I would even say that the future is for quantum mathematicians.

Mathematics is also what mathematicians do, and we do this with our brain, and so how is our brain created?

Well, many many million years of evolution. And so I think our concepts

are very much influenced by the things to see around us.

So we talk about space because we kind of move in space,

We move in time. We count because we see objects. We throw stones,

we want to describe the path of a stone and its velocity.

And so calculus, for instance, is based on the idea of speed or slope.

So I think actually our mind is very much kind of set by everyday experience.

So you might think that mathematics kind of stops being effective if we move into a new territory.

For instance, the very large, the universe, or the very small, the world of elementary particles.

But a remarkable thing: this hasn't happened. So mathematics has been

extremely successful in conquering, certainly I think, the physical experience.

General relativity,

Einstein's theory is a good example.

He needed a concept of curved space. And it turns out that Riemann and others,

in the 19th century, had already thought about it. They thought, you have two dimensions, three dimensions,

why not have an arbitrary number of dimensions?

Abstract thinking of mathematicians was kind of ahead. In quantum theory,

you know, even if you sit and meditate for a million years you wouldn't come up with the laws of quantum

mechanics because it's so bizarre. So in that sense nature had to force that upon us,

and it was like a very painful process. We had to let go of many very cherished concepts.

For instance, the fact that the world is a predictable system. Quantum mechanics is just the way nature works.

You see it very clearly at the level of atoms and elementary particles,

and if you take many many atoms together, then in some sense many of the features of quantum mechanics

are kind of washed away. We don't observe it because we are made out of like 10 to the 24 molecules typically.

There are several things that you have to give up in quantum mechanics. The first is that you cannot know the answer to all questions.

So, for instance, typically, I would say, well, this is pen, I know where it is, and if I drop it,

I know what the velocity is; that it's moving. Quantum mechanics says that's all fine,

you can know either the position or the velocity, so roughly half the questions are unanswerable.

That's really bizarre, because how would I then describe a pen?

But there's a second thing, which is that even if you want to describe

how things move, they never have a specific answer. So, like if you are a

classical object like a pen, now you would say "How do I go from A to B?"

Well there will be a shortest path. If I throw a stone it will go like this from A to B.

And I will know exactly how it went,

and this is like the optimal path, and I can calculate this, and mathematics is about this. Now, if you're a

quantum mechanical particle, you are basically free to go on any possible

path, so you know, and you could even have really very bizarre paths that goes all the way like that.

And the thing that physics will do, it will give a certain probability for all these

possible scenarios. If I for instance want to go from my work to home

I know pretty good what the path is I'm taking,

but if I'm an electron that's moving, say, from one atom to another atom,

it's more like these, these curves, so everything is in terms of chance.

The quantum probability is something, it's like an, it's a complex number, and if you take the absolute value squared,

it becomes actual probabilities.

So for instance, one thing that, because of these kind of funny things, what can happen is that

instead, you know, even if you think about where a particle is,

now it could be either here, or it could be here, and you might say, well there's a certain probability

it's an A, and a probability it's in B.

But a quantum mechanical particle can be for half be here and half be there, at the same time.

So this is very strange.

It's like you take a point, and you divide it in two fractions of a point, put one here and put one there.

And so the rules of quantum mechanics are partly captured probability,

but they have some kind of very strange phenomena built in them, and it's ... you have to kind of dig deep and basically things

happening, you know, all at the same time

makes quantum mechanics very difficult to absorb.

It's been said, you know, if you say you understand quantum mechanics that you're just lying, because this is

something that in our intuition we simply cannot understand.

Physicists were thinking about, well, we need some objects, for instance, to describe the transitions in the atom,

how we go from one state to another state,

so well, we need something, and then they discovered it's called a matrix, and it's actually known, of course, in the mathematical literature.

But to understand the

concepts that came in, like Hilbert spaces and wave functions, that was mathematics that partly was being developed at that time,

but I would say then the physicists really ran off with it,

and at this point, I would say mathematicians are

trying to catch up and

understand, you know,

not only the language, but also the power of the concept. Because there's something incredibly

natural in quantum theory which is perfect for the mathematical mind.

Because mathematicians, they're not studying one object.

They actually really always think about the whole family; the category, all possible objects. So, all numbers at the same time, or all possible

geometrical shapes. And what quantum mechanics says is, that's like perfect, because I'm not considering one path,

I'm considering the space of all paths. And in fact, I'm giving you something else. I'm giving you a way,

indeed, a probability of each possible path,

so in some way, you can sum all over these different paths. We call this the sum over histories

which is a ...

again, very funny concept, because we think of history as something, well,

certain things, certain facts that happens. For the quantum mechanical mathematician,

history is all possible scenarios at the same time. And so there's a certain way to

consider many different things at the same time. And so one

wonderful application on all of this was in, in knot theory. So if you have a knot, like this,

connected like this, so it's a ... it's a... I'm just

drawing here the projection, but there's a certain trajectory in three dimensions, and, you know,

mathematicians want to distinguish. This one is knotted, this one is not. And so, if you're a knot

theorist, you want to study knots in generality; the space of all possible knots. Actually

quantum theory comes to the rescue.

