Kind: captions
Language: en
The following is a conversation with
Terrence Tao. Widely considered to be
one of the greatest mathematicians in
history. Often referred to as the Mozart
of math, he won the Fields Medal and the
Breakthrough Prize in mathematics and
has contributed groundbreaking work to a
truly astonishing range of fields in
mathematics and physics.
This was a huge honor for me for many
reasons, including the humility and
kindness that Terry showed to me
throughout all our interactions. It
means the world. This is the Lex
Freedman podcast. To support it, please
check out our sponsors in the
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And now, dear friends, here's Terren
Tao.
What was the first really difficult
research level math problem that you
encountered? One that gave you pause
maybe. Well, I mean in your
undergraduate um education, you learn
about the really hard impossible
problems like the reman hypothesis, the
twin primes conjecture. You can make
problems arbitrarily difficult. That's
not really a problem. In fact, there's
even problems that we know to be
unsolvable. What's really interesting
are the problems just at the on the
boundary between what we can do
relatively easily and what are hopeless.
Um but what are problems where like
existing techniques can do like 90% of
the job and then you just need that
remaining 10%. Um I think as a PhD
student the CA problem certainly caught
my eye and it just got solved actually.
It's a problem I've worked on a lot in
my early research. Historically it came
from a little puzzle by the Japanese
mathematician Soji Kaya uh in like 1918
or so. Um, so the puzzle is that you you
you have um a needle um in on the plane.
Um think like like a like driving like
on on on a road something and you you
want it to execute a U-turn. You want to
turn the needle around. Um but you want
to do it in as little space as possible.
So you want to use as little area in
order to turn it around. So um but the
needle is infinitely maneuverable.
So you can imagine just spinning it
around its um as a unit needle. You can
spin it around its center. Um, and I
think, um, that gives you a disc of of
area, I think pi over four. Um, or you
can do a three-point U-turn, which is
what they we teach people in in the
driving schools to do. Uh, and that
actually takes area pi over 8. So, it's
it's a little bit more efficient than um
a rotation. And so, for a while, people
thought that was the most efficient uh
way to turn things around. But,
Mazikovich uh showed that in fact, you
could actually uh turn the needle around
using as little area as you wanted. So
0001 there was some really fancy multi-
um u back and forth U-turn thing that
you could you could do that that you
could turn a needle around and in so
doing it would pass through every
intermediate direction. Is this in the
two dimensional plane? This is in the
two dimensional plane. Yeah. So we
understand everything in two dimensions.
So the next question is what happens in
three dimensions. So suppose like the
Hubble space telescope is tube in space
and you want to observe every single
star in the universe. So you want to
rotate the telescope to reach every
single direction. And here's unrealistic
part. Suppose that space is at a
premium, which it totally is not. Uh you
want to occupy as little volume as
possible in order to rotate your your
needle around in order to see every
single star in the sky. Um how small a
volume do you need to do that? And so
you can modify basic construction. And
so if your telescope has zero thickness,
then you can use as little volume as you
need. That's a simple modification of
the two dimensional construction. But
the question is that if your telescope
is not zero thickness but but just very
very thin some thickness delta what is
the minimum volume needed to be able to
see every single direction as a function
of delta. So as delta gets smaller as
you need gets thinner the volume should
go down but but how fast does it go
down? Um and the conjecture was that it
goes down very very slowly um like
logarithmically um uh roughly speaking
and that was proved after a lot of work.
So this seems like a puzzle. Why is it
interesting? So it turns out to be
surprisingly connected to a lot of
problems in partial differential
equations, in number theory, in
geometry, comics. For example, in in
wave propagation, you splash some some
water around um you create water waves
and they they travel in various
directions. Um but waves exhibit both
both particle and wave type behavior. So
you can have what's called a wave
packet, which is like a a very localized
wave that is localized in space and
moving a certain direction in time. And
so if you plot it in both space and
time, it occupies a region which looks
like a tube. And so what can happen is
that you can have a wave which initially
is very dispersed but it all comes it
all focuses at a single point later in
time. Like you can imagine dropping a
pebble into a pond and ripples spread
out. But then if you time reverse that
that um that scenario and the equations
of wave motion are time reversible. You
can imagine ripples that are converging
um to a single point and then a big
splash occurs um maybe even a
singularity.
