# Quantum Mathematics and the Fate of Space, Time and Matter – Robbert Dijkgraaf

Hello. Hi, everybody. I would like to welcome you

all to this evening’s lecture. My name is Eleny Ionel. And I am the chair of the

mathematics department here at Stanford. Tonight’s lecture is jointly

organized and sponsored by the Stanford

Mathematics Research Center and the Stanford

Institute for Theoretical Physics. This lecture is also part of

a series of public lectures, roughly about two

per year, which are organized by the Stanford

Mathematics Department and sponsored by the Stanford

Mathematical Research Center and the friends of

Stanford mathematics. If you would like to be notified

about future events like this but you are not on

our mailing list, please send me an email at

[email protected] This evening, we are

very fortunate to have as our speaker Professor

Robbert Dijkgraaf. Before we begin, let me just

tell you briefly a few things about our speaker. Born and educated

in Netherlands, Professor Dijkgraaf obtained his

bachelor’s, master’s, and PhD degrees in theoretical physics

from Utrecht University. Since 2012, he has been

the Leon Levy professor and the director

of the Institute for Advanced Study in Princeton. He is a mathematical

physicist, who has made significant

contributions to string theory and also to the advancement

of science education. Professor Dijkgraaf is also the

president of the InterAcademy partnership since

2014, a past president of the Royal Netherlands

Academy of Arts and Sciences, and a distinguished public

policy advisor and advocate for science and the arts. His research focuses on the

interface between mathematics and particle physics. In addition to finding

surprising interconnections between metrics models,

topological string theory, and supersymmetric

quantum field theory, Professor Dijkgraaf

developed precise formulas for counting of

bound states that explain the entropy of

certain kinds of black holes. For his contributions

to science, Professor Dijkgraaf has

received the Spinoza Prize in 2003, the highest scientific

award in the Netherlands. And in 2012, he was named

a Knight of the Order of the Netherlands Lion. He is also a member of

the American Academy of Arts and Sciences and of the

American Philosophical Society. Professor Dijkgraaf also

a trained artist, writer, and popular lecturer. His annual televised

lectures on science attract more than 1.5 million

viewers in the Netherlands. And he also writes a monthly

column for Holland’s top newspaper NRC Handelsblad. Most recently, Robbert

wrote a companion essay to the classic 1939 essay,

“The Usefulness of Useless Knowledge,” by the Institute of

the Advanced Studies founding director Abraham Flexner. It provides a modern

argument for supporting the curiosity-driven basic

research and original thought that are critically important

for future innovation and societal progress. So we’re really delighted to

have Robbert with us tonight. His title is Quantum

Mathematics and the Fate of Space, Time, and Matter. So please join me in

welcoming Professor Dijkgraaf. [APPLAUSE] Thank you so much for

this very kind invitation. What a pleasure

to see all of you here, in particular

my dear friends and colleagues both of

mathematics and physics. So I’m kind of perfectly happy

to talk about the interaction of the two. And I want to begin

by saying some words about the role of

mathematics vis-a-vis, the natural sciences. So one point is

that mathematics, which, of course, strives

for kind of eternal truth, it’s also very much an

environmental science. But it’s this famous

line by, I think it’s from Jean-Pierre Serre

that said that physicists try to find out which laws got

picked for the universe, mathematicians try to find

out which laws even God has to obey. But on the other hand,

many of the concepts we use in mathematics have in

some sense natural sources. And there are wonderful

quotes of this. This is the famous

quote of Galileo. He spoke about the

Book of Nature. The Book of Nature– that’s if you cannot

understand it, you are wandering around

in a dark labyrinth. So the natural

language, in which kind of physics, of science

is written, is mathematics. This was true in

the 17th century. But there are some kind

of more modern advocates of that point of view. Here is Richard

Feynman, by the way, not known as a great connoisseur

of refined mathematics. But yet, he says, for “those

who do not know mathematics, it’s difficult to get

across a real feeling as to the beauty, the

deepest beauty of nature. If you want to

learn about nature, you have to learn the

language that she speaks it.” On the other hand,

another quote of him says that “if all mathematics

disappeared today, physics would be set

back exactly one week. [LAUGHTER] Now, don’t applaud too

soon, because this might be a very smart

remark, till I actually had a very famous mathematician

giving the perfect answer to this, “that was the week

that God created the world.” [LAUGHTER] So I would say math-physics

2-1 as intermediate score. But the question is, of course,

how are mathematical concepts illuminated by the world? And I want to bring into

a wonderful essay actually written by the

late Bill Thurston. He wrote– a famous

mathematician, of course, certainly well-known

here in this area– essentially describing

what is the best definition of a mathematical object. And he took something

that we all kind of know, the derivative of a function. So how do you describe the

derivative of a function? And he basically

says the following, he said, well, we can start

with like the technical formal definition. It’s epsilon and delta. And very tellingly in the

article, he has it wrong. [LAUGHTER] But then his point

is, well, we can also think of this as the

rate, the velocity. And if you do know that,

then you know actually it can also be a vector. It can have a certain direction. You can think of a function

in a higher dimensional space. Well, it can be the tangent. It can be tangent plane. So immediately, you

understand functions of more than one variable. Well, definition

4 is infinitesimal with infinitesimal variations. Or think about

discrete variations. Or think about symbolic

manipulation of derivatives or the linear approximation,

or the microscopic– so you zoom in

into the function. And he goes on, and on, and on. And certainly,

there’s definition 37, which is a particular Lagrangian

section of a cotangent bundle, very technical definition. But at some point, he

needed definition 37. But his main point is,

we need all of them. You are in a very, very poor

position if you cannot kind of enjoy all these

different perspectives. So in some sense, a good

mathematical subject, like a diamond, has

many different facets that illuminates itself. Now another point

I want to bring you that there is some

very important symbol in mathematical equations. So typical mathematical

equation looks like this. I want to point out a kind

of forgotten symbol, which is in the middle. It’s the equal sign. And I know philosopher

sometimes talk about this as kind of what

they call Clinton’s Principle. That’s really a 1990s concept. But there was a definition on– [LAUGHTER] –what the meaning of is is. What do we mean exactly

if you equate A and B? But it’s wonderful that some

of the most beautiful equations in mathematics, they connect

two different worlds. So I think very appropriate, the

equal sign is these two lines, because in some sense, I would

say intellectual ideas can flow from A to B and from B to A.