It's a very efficient way to describe these knots. And young physicists are not that surprised anymore as

for instance Niels Bohr or

Albert Einstein was, who were really fighting the concept. I often think, you know, at some point there will be a generation of mathematicians

that kind of grow, grow up with quantum theory, and they will probably apply these kinds of rules in a very natural way.

Brady: "They are starting to steal your stuff, arent they?"

Exactly. So, the remarkable thing is that there are some very deep

mathematical problems; abstract mathematical problems, like this classification of all knots,

or thinking of four- or six-dimensional spaces, really purely interested from mathematical arguments,

and now there is a bunch of these problems that actually have been solved using quantum theory. I think of it like a good mathematician

has a toolbox that's filled with everything: geometry,

calculus, number theory,

and, you know, the rules of modern physics should be there.

It's just a very efficient way to deal with certain problems, and I am actually ... I think many mathematicians were surprised that the

information was flowing in the other direction. Physicists are surprised, because they see the quantum

mathematicians, so to say, take these ideas and

generalise them even further.

It's like you have a very brilliant student and you're teaching the student everything you know,

and then the student takes that much further, because again,

and that's the beauty of math. It is, in some sense, rooted in reality, but then it's not. It's hard to meet a

mathematician that's not ...

doesn't think

platonic; thinks that, you know, the prime numbers are just out there, and they have nothing to do with the fact that there are

three pens here. That the number three is there.

And so, there's this kind of

interesting kind of an

oscillation between math

absorbing

facts from reality,

and then kind of take a flight of the imagination, and go in directions that the physicist says, wait a moment that was not

actually our intent to go and do, to do this kind of crazy stuff.

Brady: "Has this emergence of mathematics coming out of physics been a difficult birth? Has it been

"something that has caused problems?"

Yes. In two directions, I think.

You know, the physicists would say math is very attractive,

but it's like a black hole in which you can disappear, because you will be divorced from reality.

Odysseus, you have to tie yourself to the to the mast of the ship not to be seduced by the beauty of mathematics

because

many of the great insights in physics were looked very painful and inelegant and ugly

to begin with. It took some time to appreciate

quantum theory as something beautiful. So that's the rivalry from the physics part. From the math part,

I think it is that, you know, physicists are sometimes seen as kind of plumbers, they, you know, they make their hands dirty,

they deal with this messy thing called reality.

It's beautiful, as a mathematician, you can go this platonic world where there's a certain perfectness. So there's also

I think quite a few mathematicians that weren't originally

impressed with the physical ideas. They might say, well,

you know, you might kind of be ... this might ... one or two areas where this is relevant, but not to my field,

which is pristine. And then certainly,

out of left field,

there were some physical ideas that proved to be very very very powerful,

and I feel it's like a wave, you know, you're, you think it won't hurt you until the wave hits you, and then suddenly you are underwater.

In physics we have really a crisis, and the crisis is that we have two

wonderful theories that describe the world.

One is quantum

theory: It's perfect for elementary particles. The other one is relativity, and it's used to describe large structures in the universe.

But the point is these two things clash, and they clash because these physical laws of the large. They include

events like black holes, and perhaps more important, the big bang, where the rules of physics break down.

So this is quite remarkable that the theory that you have to describe reality has its own failure built into it; and

we think that, in some sense, the only solution is

to bring quantum mechanics into it. So we somehow have to marry these two pictures. It's not enough to do quantum mechanics.

It's not enough to do geometry. And we know this is happening, because the universe is working,

you know, right now while we speaking, the laws of nature are perfectly working,

and there's no breakdown anywhere. So that means that we have to come up with something

deeper than geometry.

Now what is that?

That's, that's, I think, the big mystery. And we have some ideas.

So, the idea could be that geometry is something like

thermodynamics, which is the theory that describes, basically, materials. So if I think of this table, it feels pretty solid to me,

but I know it's a large collection of molecules.

I feel the air around me, and I feel a certain temperature.

But there's no such thing as temperature, these are just molecules bouncing up and down.

So could it be that just as this physical sheet of paper, if I zoom in, zoom in, zoom in,

I will certainly see molecules and atoms and elementary particles, and there's no paper anymore,

could it be that if I zoom into space itself, at some point I see the pixels?

I see something else? You know, we were thinking perhaps it's pure informations, like zeroes and ones. So, I find that absolutely

breathtaking. Because if you are able to come up with a piece of mathematics

that is more fundamental than space, than geometry, I think it would be

revolutionising all of math.

Just imagine

trying to imagine these two huge objects what they're doing to the matter, to the space-time around it as

they, and the space-time must be going "oh my god what's happening here", and it's flipping up and down up and down.

And they're just generating, these waves are beginning to propagate out. 350 kilometers is about the distance,

what, to London? Then you've got two thirty solar mass black holes sort of orbiting one another in this region

And so they're going...