Um and so it's possible to do that. Uh
and geometrically what's going on is
that there's always s of light rays. Um
so like if if if this wave represents
light for example um you can imagine
this wave as a superp position of
photons um all traveling at the speed of
light. They all travel on these light
rays and they're all focusing at this
one point. So you can have a very
dispersed wave focus into a very
concentrated wave at one point in space
and time, but then it defocuses again
and it separates. But potentially if the
conjecture had a negative solution. So
what that meant is that there's there's
a very efficient way to pack um tubes
pointing different directions into a
very very narrow region of of of very
narrow volume. Then you would also be
able to create waves that start out some
there'll be some arrangement of waves
that start out very very dispersed but
they would concentrate not just at a
single point but um um there'll be a
large um there'll be a lot of
concentrations in space and time and uh
um and you could create what's called a
blowup where these waves their amplitude
becomes so great that the laws of
physics that they're governed by are no
longer wave equations but something more
complicated and nonlinear. Um and so in
mathematical physics we care a lot about
whether certain equations in in wave
equations are stable or not whether they
can create um these singularities.
There's a famous unsolved problem called
the Navia Stokes regularity problem. So
the Navia Stokes equations equations
that govern the fluid flow for
incompressible fluids like water. The
question asks if you start with a smooth
velocity field of water can it ever
concentrate so much that like the
velocity becomes infinite at some point
that's called a singularity. We don't
see that um in real life. You know, if
you splash around water on the bathtub,
it won't explode on you. Um or or have
have water leaving at the speed of
light, I think. But potentially, it is
possible. Um and in fact, in recent
years, the the consensus has has drifted
towards the uh the belief that uh that
in fact for certain very special initial
configurations of of say water that
singularities can form. But people have
not yet been able to uh to actually
establish this. The clay foundation has
these seven millennium prize problems
has a million dollar prize for solving
one of these problems that this is one
of them. Of these seven only one of them
has been solved the point conjecture by
Pelman. So the Ka conjecture is not
directly directly related to the Navis
Stokes problem but understanding it
would help us understand some aspects of
things like wave concentration which
would indirectly probably help us
understand the Navis problem better. Can
you speak to the neighbors? So the
existence and smoothness like you said
millennial prize problem right you've
made a lot of progress on this one in
2016 you published a paper finite time
blow up for an averaged threedimensional
navia stoke equation right
so we're trying to figure out if this
thing usually doesn't blow up right but
can we say for sure it never blows up
right yeah so yeah that is literally the
the million- dollar question yeah so
this is what distinguishes
mathematicians from pretty much
everybody else like it
If something holds 99.99% of the time,
um that's good enough for most, you
know, uh for for most things, but
mathematicians are one of the few people
who really care about whether every like
100% really 100% of all um situations
are covered by by um yeah, so most fluid
most of the time um water that does not
blow up. But could you design a very
special initial state that does this?
And maybe we should say that this is a
this is a set of equations that govern
in the field of fluid dynamics. Trying
to understand how fluid behaves and it's
actually turns out to be a really comp
you know fluid is yeah extremely
complicated thing to try to model. Yeah.
So it has practical importance. So this
clay price problem concerns what's
called the incompressible navio stokes
which governs things like water. There's
something called the compressible navio
stokes which governs things like air.
And that's particularly important for
weather prediction. Weather prediction
it does a lot of computational fluid
dynamics. A lot of it is actually just
trying to solve the ny stokes equations
as best they can. Um also gathering a
lot of data so that they can get they
can in initialize the equation. There's
a lot of moving parts. So it's very
important practically. Why is it
difficult to prove general things
about the set of equations like it not
not blowing up? Short answer is
Maxwell's demon. Um so exos demon is a
concept in thermodynamics like if you
have a box of two gases and oxygen and
hydrogen uh and maybe you start with all
the oxygen one side and nitrogen the
other side but there's no barrier
between them right then they will mix um
and they should stay mixed right there
there's no reason why they should unmix
but in principle because of all the
collisions between them there could be
some sort of weird conspiracy that that
um like maybe there's a microscopic
demon called Maxwell's demon that will
um every time a oxygen and nitrogen atom
collide they will bounce off in such a
way that the oxygen sort of drifts onto
one side and then goes to the other and
uh you could have an extremely
improbable configuration emerge. Uh
which we never see. Um and and we
statistically it's extremely unlikely
but mathematically it's possible that
this can happen and we can't rule it
out. Um and this is a situation that
shows up a lot in mathematics. Um a
basic example is the digits of pi
3.14159 and so forth. The digits look
like they have no pattern and we believe
they have no pattern. On the long term,
you should see as many ones and twos and
threes as fours and fives and sixes.
