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
chair@math.stanford.edu.
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]