And sometimes, these connections are really surprising. Some of the most elegant

equations in physics have the property too. One great master of

that was Einstein. Now, perhaps, E

equals m c squared is the most famous

equation in the world. But, of course, before

Einstein, E and m– energy and mass– were

totally unrelated concepts. It’s just the fact

that you find both of these elements

in one equation was incredibly powerful. Of course, it tells

you that you can– if you move, you will be

a little bit more massive. And you can take mass and

convert it in pure energy. So I think many of the

equations have this property. And actually, that’s

what basically will be the overarching

theme today in my lecture– how mathematics can

connect different worlds and what we learn for it. So you will see

many examples, which are like Rosetta

Stones, where there are two different languages

that in some sense find a dictionary. If math is the natural

language of the physical world, we’ll see that the

physical world doesn’t have yet a universal language. There are various kinds

of dialects, languages that have to be connected. In fact, if you

look at mathematics as a pure technical

thing, you can see some of the

large developments in modern mathematics. Some lists here a few will

feature in my lecture. But for instance, the Langlands

program, a very famous– or the whole studying of

number theory and geometry. There are all these

wonderful equations that somehow are marrying two

worlds that some say nobody thought would actually meet. In physics, perhaps

the most famous example is the duality that arises

in the early days of quantum mechanics– Heisenberg’s

uncertainty principle, the fact that an electron

or any other object could be both a

wave and a particle. So my favorite line

for this is actually from Werner Heisenberg– from Wolfgang Pauli. Heisenberg makes the discovery,

writes to his good friend Pauli. And then within three weeks,

Pauli has a reaction to it. He says, ah, I understand

what you are saying. If I look with my left

eye, I see a particle. If I look with my right

eye, I see a wave. If I open both eyes,

I become crazy. [LAUGHTER] And so there is something

of that what you will see it today, where we look with

the left eye, right eye, it’s not always clear

we can open up both. Here’s actually a lovely

quote of André Weil, a famous mathematician, a long

time Institute faculty member, where he says, there are

these obscure analogies, blurred reflections of

one theory and another, an illusion that two things are

the same but perhaps there are not. And it’s, in fact, these kind

of very intuitive feelings that two worlds

have to be connected that are driving mathematics

in a very, very powerful way. It’s, by the way, a

wonderful property that in one of the most

rigorous fields– intellectual fields,

mathematics– there’s kind of art forms to express kind

of uncertain and ambiguous relations. So there’s the conjecture. There’s the program. Somehow, two things have

to do something together, but we do not know exactly how. Well, an important element in

going forward is in some sense our aesthetic feelings. So the Institute for Advanced

Study was created in 1930. And the founding director,

the first thing he did was not make a building

or recruit faculty, but ask an artist to design

a seal and devise a motto. And the motto he found

was Truth and Beauty. And there have the

many reflections about the nature of

truth and beauty. Both are very important

in mathematics. I would even say that

mathematicians, and perhaps scientists, are one of the

few who talk about beauty with any sense of irony. Like in art, you can

no longer describe a painting as beautiful. It has to be

interesting, or whatever. But mathematical equations

are just still beautiful. And there’s this wonderful line

by another Institute professor, Hermann Weyl, who

was asked, how do you manage with truth and beauty. He said, always try

to combine the two. But if I had to make a choice,

I usually chose the beautiful. Here’s the famous physicist

Paul Dirac, who said, in physics, it’s “more

important for the equation to be beautiful than

to fit experiment.” [LAUGHTER] There is actually an

opposite point of view that I want to

present too tonight. And actually, I love this quote

by John Wheeler, the physicist, who said “every law of

physics pushed to the extreme will be found to be

statistical and approximate, not mathematically

perfect and precise.” He was making exactly

the opposite point. And his point of view is

that, by being approximate, there might be still some

kind of flexibility that allows the law, the

physical law, to survive. He is another famous

quote by Francis Crick from Crick and Watson fame. “Any theory that account

for all the facts is wrong, because some of

the facts are always wrong.” [LAUGHTER] So let me just say one word

about this whole notion of truth and beauty

and beauty in science. And so I would say there are

two opposite points of view, where you find the true deepest

beauty in the natural world. And this is roughly

the point of view from particle physicists or

condensed matter physicists. So particle physicists

would say, well, we look at this kind

of garbage, which is like everyday life

with all its complexities. And if you reduce to the

elementary particles, you find these incredibly

beautiful equations. So ultimately, beauty is found

at the smallest distances. Condensed matter physicists

would do exactly the opposite. If you have a

glass of water, you can describe it as the motion

of 10 to the 24th molecules. But you would miss the

laws of hydrodynamics. You would miss the

laws of thermodynamics. These laws are not present in

the microscopic description. They emerge in the

macroscopic description. So great beauty is found

at the very other end, at the largest structures

that we find in science. I actually will make

the point tonight that there’s a similar

kind of distinction in the framework, the language

that we use to describe the large and the small. The largest realm of gravity,

we use Einstein’s theory of relativity– of space

and time and curvature and gravitational forces. The world of the very small is

the world of quantum mechanics, very counterintuitive, a world

of operators and uncertainty, much more kind of

an algebraic world. And you can say that on the

reflection of mathematics are the two ways of

thinking about math– the geometrical way,

which I would say is the visual, the right

brain way of looking at math– sketching, walking

around your object– and the much more algorithmic,

algebraic way of doing math, like a computer

code– you do steps. I note in math education,

these are two completely different ways to

learn the field. And perhaps some of you– if I want to give you

directions, some of you would love to see a map. Others would like to

see an algorithm– left, left, right, left. And you might be

either more literary or a more visual person, and

both these ideas are around. In fact, what I

will try to explain to you is that in some

sense, in mathematics, physics is giving us a way to

go from a geometrical object to algebraic objects– so shapes and going to numbers. And in fact, there

will also be some way to go the other way around. We can also go from algebra,

and geometry will appear. So if you will

have a discussion, what is more fundamental? The conclusion tonight is

that there isn’t really this kind of perfect symmetry. And in some sense, we

have to find a synthesis from these two points of view– the microcosm and in the

macrocosm, or if you will, the algebra or geometric

way of looking at the world. Now, to begin, I want to

give the mathematicians a kind of little crash

course in physics. So this is roughly the ABCs. So I would say there

are various stages in the evolution of physics. The first one I would say

is the classical world. So in classical

mechanics, you want to go from A to B.

And the question is, what is the way you go? And typically, there’s

some preferred path. Oh, it would be the

minimum of some action. You find some

geodesic in the space or the solution of a

differential equation. And of course, this is

classical mechanics. It was born in the 17th century. And it led to terrific

applications in math– as I said, to calculus,

to analytic geometry, to dynamical systems. And then in the 20th century,

we got quantum mechanics. And in quantum

mechanics, it’s not about the question of how you

go from A to B. In some sense, in quantum mechanics,

all possible paths are being explored. Quantum mechanics has this

phenomenal deep concept of a sum over histories– that you sum about all

the possibilities waiting by the action of

the specific path. And in some sense, that leads

to a completely different realm in mathematics, many topics

that were kind of discovered in the 20th century. The third level would

be quantum field theory. In quantum field

theory, you start in A and you can end up in

both B and C. Particles can be created and

annihilated, as they have been done in particle accelerators. So it’s much more about summing