There should be no preference in the
digits of pi to favor let's say 7 over
8. Um, but maybe there's some demon in
the digits of pi that that like every
time you compute more digits, it sort of
biases one digit to another. Um, and
this is a conspiracy that should not
happen. There's no reason it should
happen, but um there's there's there's
no way to prove it.
uh with our current technology. Okay. So
getting back to Nabia Stokes, a fluid
has a certain amount of energy and
because a fluid is in motion, the energy
gets transported around and water is
also viscous. So if the energy is spread
out over many different locations, the
natural viscosity of the fluid will just
damp out the energy and will it will go
to zero. Um and this is what happens um
in um uh when we actually experiment
with water like you splash around there.
there's some turbulence and waves and so
forth. But eventually it it settles down
and and and the the lower the amplitude,
the smaller the velocity, the the more
calm it gets. Um but potentially there
is some sort of a demon that keeps
pushing the uh the energy of the fluid
into a smaller and smaller scale and it
will move faster and faster and at
faster speeds the effective viscosity is
relatively less. And so it could happen
that that it it creates a some sort of
um um what's called a self similar
blowup scenario where you know um the
energy of fluid starts off at some um
large scale and then it all sort of um
transfers it energy into a smaller um
region of of of the fluid which then at
a much faster rate um moves into um an
even smaller region and so forth. Um and
and each time it does this uh it takes
maybe half as as long as as the previous
one and then you you could you could
actually uh converge to all the energy
concentrating in one point in a finite
amount of time. Um and that that's uh
that scenario is called finite blow up.
Um so in practice this doesn't happen.
Um so water is what's called turbulent.
Um so it is true that um if you have a
big eddy of water it will tend to break
up into smaller eddies but it won't
transfer all the the energy from one big
eddy into one smaller eddy. It will
transfer into maybe three or four and
then those must split up into maybe
three or four small edies of their own
and so the energy gets dispersed to the
point where the viscosity can can then
keep that thing under control. Um but if
it can somehow um concentrate um all the
energy keep it all together um and do it
fast enough that the viscous effects
don't have enough time to calm
everything down then this blob can
occur. So there were papers who had
claimed that oh you just need to take
into account conservation energy and
just carefully use the viscosity and you
can keep everything under control for
not just Navia Stokes but for many many
types of equations like this and so in
the past there have been many attempts
to try to obtain what's called global
regularity for Navio Stokes which is the
opposite of final time blow up that
velocity say smooth and it all failed
there was always some sign error or some
subtle mistake and and it couldn't be
salvaged. Um so what I was interested in
doing was trying to explain why we were
not able to disprove um planet time blow
up. I couldn't do it for the actual
equations of fluids which were too
complicated. But if I could average the
equations of motion of naval basically
if if um if I could turn off certain
types of of ways in which water
interacts and only keep the ones that I
want. Um, so in particular, um, if, um,
if there's a fluid and it could transfer
energy from a large Eddie into this
small Eddie or this other small Eddie, I
would turn off the energy channel that
would transfer energy to this this one
and and direct it only into um, this
smaller Eddie while still preserving the
law of conservation of energy. So you're
trying to make it blow up. Yeah. Yeah.
So I I I basically engineer um, a blow
up by changing the laws of physics,
which is one thing that mathematicians
are allowed to do. We can change the
equation. How does that help you get
closer to the proof of something? Right?
So, it provides what's called an
obstruction in mathematics. Um, so, so
what I did was that uh basically if I
turned off the um certain parts of the
equation, so which usually when you turn
off certain interactions make it less
nonlinear, it makes it more regular and
less likely to blow up. But I found that
by turning off a very well-designed set
of of of of interactions, I could force
all the energy to blow in finite time.
So what that means is that if you wanted
to prove um global regularity for Navia
Stokes um for the actual equation you
had you must use some feature of the
true equation which which my artificial
equation um does not satisfy. So it it
rules out certain um certain approaches.
So um the thing about math is is it's
not just about finding you know taking a
technique that is going to work and
applying it but you you need to not take
the techniques that don't work. Um and
for the problems that are really hard,
often there are dozens of ways that you
might think might apply to solve the
problem. But uh it's only after a lot of
experience that you realize there's no
way that these methods are going to
work. So having these counter examples
for nearby problems um kind of rules out
um uh it saves you a lot of time because
you you're not wasting um energy on on
things that you now know cannot possibly
ever work. How deeply connected is it to
that specific problem of fluid dynamics
or just some more general intuition you
build up about mathematics? Right. Yeah.
