NP problems — can be proved by ‘zero-knowledge proofs of this kind’. These problems include every problem that a computer can solve efficiently. That means zero-knowledge proofs are exceptionally useful in, among other things, security — it goes without saying that the designers of security systems want people to have confidence in them without knowing how they work.

Wigderson and his collaborators proved this fact in 1991 – it is one of the main reasons for his Abel Prize.

He acknowledges it seems counterintuitive — it seems impossible that you can change someone’s mind without giving them information. “We somehow associate conviction with information transfer,” he says.

More to Come This year’s Abel Prizes are special because discrete mathematics is an area where the questions can be understood even by non-experts, although the answers can stump non-experts and experts alike.

Keep in mind — none of what I’ve written about are new discoveries! But, thanks to the connection of maths and computer science, the work of Lovász and Wigderson continues to have practical applications today, including but not limited to Lovász’s work in graph theory and Wigderson’s work on zero-knowledge proofs.

If you’re interested in groundbreaking present-day maths — keep an eye out for the Fields Medal, the other ‘Nobel of maths’, which is due to be awarded in 2022.