over graphs, the famous Feynman graphs. And actually, those

areas in mathematics are relatively recent. They are being developed

over the last few decades and are extremely fruitful

in mathematical results. Another phase would

be string theory. String theory– we’ll

meet it very soon– instead of thinking of

points, we think of loops. And instead of thinking

of summing over graphs, we sum over surfaces. And again, this gave rise

to a whole different set of mathematical subjects

that are very, very new. And I would say, we’re

still scratching the surface of that mathematical area. And the highest

level– and we’ll end my lecture with

this– is quantum gravity. In quantum gravity,

space itself becomes a quantum mechanical object. So we just celebrated

this year– actually, last Tuesday– the

Nobel Prize to gravitational waves– ripples in

space and time– a discovery exactly 100

years after Einstein first proposed this. So space can ripple

and can move. And we think that, at

the smallest distances, the incredibly small distance

of Planck length, space itself will become a

quantum mechanical object. That is to say, at the

smallest distances, it’s not even clear

where a point is. So geometry cannot disappear. It gets replaced by something

completely different. And then the question is, what

kind of mathematical language is able to capture these

very foreign ideas? So my prediction from physics

is that the classical field of geometry, as we study in

mathematics, has to be altered. There must be some way in which

we can generalize geometry to something that really

uses the insights of quantum mechanics. And as I will show you,

there is like two steps. We can use string theory. There’s some wonderful examples

of generalized geometries, stringy geometry. But in the end,

we need something like a quantum geometry that

is really an emergent quantity. So like you take a picture

and zoom in and more in and you see the

individual pixels, in physics in some

sense we’re are looking for the atoms

of space and time. Now, I want to start the

story with the classical story and slowly move to

these various stages of this intellectual development

that’s been happening in the last 50 years. And I want to start

with particles. And actually, one of

my favorite stories, which is a question that some of

you might not have asked– you learn about the

properties of particles. But these particles

are all identical. If you have two

electrons, there’s no way in which one is a little

bit heavier than the other. So if you think about

it, it’s quite remarkable that nature has a facility

to produce particles at exactly the same property. It’s something that modern

physics is able to explain. And the origin of

idea was actually a telephone conversation

between John Wheeler and Richard Feynman. Feynman was at that

point a grad student in physics at Princeton. And Wheeler calls him up

in the middle of the night. And he asked, do we

know why all electrons are the same, why they have

exactly the same property? And his line is, well, because

there’s only one electron in the whole universe– so a general joke if

your thesis advisor calls you in the middle of

the night with a crazy idea. But in fact, it’s in

some sense this idea that led to really big

breakthroughs of Feynman. And this was Wheeler’s idea. So this is space-time. So you think of this

as slices of space moving upwards in time. Here is an electron. And Wheeler had the point,

suppose the election could go, not only up in time,

but back in time. Now, if I could time travel,

I could reappear here as an exact copy of myself. So there would be two Robberts. And they would be

exactly identical. So if instead of

going up and down, this electron is able to weave

a big knot in space-time, and you think of this as

slices of space– so like, pictures that you take at

certain moments in time– there would be one

electron in the initial. But if you slice it in the

middle, you see many electrons, and, in fact, antielectrons. And they would all

be exactly the same, because it’s the same particle. So this was his idea. And Feynman basically

forgot about it. And then in the 1940s

when he was kind of bored with his other

physics, he started to come back to this idea. And I love this page

from his notebook, because you can see

him here literally sending these

elections back in time and seeing, well, how do the

rules of quantum field theory behave if you try to do this? And it’s actually

worked out magically. So this actually seemed to work. And now, we know these

objects as Feynman diagrams. And they are allowed– in this formalism, particles

are allowed to go back in time. And Feynman had this

vision at that point. Most people were doing very

difficult calculations. Perhaps you can

just do drawings. And he has this image

that perhaps at some point physics books would

be full of drawings. Wouldn’t it be that lovely? Of course, now they are. In fact, they are so popular

that my favorite story is about this van. So this van drove around in LA. And there’s a story that physics

students followed the van. And it was a lady

driving the van. And he stops her at

the traffic light. And he goes up to

her and said, Mrs., do you know that these

symbols on your van are actually called

Feynman diagrams and that we use them

in physics every time. And she said, yes, I

know, I’m Mrs. Feynman. [LAUGHTER] And here is the Feynman

family with the van– I think very 1960s, I would say. And, of course, it’s

incredibly successful. In fact, the role of mathematics

in this was not always clear. In the 1960s, actually,

at some point, physicists were

giving up this idea that these Feynman

diagrams were still able to capture everything. And at that point, we

had kind of a black box model of physics, where you

see physics something coming in and coming out, but very

difficult to connect the two concepts. And actually, Freeman Dyson

famously gave a lecture in 1972, where he

said– basically, he declared the marriage

between physics and mathematics to be divorced and broken

up, because at that point, physicists and mathematicians

weren’t talking to each other. There’s a famous

letter that Feynman is invited to a math conference. And he writes back, I

think a one line answer, I’m not sure why– I see no reason why I should

attend a math conference. But very soon thereof,

I would like to say, the black box was opened. And inside was a

very tiny formula. So this is the formula,

used very clever mathematical symbols

that describes the particle– the standard

model of elementary particle physics. And the symbols

that are used there and the mathematical

language that’s being used is incredibly natural and

elegant and beautiful, I would say, from a perspective

of geometry and mathematics. In fact, in some sense,

the unifying theme connecting modern

particle physics is symmetry and in

all its realizations, going back to original

ideas again of Hermann Weyl. In fact, talking

about dictionaries, this is a famous

paper from the 1970s. It was actually the

outcome of a seminar at Stony Brook between the

Nobel Prize winner CN Yang and the mathematician

Jim Simons. And they were just

comparing notes. And they noticed

that, wait a moment, all the objects I’m

studying you have too, although you give

them different names. And this was an example, I

think, where in some sense kind of a Rosetta

Stone was discovered. And nowadays, symmetry,

I think, is really our guiding principle. So this is my kind of cartoon

version of introducing it. So for instance, if you

think about the theory of quarks, quarks, you

have an internal label. So you can think of them

as having a little– I hope you admire my

PowerPoint skills here– there’s a little kind of

arrow that moves around in an abstract three

dimensional space. And that symmetry is

not only present at one particular point, it’s like

present at all possible points. In fact, there’s not this kind

of Stalinist point of view. There’s a much more kind of

a democratic point of view, where in some sense

these arrows are able to wiggle at each

point in space and time. So there’s huge symmetry

group underlying our particle physics. And in some way, you can

think of modern fields as kind of waves in these

kind of little arrows. And if you bring

this whole concept into the language of quantum

theory and Feynman diagrams, then these become particles– colored particles. And in fact, the particles