So the key phenomenon that uh my my
technique exploits is what's called
superc criticality. So in partial
differential equations often these
equations are like a tugof-war between
different forces. So in Navia Stokes
there's the dissipation um force coming
from viscosity and it's very well
understood. It's linear. It calms things
down. If if viscosity was all there was,
then then nothing bad would ever happen.
Um but there's also transport um that
that energy from in one location of
space can get transported because the
fluid is in motion to to other
locations. Um and that's a nonlinear
effect and that causes all the all the
problems. Um so there are these two
competing terms in the Davis equation
the dissipation term and the transport
term. If the dissipation term dominates,
if it's if it's large, then basically
you get regularity. And if um if the
transport term dominates, then uh then
we don't know what's going on. It's a
very nonlinear situation. It's
unpredictable. It's turbulent. So
sometimes these forces are in balance at
small scales, but not in balance at
large scales or or vice versa. Um so
Navis Stokes is what's called
supercritical. So at at smaller and
smaller scales, the transport terms are
much stronger than the viscosity terms.
So the viscosity are the things that
calm things down. Um and so this is um
um this is why the problem is hard in
two dimensions. So the Soviet
mathematician ladish skaya she in the
60s shows in two dimensions there is no
blow up and in two dimensions the nav
equations is what's called critical the
effect of transport and the effect of
viscosity about the same strength even
at very very small scales and we have a
lot of technology to handle critical and
also subcritical equations and proof um
regularity but for superc critical
equations it was not clear what was
going on
and I did a lot of work and then there's
been a lot of follow-up showing that for
many other types of superc critical
equations you create all kinds of blow
up examples. Once the nonlinear effects
dominate the linear effects at small
scales, you can have all kinds of bad
things happen. So this is sort of one of
the main insights of this this line of
work is that superc criticality versus
criticality and subcriticality. This
this makes a big difference. I mean
that's a key qualitative feature that
distinguishes some equations for being
sort of nice and predictable and you
know like like planetary motion and I
mean there are certain equations that
that you can predict for millions of
years and or thousands at least. Again,
it's not really a problem, but but
there's a reason why we can't predict
the weather past 2 weeks into the future
because it's a super critical equation.
Lots of really strange things are going
on at very fine scales. So, whenever
there is some huge source of
nonlinearity,
yeah, that can create a huge problem for
predicting what's going to happen. Yeah.
And if the nonlinearity is somehow more
and more featured and interesting at at
small scales. Um I mean there's there's
many equations that are nonlinear but um
in in many equations you can approximate
things by the bulk. Um so for example
planetary motion you know if you want to
understand the orbit of the moon or Mars
or something you don't really need the
micro structure of like the seismology
of the moon or or like exactly how the
mass is distributed. um you just
basically you can almost approximate
these planets by point masses and just
the aggregate behavior is important um
but if you want to model a fluid um like
like the weather you can't just say in
Los Angeles the temperature is this the
wind speed is this for super critical
equations the finance confirmation is is
really important if we can just linger
on the narto's uh equations a little bit
so you've suggested maybe you can
describe it that one of the ways to uh
solve it or to negatively resolve it
would be to
sort of to construct a liquid a kind of
liquid computer, right? And then show
that the halting problem from
computation theory has consequences for
fluid dynamics. So uh show it in that
way. Can you describe this this Yeah. So
this came out of of this work of
constructing this this this average
equation that that blew up. Um so one um
as as part of how I had to do this. So
there this naive way to do it. You you
just keep pushing um um every time you
you get energy at one scale you you push
it immediately to the next scale as as
fast as possible. This is sort of the
naive way to to to to force blow up. Um
it turns out in five and high dimensions
this works. Um but in three dimensions
there was this funny phenomenon that I
discovered that if you if you keep if if
you change the laws of physics you just
always keep trying to push um the energy
into smaller smaller scales. Um what
happens is that the energy starts
getting spread out into multi many
scales at once. Um so that you you have
energy at one scale you're pushing it
into the next scale and then um as soon
as it enters that scale you also push it
to the next scale but there's still some
energy left over from the previous
scale. um you're trying to do everything
at once. Um and this spreads out the
energy too much. Um and then it turns
out that that um it makes it vulnerable
for viscosity to come in and actually
just damp out everything. So um so it
turns out this this direct bush doesn't
doesn't actually work. There was a
separate paper by some other authors
that actually showed this um in three
dimensions. Um so what I needed was to
program a delay. Um so kind of like air
locks. So um I needed an equation which
would start with a fluid doing something
at one scale. It would push this energy
into the next scale but it would stay
there until all the energy from the from
the larger scale got transferred and
only after you pushed all the energy in
then you sort of open the next gate and
and then you you push that in as well.
