Thomas Lin is the Editor In Chief of Quanta Magazine and the editor of two math and science books, “The prime number conspiracy” and “Alice and Bob meet the wall of fire”. He joins Hari Sreenivasan to discuss a unique mathematical phenomenon across the world known as Universality.

# Breaking Down Universality

## TRANSCRIPT

Thomas Lin is the Editor in Chief of magazine and the editor of two math and science books, 'The Prime Number Conspiracy' and 'Alice and Bob Meet the Wall of Fire.'

He joins us now to discuss a unique mathematical phenomena across the world known as universality.

All right.

I'm intrigued.

Universality means what?

Okay, so universality, as the name suggests, means that it's something that's found in a lot of different places and settings, and, essentially, it's this mysterious mathematical pattern that scientists have found in everything from heavy atomic nuclei to the distribution of prime numbers to models of the Internet to chicken eyes and independent bus systems, and so you wouldn't think these things have any real connection with each other, and so, you know, as scientists do, they want to understand what's at the heart of these different systems and why would this pattern emerge?

And so, in the 1950s, the Nobel Prize-winning physicist Eugene Wigner discovered this pattern in the energy spectra of the uranium nucleus...

Okay.

...and he had this idea that, well, maybe this is a pattern that is universal and that emerges in lots of different complex correlated systems -- 'complex' just meaning that there are many, many elements, and 'correlated' means that they strongly interact with each other, and, subsequently, it was found in all these different places, including in this bus system in Mexico, which is just strange and fascinating.

So, give me an idea.

How do you describe this pattern?

I mean, you know, in music it winds up 'A,' 'A,' 'B,' 'A,' and we repeat that over and over again...

Right.

...or, you know, fractal theory says, okay, look at this nautilus shell and the spiral galaxy...

Yeah.

...but what does this pattern look like?

That's a great question, and so I think the best way to describe it is that it sits somewhere between order and randomness, and so it's neither completely regular or periodic like in order.

If you think of a barcode or something that's very periodic, that would be a very regular order structure, and random would just be all over the place, and a random pattern would have, potentially, a lot of these lines in it in a barcode closer together, and some could be really far apart and be all over the place, but this has -- this pattern, universal pattern -- essentially, because the different elements interact with each other, they sort of push off from each other, and so it's disordered, and yet it also has a certain amount of regularity because you'll never get them too close to each other or too far apart, and what mathematicians found, which is really surprising, and Eugene Wigner, as well, way back decades ago, was that these kind of systems could be modeled by a kind of math called matrices, but not just any matrices, but by random matrices.

A matrix is, essentially, a rectangular array filled with numbers or elements and somehow, just with random elements, the characteristic values or the eigenvalues of these matrices form this exact same pattern that we're seeing in uranium nuclei, in bus systems, in chicken eyes, and other places.

So, okay, if I have a rectangular box, and I fill it with random numbers, you're telling me that there is a pattern in there even though I can't see it?

I'm specifically saying I'm gonna heads-and-tails, I'm gonna keep flipping coins, I gonna write down these numbers, and there's a pattern in there?

Exactly.

And, so, to just make clear, it's not the random numbers themselves that are the pattern, it's the eigenvalues that come out of the matrix that form this same sort of special pattern, and the reason for that is that if you look at very simple atoms, if you look at hydrogen and helium, physicists can come up with exact matrices that describe their energy levels because they're simple enough, but when you get to heavier atoms like uranium, the nucleus is so big and complex, and there's so many interacting parts, and there's so many different energy levels that you can't do that.

You can't find the exact matrix.

And so what was surprising was that a physicist decades ago discovered that you could just use a random matrix to model it because say you're in a room full of people...

Mm-hmm.

...and say there's only a few people.

That would be like a simple atom, and one person could really become -- his or her personality could just take over the conversation.

Sure.

But in a room with many, many more people all talking to each other, no one person is gonna stand out or control the conversation, and it will just sort of get washed out almost like noise, and that's, essentially, what a random matrix is, and that's why this universal pattern emerges.

So, let me give you another example of the bus system in Mexico.

This is kind of a funny situation where a physicist who happened to be in a city called Cuernavaca, Mexico, 1999, happened to be waiting for a bus.

They're independent buses, so different bus drivers come and go as they please, and they want to have as much business as possible.

So they want to arrive at a stop where there are enough people so that they can have enough business to take them to the next destination, and so what this physicist saw and noticed was that the bus drivers, these independent bus drivers, were paying somebody at each stop and getting a piece of paper from them, and later, after a lot of cajoling, and I think he had to give some tequila to some of the drivers, as well, to find out what was happening -- or to the spies that they were paying.

Essentially, the drivers were paying spies to ask to sit there and see when the previous bus had left so that they could then decide, 'Well, should I slow down because the other one just left, and I don't want to go too soon because then there'll be nobody at the next stop, or should I speed up if the previous bus had left a long time ago to try to beat the other buses to the next stop and get all the passengers?'

And so because of the spies and because of that knowledge and information, that created this correlation between the bus arrivals...

Mm-hmm.

...and the most surprising thing about this is when the physicists gathered all the pieces of paper and all this data, they found this exact same universal pattern that was found in the nuclei of uranium atoms.

So, how do we make these patterns work for us?

What are applications that you can take?

If this universal pattern is in so many things, is there a way that we can manipulate it to our own interests?

So, the way a lot of basic science works, first you have to understand things, right, before you can think about applications, but one thing, because of this connection to random matrices, one thing that it does help scientists do is that when they find other complex correlated systems that they want to understand, but they can't model it precisely, they can use different types of random matrices to model them very accurately, and they can learn things from that.

Let me give you another example that may be a little bit more sort of practical and connected to our daily lives.

This universal pattern has also been found in human bones, and so you take a special measurement of human bones, and a bone that has deteriorated through osteoporosis might not be correlated enough to exhibit this universality pattern, but ones that are healthy bones would have those connections and would exhibit the pattern.

So it could, potentially.

It hasn't yet, I don't think, but it could, potentially, lead to ways to detect whether your bone is healthy or not.

So, is there a school of mathematics that is chasing down more examples of where these patterns exist and where they don't?

Absolutely. Yes.

This is a fundamental interest, I think, to both mathematicians and physicists and people studying climate, for example.

Arctic melt ponds are another example.

If, as the Arctic ice melts, and you get these pools of water, if they are separated enough, then they don't exhibit universality, but once the pools start sort of sloshing into each other and connecting and forming these interconnected melt ponds, then they start exhibiting this universal pattern, and, again, this is a way to understand these very complicated systems, which can then potentially lead to better models for things like understanding climate and a whole host of other phenomena out there.

All right.

Thomas Lin of magazine, thanks so much.