that mediate these forces in the gauge theory,

we can think of them as a very natural way

as kind of matrices. They are labeled by

two kind of colors, if you want to, say, turn a

red quark into a green quark. Now, the rules of quantum

theory are even more bizarre, because the assertion, as

Feynman was expressing, particles can both– they

can kind of interact, they can be decayed, but they

can go forwards and backwards in time. So there are these so-called

virtual particles that exist for a very brief time. My favorite joke is

that, in the Netherlands, we have a rule of this

that basically anything you do fast enough before it’s

being observed is fine. And this is somehow

a very natural way to describe quantum mechanics,

because these particles live for a very brief time. In fact, kind of an ultimate

process in quantum theory, there’s this process, which you

can either think of a particle and antiparticle being

created out of nothing and then decaying again; or

taking a realist point of view, a particle going up and down

in time in an endless loop. So these diagrams exist. In fact, they exist

right as we speak– empty space according to

modern quantum field theory is filled with these

so-called vacuum fluctuations. So again, this is my animated– my visualization of

the vacuum, where you have a kind of a

boiling pot of particles being created and annihilated

at a very concentrate. And so this is, again,

an underlying principle of quantum theory. As Murray Gell-Mann said,

“everything that is allowed is obligatory” in

quantum mechanics. Anything that can happen

will happen, perhaps with a very small

probability, but it does. And I think, this is the first

point I want to emphasize, that the quantum theory

has a very different point of view on life. It’s not looking at

one specific instance. It’s looking at all

instances at the same time. And this fits very much the

modern view of mathematics. The modern view of

mathematics, you do not consider one particular object. But you always

consider there’s kind of the universe of all possible

objects in that category. So it could be whether it’s

sets or symmetry groups, or whatever. And so we always think

about the whole universe and the relations among the

inhabitants of that universe. So from that point of

view, quantum theory is a very natural way to look

at everything at the same time and kind of bring

some kind of order in that particular universe. And that has to be very

powerful in this way. So one of the earliest successes

of the application of quantum theory in pure math

actually appeared in knot theory,

typically a subject that happened in the 1980s and ’90s. So what is a knot? A knot is a curve in

three dimensional space that closes on itself. And one of the

problems in mathematics was to kind of give a list

of all possible knots. So you could have

imagine there’s like a universe–

there’s a book of knots with infinite number

of pages, where all these different knots are. And how do you distinguish them? And the way this

problem was solved by kind of seeing this in the

language of Feynman diagrams. So you can think of a

knot as a trajectory, a trajectory of a particle

that goes up and down in space. In this case, you have to make a

space-time of two space and one time direction. So you have to make

something with three, not four, dimensions. And for each of these

diagrams, the laws of physics will give you a certain

number– namely, the probability amplitudes that this

actually will happen. So in fact, you can do

even more precisely. You can think of this as,

say, a quark going around in space in one of

these virtual diagrams. And now, it can interact

by kind of shooting gluons as ordinary particle models do. And for each of these

kind of Feynman diagrams, you will get a specific

number that turned out to be incredibly powerful in

solving the mathematical issue. So in some sense,

the knot theory was crying out to

be reinterpreted as a theory of particles

moving in space-time. My second example is an example

from a kind of esoteric subject that’s related to string theory

that was very fashionable in the 19th century. And then it had this incredible

revival in recent years. And it’s called

enumerative geometry. And it’s studying

certain spaces. The most famous is

the so-called quintic. It’s called the

quintic because it’s a function of five

variables that all are raised to the fifth power. If you think about this, these

are five complex variables. You have one equation,

so you have four left. And then there is an obvious

scaling relation to get out. Turns out to be a space of

three complex dimensions. And one thing that

mathematicians were interested in the old days, and still

are, is counting objects. And it turns out, it’s

an interesting question to count the number of curves

you can draw on this three dimensional space. So drawing a curve means that

you have the five variables. Each of them are polynomials

in some extra variable z, so polynomials of degree d. And then you can

study these questions. So how many of these curves are

there in degree 1, in degree 2, in degree 3, et cetera. And you can imagine

that these problems soon become very, very complicated. The case of degree 1– so the

number of lines that you can draw in that space– is a classical result

from the 19th century. So I think every real

algebraic geometry will know there are 2,875

lines on the quintic. Then to go to the next level, so

conics, so the degree 2 curves. It’s already a huge

number, 609,000. And this was computed

in the 1980s. And the next case, there’s

an interesting story to it. The story is it was

computed by mathematicians, because physicists wanted

to know the number. So they asked, can

you compute it? It was a very

complicated calculation, a lot of computer

algorithms involved. And they came out

with a number– not this number. And then the physicists

said, are you sure it’s the right number? Didn’t you make a mistake? And they went back. And, indeed, the mathematicians

found that there was a mistake. And they got this number. But then, of course, how did

you know we made a mistake? And then the physicists,

well, you know, we have all the numbers. And– [LAUGHTER] So it’s hard to bring home the

shock when this first happened, because it’s like this has

become exponentially difficult with each step. So what was going on? How were physicists able to do

all the calculations at once? And of course, the answer

is, again, quantum theory, using the sum of

histories– in this case, thinking of counting curves as

the movement of a string that takes all different shapes

in this particular space. And what quantum

mechanics will do, it will calculate this

amplitude, this probability, by summing over all histories. And so there’s this

kind of nice function. And the coefficients

in the function were the numbers that

you want to compute. So this was the first

insight, that these numbers, you should not look

at them individually. You should see them

all at the same time. But there was a second

thing that has happened, which was like very,

very much a surprise. Physicists were interested in

these spaces in string theory, because they’re used to

wrap up internal dimensions. I won’t go into this, but

so they wanted to know, what kind of these

kind of spaces exist, these so-called

Calabi-Yau manifolds? In fact, in physics, the

choice of such a manifold would lead to a

particular particle model. So part of understanding

the physics is understanding all

the models, and so also understanding

all the spaces. And originally, they made a map. So what physicists

do, they make a plot. These spaces were

characterized by two numbers. And so we could make this

kind of scatter plot. And looking at the

plot, they said, well, there’s a symmetry there. It looks like,

for every space X, there is another space

on the other side that’s very different. But it seemed to

share some properties. And this was a major discovery. It’s called mirror

symmetry, because there’s like a reflection that goes

from the left to the right. And it turns out that with

these two spaces, which really couldn’t be more different,

seen as real spaces through the eyes

of quantum theory, they certainly became the same. It turns out that the

quantum mechanical properties of the two spaces

were identical. And so it turns out

that on the one hand you were doing these very

complicated calculations– these were the numbers that

the mathematicians were trying to prove one by one. The same calculation on the

other space, the mirror space, turned out to be a very

simple calculation– a classical calculation

that people could do. So certainly, by using

the Rosetta Stone and going to the other side,

using the other language, a very difficult problem

was actually solvable. So this was a great success. In fact, this is the beginning

of a wonderful program that led to very terrific results. And so we could declare victory. But now, there’s kind