So um by doing that it kind of the
energy inches forward scale by scale in
such a way that it's always um localized
at one scale at a time. Um and then it
can resist the effects of viscosity
because it's not dispersed. Um so in
order to make that happen um yeah I had
to construct a rather complicated
nonlinearity. Um and it was basically
like um you know like was constructed
like electronic circuit. So I I actually
thank my wife for this because she was
trained as a electrical engineer. Um and
um you know he talked about um uh you
know he had to design circuits and so
forth. And you know if if you want a
circuit that does a certain thing like
maybe have a light that that flashes on
and then turns off and then on and then
off. You can build it from from more
primitive components you know capacitors
and resistors and so forth and you have
to build a diagram and you um and these
diagrams you can you can sort of follow
your eyeballs and say oh yeah the the
current will build up here and then it
will stop and then it will do that. So I
knew how to build the analog of basic
electronic components, you know, like
resistors and capacitors and so forth.
And and I would I would stack them
together um in in such a way that that I
would create something that would open
one gate and then there'll be a clock
that would and then once the clock hits
a certain threshold it would close it
kind of a rude Goldberg type machine but
described mathematically and this ended
up working. So what I realized is that
if you could pull the same thing off for
the actual equations. So if the
equations of water support a computation
so um like if you can imagine kind of a
steampunk but really water punk uh type
of thing where um you know so modern
computers are electronic you know they
they they're powered by by electrons
passing through very tiny wires and
interacting with other electrons and so
forth. But instead of electrons, you can
imagine these pulses of of water moving
at certain velocity and maybe it's
they're two different configurations
corresponding to a bit being up or down.
Probably if you had two of these moving
bodies of water collide, it would come
out with some new configuration which is
which would be something like an ANDgate
or orgate. you know that if the the the
output would depend in a very
predictable way on on the inputs and
like you could chain these together and
maybe create a touring machine and and
then you could you have computers which
are made completely out of water um and
if you have computers then maybe you can
do robotics so I you know hydraulics and
so forth um and so you could create some
machine which is basically a fluid
analog what's called a vonomian machine
so vonomian proposed if you want to
colonize Mars. The sheer cost of
transporting people machines to Mars is
just ridiculous. But if you could
transport one machine to Mars and this
machine had the ability to mine the
planet, create some more materials to
smelt them and build more copies of the
same machine. Um, then you could
colonize a whole planet um over time.
Um, so uh if you could build a fluid
machine, which uh yeah, so it's it's
it's a it's a robot. Okay. And what it
would do it its purpose in life, it's
programmed so that it would create a
smaller version of itself in some sort
of cold state. It wouldn't start just
yet. Once it's ready, the big robot
configuration water would transfer all
his energy into the smaller
configuration and then power down. Okay?
And then like I clean itself up. And
then what's left is this newest state
which would then turn on and do the same
thing but smaller and faster. And then
the equation has a certain scaling
symmetry. Once you do that, it can just
keep iterating. So this in principle
would create a blow up uh for the actual
Navia Stokes and this is what I managed
to accomplish for this average Navia
Stokes. So it provided the sort of road
map to solve the problem. Now this is uh
a pipe dream because uh there are so
many things that are missing for this to
actually be a reality. Um so um I I I
can't create these basic logic gates. Um
I I don't I don't have these in these
special configurations of water. Um, I
mean there's candidates there things
called vortex rings that might possibly
work but um um but also you know analog
computing is really nasty um compared to
digital computing. I mean because
there's always errors um you you have to
you have to do a lot of error correction
along the way. I don't know how to
completely power down the big machine so
that it doesn't interfere with the the
running of the smaller machine but
everything in principle can happen like
it doesn't contradict any of the laws of
physics. Um so it's sort of evidence
that this thing is possible. Um there
are other groups who are now pursuing
ways to make navis blow up which are
nowhere near as ridiculously complicated
as this. Um um they they actually are
pursuing much closer to the the direct
self similar model which can it doesn't
quite work as is but there could be some
simpler scheme than what I just
described to make this work. There is a
real leap of genius here to go from
Navia Stokes to this touring machine. So
it goes from what the self similar blob
scenario that you're trying to get the
smaller and smaller blob to now having a
liquid toying machine gets smaller and
smaller and smaller and somehow seeing
how that
could be used
to say something about a blowup. I mean
that's a big leap. So there's precedent.