of a thing that’s kind of unfortunate,

which is that somehow physics and mathematics

doesn’t seem to commute. That is to say, these

things have been proven. But the mathematical

proofs didn’t make the physical intuition precise. In fact, the mathematical

proofs do not use the other side

of the equation. They basically stay on one side

and just prove the results. They just prove that the

numbers are what they are. And so it’s very difficult

in some sense to– physicists had hoped that

mathematicians would somehow provide firm footing of

their very intuitive and kind of shaky ideas. But it hasn’t happened. So one important theme

across this lecture is that is it true that physical

intuition might point you to certain equations,

certain relations. I’m not quite sure whether

the physical intuition is enough to finally

make these proofs. And this is particularly

relevant for the third topic that I want to discuss, which is

the realm of quantum geometry. So I think basically physicists

are telling mathematicians, this is only the beginning. You have to more thoroughly

and deeply question what you mean by

geometry and by algebra and really think of something

very, very different. And the reason for this is

that essentially what is, I would say, an existential

crisis in physics and has to do with the

really violent phenomenon that we study in the cosmos– so I would say black holes

and the big bang singularity in our cosmological evolution. So the remarkable

thing of this is we have this very beautiful

theory of Einstein’s describing space and time

and its curvature. But it has some

kind of mistakes, or it has some kind of gaps– very important gaps,

which are singularities. So if you think of

the cosmic evolution, there are two points

that we worry about– the big bang, which is the

beginning of time and part where in some sense

Einstein’s theory breaks down. And in a similar way,

Einstein’s theory breaks down in places

where time ends– that is, inside black holes. Now, again, there’s

a great quote of John Wheeler about this. He said, and this

is from the 1960s, “the existence of

spacetime singularities presents an end to the principle

of sufficient causation”– what happened before the

big bang, essentially– “and so to the predictability

gained by science.” How could physics lead

to a violation of itself? How could physics

lead to no physics? And so that’s one of the

fundamental questions that modern physics

is involved with– how could it be that our

great theory has this kind of built-in deficiencies? And we feel that black holes

are, in some sense, the way forward. Many of us feel

that the black hole has the same kind

of center stage now that the atom or the

particle had 100 years ago. And that led to the birth

of quantum mechanics, a great revolution

of our thinking, that, as I tried

to indicate, also revolutionized our

mathematical language. I think we are now

at a stage where we need a further

revolution of that kind. And black holes are interesting,

because I know as it’s said, they are very

paradoxical subjects. On the one hand, they are

the most simple objects. It’s just a hole in space. On the other hand,

they’re the most complex, because it’s the most efficient

way to condense information. In physics, we

think we have a way to calculate the information

content of a black hole. And we think it’s equal to

the size of the horizon– so that’s kind of

the no-go area that surrounds the space-time

singularity– measured in this Planckian unit. So in the bits of it– so we think roughly that is kind

of the schematic view of one bit of information placed over

each kind of square Planck length unit on the

surface of this horizon. So black holes

should kind of marry quantum mechanics and gravity

in a very meaningful way. And again, black holes

provides us a dictionary. It’s not a Rosetta Stone. It’s a Rosetta

Stone that compares the laws of thermodynamics

and the laws of general relativity– quantum gravity in the

presence of black holes. So as we said,

there is the concept of entropy, the

amount of information that you can store in a

statistical mechanical system as measured by the

horizon of a black hole. We feel that the second

law of thermodynamics, that entropy always increases,

has an equivalent in black hole physics, if two

black holes merge that they create a black

hole whose horizon is larger than the sum of the two. And of course, with this

recent LIGO discovery, we actually have physical

evidence of this process. And indeed, it’s kind

of worked, so to say. And finally, I think there is

something like a temperature concept too in black

holes, because there’s the concept of

Hawking radiation. Black holes, under the

laws of quantum theory, are not strictly speaking black. They are some way

to emit particles. So again, this is

an example where there seems to be an analogy. And this analogy could

be more than an analogy. We are essentially looking

for the equal sign that puts these two worlds together. Now, there are, again,

from string theory some very interesting ideas. And so I want to share

a few of these with you. The way apparently forward

to describe these black holes is using a technology using

so-called branes, which are objects inside space,

where strings can end. So in string theory, apart

from the closed strings that I described before,

there could also be open strings– little open lines–

that can be end on something. And that’s what

you call a brane. And these branes are

terrific, because the mathematical language

that you need to describe them turns out to be the language

of Yang-Mills theory, of gauge theories of matrices. So in some sense,

again, we are given a gift, namely a

mathematical framework that describes these

objects that looks very different from geometry. It’s actually an

algebraic framework. It’s studying the

same matrices that are responsible for the

symmetries, for instance, in the standard model– the quarks that you saw. So here’s my kind of

cartoon version of what is kind of happening here. So you can think of this

a little bit as follows. So we think of this

kind of branes, it’s not an accurate

description, but for a cartoon

version, it will suffice. Think a little bit as

the branes at the horizon of the black hole. So in string theory, we have

kind of these closed strings that essentially they should be

responsible for the space-time and its curvature. And then you can have these

so-called open strings. And well, intuitively,

you can think of them as kind of little strings

that have been falling halfway through the horizon. So they’re kind of just sticking

out and waving, help, help. And it’s these kind

of open strings on the surface of the

horizon does some capturing– that’s the string

theory cartoon version of capturing the information

inside the black hole. In fact, here’s my

kind of animation that would look like how Hawking

radiation could look like. You can think of two

of these open strings meeting and touching

and then forming a closed string that

kind of lifts off and is able to escape. So remarkable that within the

framework of string theory, there have been very precise

calculations, only very specific examples of these

kind of quantum black holes. So we feel that we

are at least pointing towards the right

language to capture the singularity of space-time. This was all kind

of brought together by a really fascinating

paper that somehow changed a lot of physics for the

last 20 years, the paper by Juan Maldacena, who

proposed something else. He said, you know, there

is actually another way to capture not

only the black hole but capture actually

space-time itself in terms of a theory

that in his language was kind of living on the

boundary of space-time. So this is the famous

AdS/CFT correspondence. And from a mathematical

point of view, the statement is

actually the following. It saying, well, again,

there is a Rosetta Stone. There are two

languages that should be talking to each other. On the one is the

language of gravity. So it’s the language of

space-time and curvature, and gravitational waves, and

black holes, and all you have. On the other hand, there’s the

theory of quantum gravity– of quantum mechanics,

sorry, that’s living on the

boundary of that space and should be

entirely equivalent. So on the left hand side,

you wouldn’t see anything like space-time curvature. You would typically have

quantum mechanical effects. On the right hand side,

you do find these objects. You would find space-time and

curvature, and everything. And what we had been

doing in the last, say, two decades in

physics is really constructing this dictionary

and extending the two– the left side and

the right hand side– and finding all these

magical equations. Again, the equal sign is

incredibly important here. In fact, this kind of idea

of Clinton’s Principle, what do we exactly mean by

is, the equality here, is taking a lot of

mental energy in physics, because what do we mean

exactly if these two are equal? Is the left hand side

define the right hand side? Or to which extent

are both well-defined? Who is helping whom in

this particular situation? But I think it’s

entirely fascinating, because it’s all pointing

out to something very deep. It’s pointing out

something that perhaps the ultimate physical laws

that we’re looking for are not at the– I mean, reductionism isn’t

somehow our guiding principle. We have to think in terms of

the other kind of beauty– the emergent kind of beauty. Of course, for

Einstein, in some sense, for him the ultimate dream was

to make the whole world out of geometry. I think he really

thought geometry is such an essential way to

look at nature that he felt it was not only the

foundation, but in the end, everything should be

created out of geometry. The later years in his life, he

spent a lot of time producing particles, in some sense, as

little knots in space and time. I think the modern

point of view of physics is that geometry is not

the ultimate foundation. But it’s like a

basement below that. You can go even deeper. And in some sense, this

kind of quantum theories, by themselves, are perhaps

a much more fundamental description of what nature is. And so in some sense, the space,

time, and gravity, and perhaps also the objects

moving in space-time aren’t a fundamental

description, but in some sense

are emergent out of something more

fundamental that I think we honestly are lacking

the mathematical language to find that. One thing which is interesting,

there were two theories that Einstein thought

the most beautiful. One was the laws

of thermodynamics. And the other were his own

laws of general relativity. And so, in fact, if this kind

of analogy becomes an equality, it will be very satisfying,

because there would, in some sense, be the same. Now, in mathematics,

I would say, though, there are small hints of

what the kind of language is that we need here. And I just want to end by

giving a very tiny example of something which

presents something of this kind of

emergent geometry. In this example are the

statistical analysis. So there’s a famous

field in mathematics that studies random matrices. And in fact, its birth came in

the study of nuclear spectra, where people would think, well,

it’s very difficult to exactly compute the spectrum

of a specific system, why not make a kind of

statistical ensemble, thinking of, in

some sense, a model, where you have random matrices

with a certain distribution, a Gaussian distribution. Now, if you have such a

matrix, it has eigenvalues. And these eigenvalues will be

somewhere on the real line. So these are the eigenvalues

of this particular matrix. And if you have

matrix of rank N, they are called distributed

with a certain width. And we have found that if

you make the matrix larger and larger and larger,

these eigenvalues start to kind of cluster. In fact, they form

a very natural kind of a eigenvalue

density, in fact that has a very beautiful shape. That’s the shape of a circle. So a very elementary

geometrical object, the circle, is emerging by studying

this system, which you can think of as a

completely caricature quantum mechanical model

that has a variable, namely the rank of the matrix. And if you increase

the rank of the matrix, so increase the

complexity of the system, then you slowly

approach this kind of classical geometrical shape. In fact, this is a very

precise mathematical field. And there are many

other similar models. And indeed, in

this realm, you can find something which

is a very nice example of emergent geometry. So if you study the

statistical mechanical models and you take the limit,

where essentially the rank or the number of particles

becomes infinite, you see that the model

is captured by very classical geometrical shapes. If you make N large

but not infinite, you can do a 1/N approximation. And you can look also at

these systems for finite N. And you find something

which I think is a good example of what

would be quantum geometry. But I think we

have to be honest, that we are in some sense still

lacking the right language. So my grand vision

is that by studying many of these examples,

like we have had before in the time of

classical mechanics, that perhaps a new

vocabulary will emerge, something perhaps even a new

breed of mathematicians that feel at home in a natural

way in quantum mechanics. Now, if you look at

the system right now, we see there are all

these wonderful examples. There’s tons of

exciting developments. There are many, many areas in

mathematics that are involved. But it’s not yet a

complete picture. Reminds me of one of my– there are many jokes about

mathematicians and physicists. And often, they are at the

cost of the mathematician, I’m afraid. But– [LAUGHTER] –one of my favorites

was actually told by Alain Connes,

the Fields Medalist. And he said, well,

there was the physicist, and he had the very big

bag of dirty laundry. And he went into town. And he was looking to a

place to do his laundry. And suddenly, he’s a shop,

and it says laundromat. In fact, it not only

says laundromat, it says restaurant and cafe,

and lots of other– hotel. And he comes in. And of course, the shop

is run by a mathematician. And he says, can

do my laundry here? He said, no, no, no,

fortunately, you cannot. He said, but I saw the sign. No, no, no, this is a

shop that only sell signs. [LAUGHTER] And sometimes, some

physicists feel there’s all these grand

topics, but it’s like, can I do my laundry somewhere? But I would say there’s

another, perhaps more a deeper philosophical point of view. As I said, in some sense, we

have these different languages. And we are finding

dictionaries– dictionaries between

algebra and geometry, between quantum theory

and classical theory, between general relativity

and quantum mechanics. And it’s a little bit like

ordinary natural languages. There’s certain things you can

say in English that I cannot say in Dutch, and vice versa. Perhaps, it’s like

describing the Earth in terms of an Atlas and maps. So you have various

maps, and you have ways to go from

one map to the other. So this could be,

in the end, the way we have to describe

the world, so to say. Of course, that’s

one point of view. But I think that the real

intellectual challenge is to kind of find

some kind of a unifying theme with all these

examples and all these ideas. It would be incredibly powerful. It will be the

language that would be helpful to capture

the fundamental equations of physics. But it also would be a great

unifier in mathematics itself, because one thing that we see

that physics has been doing– quantum physics in particular– it has been kind

of crisscrossing these various mathematical

fields with kind of very little respect

for the natural boundaries of these various topics. So if I want to end,

it’s with a dream that perhaps at

some point, there is somebody will

find me a globe, and we have quantum mathematics

as one specific subject. Thank you very much. [APPLAUSE] So now, we can have a question

and an answer session. There are two microphones

on both sides. So you can line up

and have questions. Yes. So you mentioned

atoms in space-time. And then you never

really got back to it. But you sort of

brought in black holes as maybe the equivalent of the

atoms of quantum mechanics. I mean, is it inevitable that

space-time, or space or time, is quantized and that there

are quanta of space-time, or space or time? And is that role filled

somehow by black holes? Is that something like

what you’re saying? It’s very confusing. Well, I think you’re

quite right pointing out. I used kind of the

concept of atom twice. So I think that

the first point is that we do feel that the

classical model of space as having kind of this

infinite number of points. Now, you can zoom in endlessly

to the smallest structure. There’s no limit to it. That actually is an

unphysical model, because it’s like strictly

impossible to measure differences between space

and time with experiments. At some point, you will

have to put so much energy in such a small area that

actually the laws of physics will prevent you

from doing this. So I think we are

in general agreement that there is nothing– the classical mathematical

model of space, Euclidean space, is not the right one to

describe physical space, because there is some

kind of natural cutoff. So when I’m talking

about atoms of space, I’m not thinking of

literally that there are all little blobs or something. So the analogy of the

pixels in a computer screen is not a right one. I think it’s– a better analogy

would be the 0’s and 1’s that encodes the picture

as a computer file. So I think we do

feel that there’s like a finite amount of

information that can be stored in a certain area of space. And a black hole is one