I mean um so the the thing about
mathematics is that it's really good at
um spotting connections between what you
think of what you might think of as
completely different um problems. Um but
if if the mathematical form is the same
you you can you you can you can draw a
connection um so um there's a lot of
work previously on what called cellular
automator um the most famous of which is
Conway's game of life. there's this
infinite discrete grid and at any given
time the grid is either occupied by a
cell or it's empty and there's a very
simple rule that uh tells you how these
cells evolve. So sometimes cells live
and sometimes they die. Um and this um
you know um when I was a a student it
was a very popular screen saver to
actually just have these these
animations going and and they look very
chaotic. In fact they look a little bit
like turbulent float sometimes. But at
some point people discovered more and
more interesting structures within this
game of life. Um so for example they
discovered this thing called a glider.
So a glider is a very tiny configuration
of like four or five cells which evolves
and it just moves at a certain direction
and that's like this this vortex rings
this um yeah so this is an analogy the
game of life is kind of like a discrete
equation and and um the flu navis is a
continuous equation but mathematically
they have some similar features um and
um so over time people discovered more
and more interesting things you could
build within the game of life. The game
life is a very simple system. It only
has like three or four rules um to to do
it, but but you can design all kinds of
interesting configurations inside it. Um
there's something called a glider gun
that does nothing to spit out gliders
one at a one one at a time. Um and then
after a lot of effort, people managed to
to create um and gates and or gates for
gliders. Like there's this massive
ridiculous structure which if you if a
if you have a stream of gliders um
coming in here and a stream of gliders
coming in here then you may produce a
stream of gliders coming out. If so
maybe if both of of the um streams um
have gliders then there'll be an output
stream but if only one of them does then
nothing comes out. Mhm. So they could
build something like that. And once you
could build and um these basic gates
then just from software engineering you
can build almost anything. Um you can
build a touring machine. I mean it's
like an enormous steampunk type things.
They look ridiculous. But then people
also generated self-replicating objects
in the game of life. A massive machine a
bon machine which over a huge period of
time and it always look like glider guns
inside doing these very steampunk
calculations. it would create another
version of itself which could replicate.
It's so incredible. A lot of this was
like community crowdsourced by like
amateur mathematicians actually. Um so I
knew about that that that work and so
that is part of what inspired me to
propose the same thing with Navia
Stokes. Um which is a much as I said
analog is much worse than digital like
it's going to be um you can't just
directly take the constructions in the
game of life and plunk them in. But
again it just it shows it's possible.
You know, there's a kind of emergence
that happens with these cellular automa.
Local rules.
Maybe it's similar to fluids. I don't
know. But local rules operating at scale
can create these incredibly complex
dynamic structures. Do you think any of
that is amendable to mathematical
analysis?
Do we have the tools to say something
profound about that? The thing is you
can get this emerg in very complicated
structures but only with very carefully
prepared initial conditions. Yeah. So so
these these these glider guns and and
gates and and so forth machines if you
just plunk down randomly some cells and
you and you will not see any of these.
Um and that's the analogous situation
with Navia Stokes again you know that
that with with typical initial
conditions you you will not have any of
this weird computation going on. Um but
basically through engineering you know
by by by specially designing things in a
very special way you can make clever
constructions. I wonder if it's possible
to prove the sort of the negative of
like basically prove that only through
engineering can you ever create
something interesting. This this is a
recurring challenge in mathematics that
um I call it the dichotomy between
structure and randomness. That most
objects that you can generate in
mathematics are random. They look like
rand like the digits of pi. Well, we
believe is a good example. Um, but
there's a very small number of things
that have patterns. Um, but um, now you
can prove something has a pattern by
just constructing, you know, like if
something has a simple pattern and you
have a proof that it it does something
like repeat itself every so often. You
can do that. But um, and you you can
prove that that for example, you can you
can prove that most sequences of of
digits have no pattern. Um, so like if
you just pick digits randomly, there's
something called low large numbers. It
tells you you're going to get as many
ones as as twos in the long run. Um but
um we have a lot fewer tools to to to if
I give you a specific pattern like the
digits of pi how can I show that this
doesn't have some weird pattern to it.