way to capture the amount. So you can think you create

a little hole in space, and you ask, how much

information is there? And the black hole

will compute for you the amount of information. Of course, if you

have classical space, you could put an infinite

amount of information, because you could create units

that are infinitesimally small. So in that sense, I think the

two concepts are connected. More questions? Any others? [INAUDIBLE] Perhaps, you can hand the

mic or something, yeah. Can you comment on two

recent developments– one that you call

amplitudology and the other, the Erik Verlinde’s

work on volume law and its relation to dark matter? OK. Let’s start with the first. So one part of this

struggle, I think, to find the right

mathematical framework is that even in

the physical models that we use, like

quantum field theory, I would say that there’s

kind of general agreement that although we all love the

Feynman diagrams and the gauge fields and all of

that, that’s only a partial description of

these physical models, because in some sense, they

have a much richer structure and deeper symmetries. So the amplituhedron

and this amplitudology is another approach,

where you say, well, it’s almost going

back to the black box that I started with. I joked that the

black box could be opened, that the formula of

the standard model was there. But I mean, perhaps, this

kind of abstract point of view and saying, what is

a physical theory, it’s a machine that computes

certain quantities for us. So there’s an input,

there’s an output, there are certain

consistency conditions. That idea totally resurfaced. And so, in some sense, physics

is now trying to kind of map out what is the space of

possible physical theories. And so this is one

particular approach to it. It’s a very top-down approach. And it’s very exciting

to see to which extent we can push that approach,

and whether, I mean, perhaps– it’s almost like understanding

physics without understanding physics, because you just

describe all the objects, but you literally are

not able to open them up. We’ll see. The second one, I think,

Erik Verlinde’s work, I think there is– in some sense, intuitively, I

feel what he’s going to say– trying to say. And I think he’s making the

point that many of us make, that space and time should

be emergent quantities. The relation with dark

matter is very tempting. Actually, I don’t

really understand the details of his argument. It would be terrific in some

sense, if something like that would be true. But I’m not yet there that

I would claim I understand. More questions [INAUDIBLE]? [INAUDIBLE] Yes, there. Yeah. So in the slide where you

compare the quantum theory and gravity theory,

I just noticed that you use a illustration by

MC Escher on the gravity side. So I just want to

know if there is any meaning behind this detail? Sorry, which equation? Sorry, I missed that. The slide where you compare

the quantum with gravity. Yes. Yeah, I think you used

an art by MC Escher. Oh, OK, so that’s, yeah, why

was the picture of Esther there? Because the space

that that particular– is the so-called

anti-de Sitter space, which is negatively

curved space. So this particular–

the original application of this idea was

in space-times that essentially have a boundary. So they are negatively curved. And so that’s why

you get the Escher picture, where you see more and

more objects on the boundary. So one of the issues in

studying this kind of dictionary is, exactly what kind of spaces

can you have– can you produce in that way? And so I think this is something

that’s actively investigated, whether you can have,

for instance, de Sitter spaces, which are the

spaces that actually are relevant for our cosmology? And what would be the

appropriate object on the left hand side? So again, I think

it’s a good example of these kind of deep

identities that we also see in math, that my equal sign,

certain things are connected. But on both worlds,

there are many models yet where we don’t have

the specific connection. Yes. So please excuse me if

this question is completely missing the point. [LAUGHS] But you described these concepts

of particles and antiparticles created by moving

forward and back in time. Yeah. And so two questions

arise from that. The first is, can

that model explain– does it correlate with why there

is more matter than antimatter in the universe? No, and actually that– Or is it sort of

inconsistent with that idea? Exactly, yeah. And then the other

question is, when a particle and

antiparticle collide, they destroy each other

with an amount of– there’s a release

of energy which is more than the energy

of just the particle. So where does that

energy come from, if it’s a particle that’s

moving backwards in time? It seems like creating

energy out of nowhere. So your first question

was apparently exactly the answer that Feynman

gave when Wheeler him up. So he said, well,

why aren’t there equal number of particles

and antiparticles? And of course, this

is why it’s wrong. There’s not only one

electron in the universe. There are many. I’m composed of many, you are. So there’s a net

amount of electrons. It’s more about–

so the right way, if you think of the movement

of a single electron, actually even the

ones in our atoms, they can make these kind

of wiggle movements. And they do. Actually, you can see this

in physical experiments. And what it does

is create a cloud of particles and antiparticles

around a physical particle. So it’s not true that

there’s one electron. But still, this idea

of going back in time is part of physics. The second question is that

these particles are not real particles. They’re so-called

virtual particles. They’re particles that are

used in our calculations. So in this process where

you create a particle and antiparticle,

you don’t violate the conservation of energy. So one particle is

positive energy, the other one has negative. The one with negative

is, therefore, not a physical particle. So if they annihilate, they

annihilate give zero energy. So again, you can think of them

more as calculational schemes. They’re not

observable particles, although the indirect effect of

these diagrams, these Feynman diagrams, we can measure. So I did have the same

response as Richard Feynman? Yes, yes. [LAUGHTER] [APPLAUSE] Now, apply it and

earn a Nobel Prize. That’s– [LAUGHTER] [INAUDIBLE] Yes, step one,

reproduce step one. More questions? Yeah. Yes. So what evidence to

axiomatic are usually in this quantum matter is like? So do we have axiomatic basis? Like, is this activity

category theory? So what happens

to axiomatization. Oh, that’s the golden

question– what are the axioms? What is an actual

way to build this? I think, we– I don’t know, I think, you know? So one thing I would love

to see is a generalization of the concept of a geometrical

space, that in some sense includes these kind of quantum

corrections in a natural way. It’s not even clear, I

think here, which has to go. It’s not clear that we

have to adapt to a quantum point of view on everything,

because as I said, the quantum theories that we

know are themselves limited. And our own understanding

of these theories is wrong. So I think we’re looking

kind of a concept that in some sense kind of

unify these two concepts. So but there is the

current axiomatic basis, like bottom-up of this

quantum [INAUDIBLE].. There are certain axiomatic

ways to approach quantum theory. But I think, in general, they

have been kind of failing. And so the question

just got from kind of top-down approach,

which is looking at this kind of

bootstrap or amplitude, is some way to have

other information. It’s very difficult to give

an axiomatic definition of these quantum systems,

because easily your axioms include lots of stuff that

you don’t want to have, or they exclude the

interesting cases. So I think if we would

know the right axioms, we’d be in a great position. But I think perhaps that’s

what we’re aiming for. And so the attempts

that the physicists do of kind of mapping

out a landscape– which is a very

fragmented landscape. It has many kind of islands and

mountain ranges and everything, and lots of surprise new

areas that we didn’t– it’s all about understanding,

in some sense, what is this kind of universe

of physical theories that we’re trying to study? Yes. Hi. Professor– sorry–

Professor Dijkgraaf, thank you very

much for your talk. So my question is not going

to be very well-formulated. But it’s generally

regarding your comment on mathematical rigor

versus physical intuition. Yes. So in your few examples,

the calculation in Gromov-Witten

theory is very precise. And you said there was

a mathematical proof that I can actually show

you that this works. However, on the side

of, for example, the AdS/CFT correspondence

that you mentioned, I think the

mathematical community, or the physical

community, don’t have agreement on how well

this correspondence is really established. And I have heard a lot

of debates, personally, about the subject. So I’m just wondering if

you have any comment on that as to how should physical

intuition work well with mathematical rigor? And to what extent

should physicists care about mathematical

rigor without satisfying, say, their productivity

and imagination on the physical side

in generating these– [LAUGHS] [LAUGHTER] –if that’s the case at all. These are discussions