Some other work that I spend a lot of
time on is to prove what are called
structure theorems or inverse theorems
that give tests for when something is is
very structured. So some functions are
what's called additive like if you have
a function that maps natural numbers
with natural numbers. So maybe um you
know two maps to four three maps to six
and so forth. um some functions what's
called additive which means that if you
add if you add two inputs together the
output gets gets added as well uh for
example multiplying by a constant if you
multiply a number by 10 um if you if you
multiply a plus b by 10 that's the same
as multiplying a by 10 and b by 10 and
then adding them together so some um
functions are additive some are kind of
additive but not completely additive um
so for example if I take a number n I
multiply by the square root of two and I
take the integer part of that So 10 by
square of two is like 14 point
something. So 10 up to 14. Um 20 up to
28. Um so in that case additively is
true then. So 10 + 10 is 20 and 14 + 14
is 28. But because of this rounding
sometimes there's roundoff errors and
and sometimes when you um add a plus b
this function doesn't quite give you the
sum of of the two individual outputs but
the sum plus minus one. Um so it's
almost additive but not quite additive.
Um so there's a lot of useful results in
mathematics and I've worked a lot on
developing things like this to the
effect that if if a function exhibits
some structure like this then um it's
basically there's a reason for why it's
true and the reason is because there's
there's some other nearby function which
is actually um completely structured
which is explaining this sort of partial
pattern that you have. Um and so if you
have these so inverse theorems it um it
creates this sort of dichotomy that they
either the objects that you study are
either have no structure at all or they
are somehow related to something that is
structured. Um and in either way in
either um in either case you can make
progress. Um a good example of this is
that there's this old theorem in
mathematics called sim theorem proven in
the 1970s. It concerns trying to find a
certain type of pattern in a set of
numbers. the patterns that have make
progression things like 3 five and seven
or or or 10 15 and 20 andreli
proved that um any set of of numbers
that are sufficiently big um what's
called positive density has um
arithmetic progressions in it of of any
length you wish um so for example um the
odd numbers have a set of density 1/2 um
and they contain arithmetic progressions
of any length um so in that case it's
obvious because the the odd numbers are
really really structured I can just take
11 13 15 17 I just I can I can easily
find arithmetic progressions in in in
that set. Um but um zerminism also
applies to random sets. If I take the
set of odd numbers and I flip a coin um
and for each number and I only keep the
numbers which for which I got a heads
okay so I just flip coins. I just
randomly take out half the numbers I
keep one half. So that's a set that has
no no patterns at all. But just from
random fluctuations, you will still get
a lot of um um of arithmetic
progressions in that set. Can you prove
that
there's arithmetic progressions of
arbitrary length within a random? Yes.
Um have you heard of the infinite monkey
theorem? Usually mathematicians give
boring names to theorists, but
occasionally they they give colorful
names. Yes. The popular version of the
infinite monkey theorem is that if you
have an infinite number of monkeys in a
room with each with a typewriter they
type out uh text randomly almost surely
one of them is going to generate the
entire screw of Hamlet or any other
finite string of text. Uh it will just
take some time quite a lot of time
actually but if you have an infinite
number then it happens. Um so um
basically the the if you take an
infinite string of of digits or whatever
um eventually any finite pattern you
wish will emerge. Um it may take a long
time but it will eventually happen. Um
in particular arithmetic progressions of
any length will eventually happen. Okay.
But you need that but you need an
extremely long random sequence for this
to happen. I suppose that's intuitive.
It's just infinity. Yeah. Infinity
absorbs a lot of sins. Yeah. How are we
humans supposed to deal with infinity?
Well, you can think of infinity as as as
just an abstraction of um a finite
number for which you you do not have a
bound for um that uh you know I mean so
nothing in real life is truly infinite.
Um but you know you can um you know you
can ask yourself questions like you know
what if I had as much money as I wanted
you know or what if I could go as fast
as I wanted and a way in which
mathematicians formalize that is
mathematics has found a formalism to
idealize instead of something being
extremely large or extremely small to
actually be exactly infinite or zero. Um
and often the the mathematics becomes a
lot cleaner when you do that. I mean in
physics we we joke about uh assuming
spherical cows. um you know like real
world problems have got all kinds of
real world effects but you can idealize
send certain things to infinity send
certain things to zero um and um and the
mathematics becomes a lot simpler to
work with there. I wonder how often
using infinity
uh forces us to deviate from um the
physics of reality. Yeah. So there's a
lot of pitfalls. Um so you know we we
spend a lot of time in undergraduate
math classes teaching analysis. Um and
analysis is often about how to take
limits and and and and whether you you
know so for example a plus b is always b
plus a. Um so when you have a finite
number of terms you add them you can
swap them and there there's no problem.