for over drinks, I think. But I think, one thing– the

one way I see it is that, so in physics, we

have these very kind of intuitive but very grand

claims of what the two worlds should be that we unify. What has happened often is that

you take these grand visions, and then you condense

them in a very specific mathematical equation that

should be the consequence of that grand framework. And I think it’s

the great progress that we make in mathematical

proof is proving some of these consequences. So I think even, for

instance in AdS/CFT, there are some specific

cases where this boils down to a certain identity. There’s a certain model or

form, and it can be expanded. I worked on this Fourier

series, and there’s like an identification. And so there are some

cases where there is– I would say, often a

[? framework ?] follows. You know, you had Plato’s

cave, where there are all these mathematical deep

truths and we could only see the reflections. I think this is something

like the opposite, in the sense there is like

these deep insights in physics, which are very hard to access. What we see is the

projections in terms of concrete

mathematical conjectures that then can be proven or not. And I think it’s

a good first step. But the physics, I mean,

they are wonderful results. And people have been

celebrating them. But now, for physicists,

I think we really want to have a conceptual

understanding why this is true. And I think we feel this is

only true the moment you really analyze the underlying

beast, so to say, which is the physical system. If you understand that system,

then this equality, A equals B, will be obvious. But now, it’s just

a lot of hard work. So in that sense, it’s– For physicists, I think this

intuition is more than just a way to get interesting

mathematical results. It’s actually a deep question

that we want to answer, because it will point us to

the nature of the objects– of reality perhaps. So I think the role of

these mathematical equations is that it sharpens

the physical thinking. And I think it has. And another really

remarkable development is that mathematicians

have taken these ideas and generalized them, sometimes

in a very spectacular way, which is kind of fascinating. So in some sense, a little

bit like a tennis game. So perhaps, this other case is

the service was the physicist. But then mathematicians

return the ball and by generalizing

some of these concepts in ways that the

physicists are now scrambling to find whether

their intuition can kind of cope with these next steps. More questions? Yes. Hi. What is the source of the

almost perfect connection between mathematics and physics? And can the universe

really be described purely mathematically? OK, that’s another– I think, we’re getting to the

ends of the discussion session, I think. [LAUGHTER] Well, I can just answer

with some further question, because there’s a huge

debate about this. So of course, the– it’s a great gift that we have

been able to capture so much of physics by

mathematical equations. And there’s a deep

debate whether this is a reflection of the nature

of reality or it’s a case, well, our brain is only

able to capture those things and understand those

things that we can describe in mathematical language. So a large part of it, we might

just totally be unaware of. My own personal point of view

is that, actually going back, I would subscribe what

Galileo and others said, that mathematics is

probably the right language to capture nature. However, as I tried to indicate,

it’s an evolving language. And like also

natural language, it grows– there are new

words, new concepts. So I think, in

its present state, it only captures a

small part of reality. But no, it has been

a terrific gift. And perhaps it’s a

good closing line, that mathematics has

been evolving whenever it interacted with

other fields of science, yet keeping its unity. That’s a remarkable

thing, that in some sense it has never really

turned to something else. It just has expanded, became

a more richer language with richer grammar,

richer subjects, richer words, so to say. And I think it’s, I

think, the deep belief of many mathematicians that

this will continue, whether it’s interactions with quantum

theory or any other field, that in the end

mathematics will turn out to be the right

language to capture it. One more question? Yes. OK, last question. OK, sorry. So something I found

interesting that you said was black holes are

the most efficient way of storing information. So I was thinking, if a black

hole can store information, is there any way for that

information to be read? [LAUGHS] I apologize– That’s a very good question. There’s some great

experts here in the room that can talk about it. I think it would go too far too. But let me just say

that one thing that has happened in the last,

again, a couple of years is that the whole field of

quantum information as it’s used in very practical

applications– in designing computers

or quantum computers– in some sense turned out to

be, again, a subject that’s very relevant for

studying all of this. So I would say, it’s, again,

a certain area of science, of mathematics, being

brought into the fold. And it led to some, again,

remarkable equations. I’m not sure whether

this is the ultimate way to understand reality, but

it’s certainly part of it. So these things are

actively discussed. Well, let’s thank– oh. Let us– oh, you want to

say something [INAUDIBLE].. No, I want to ask a question. [LAUGHS] He has to take a plane. He’s taking the red-eye. But I will get [INAUDIBLE]. Talk to Professor Susskind. He will give you

all the answers. [LAUGHS] Robbert? Yes. It seems to me

mathematics [INAUDIBLE] at least two different ways. One way is just to very– alter

very, very simple principles. Take Newton, Newton had– was

just a couple of sentences that could find basic

Newtonian physics– Yes. –all of it. And then there’s all the

mathematics which comes out from solving those things. Yes. And those mathematics

can often be very subtle, very complicated, and elliptic

integrals and all sorts of stuff [INAUDIBLE]. Right. How do you know which is which? [LAUGHS] Calabi-Yau manifolds,

do they have to do with the

fundamentals principles? Or do they have to do with

very complicated solutions and some very [INAUDIBLE]? You make a terrific point. And I would say the best

example is number theory, right? So it’s very easy

define numbers. But then number theory is

this incredibly rich field, where almost any

problem is insolvable. And there are very,

very deep themes. I think, actually, we are

confused at this point. And I think somehow

the two ways– the kind of reductionist

and emergent point of view– might be even confusing

these two points of view. So I would say,

at this point, we have no idea what the actions

are, the laws that Newton posed with very, very simple math. And what are the applications,

what are the technology? I personally feel we are in the

technological point of view. I think, we are

solving equations, but we haven’t yet discovered

the laws themselves. I think that would be my guess. So we , in some sense

work backwards, I think. We’re studying– We develop

all these very refined tools to analyze solutions,

but we haven’t really discovered solutions

to what equation. [INAUDIBLE] rules and not

knowing Newton’s laws. Exactly, exactly. Yes. Well, it could be a grandmaster

in elliptic integrals in the 19th century, right? [LAUGHS] OK, with that,

let’s thank Robbert. [APPLAUSE]