But when you have infinite number of
terms there these sort of shell games
you can play where you can have a series
which converges to one value but you
rearrange it and it suddenly converges
to another value. And so you can make
mistakes. You have to know what you're
doing when you allow infinity. Um you
have to introduce these epsilons and
deltas and and this there's a certain
type of way of reasoning that helps you
avoid mistakes. Um
in more recent years um people have
started taking results that are true in
infinite limits and what's called
finetizing them. Um so you know that
something's true eventually but um you
don't know when. Now give me a rate.
Okay. Okay, so it's such a if I have
don't have an infinite number of monkeys
but but a large finite number of
monkeys, how long do I have to wait for
H to come out? Um and that's a more
quantitative question. Um and this is
something that you can you can um attack
by purely finite methods and you can use
your finite intuition. Um and in this
case it turns out to be exponential in
the length of the text that you're
you're trying to generate. Um so um and
so this is why you never see the monkeys
create Hamilton. you can maybe see them
create a four-letter word, but nothing
that big. And so I personally find once
you finitize an infinite statement, it's
it does become much more intuitive and
it's no longer so so weird. Um so even
if you're working with infinity, it's
good to finitize so that you can have
some intuition. Yeah. The downside is
that the finite groups are just much
much messier and and uh yeah. So so the
infinite ones are found first usually
like decades earlier and then later on
people finize them. So since we
mentioned a lot of math and a lot of
physics uh what is the difference
between mathematics and physics as
disciplines as ways of understanding of
seeing the world maybe we can throw in
engineering in there you mentioned your
wife is an engineer give it new
perspective on circuits right so this
different way of looking at the world
given that you've done mathematical
physics so you you've you've worn all
the hats right so I think science in
general is interaction between three
things um there's the real world um
there's is what we observe of the
reward, our observations and then our
mental models as to how we think the
world works. Um so um we can't directly
access reality. Okay. Uh all we have are
the observations which are incomplete
and they they have errors. Um and um
there are many many cases where we would
um uh we want to know for example what
is the weather like tomorrow and we
don't yet have the observation we'd like
to a prediction. Um and then we have
these simplified models sometimes making
unrealistic assumptions you know
spherical cow type things. Those are the
mathematical models. Mathematics is
concerned with the models. Science
collects the observations and it
proposes the models that might explain
these observations. What mathematics
does we we stay within the model and we
ask what are the consequences of that
model? what observations would what
predictions would the model make of the
of future observations um or past
observations does it fit observed data
um so there's definitely a symbiosis um
it's ma I guess mathematics is is
unusual among other disciplines is that
we start from hypothesis like the axims
of a model and ask what conclusions come
up from that that model um in almost any
other discipline uh you start with the
conclusions you know I want to do this I
want to build a bridge, you know, I I
want to to make money. I want to do
this. Okay. And then you you you find
the path to get there. Um
a lot there there's a lot less sort of
speculation about suppose I did this,
what would happen? Um you know, planning
and and and modeling um uh speculative
fiction maybe is one other place. Uh but
uh that's about it actually. Most of
things we do in life is conclusions
driven including physics and science.
You I mean they want to know you know
where is this asteroid going to go? What
was what what is the weather going to be
tomorrow? Um but um Bathe also has this
other direction of of going from the uh
the axioms. What do you think there is
this tension in physics between theory
and experiment? Mhm. What do you think
is the more powerful way of discovering
truly novel ideas about reality? Well,
you need both top down and bottom up. Um
yeah, it's it's a real interaction
between all these things. So over time
the observations and the theory and the
modeling should both get closer to
reality. But initially and it is I mean
this is um this is always the case. You
know they're always far apart to begin
with. Um but you need one to figure out
where to push the other you know. So um
if your model is predicting anomalies um
that are not picked up by experiment
that tells experimenters where to look
you know um to to to to find more data
to refine the models. Um yeah so it it
it goes it goes back and forth. Um
within mathematics itself there's
there's also a theory and experimental
component. It's just that until very
recently theory has dominated almost
completely like 99% of mathematics is
theoretical mathematics and there's a
very tiny amount of experimental
mathematics. Um I mean people do do it
you know like if they want to study
prime numbers or whatever they can just
generate large data sets and with a so
once we had computers um we be to do it
a little bit. Um although even before
well like Gaus for example he discovered
he conjectured the most basic theorem in
in number theory to call the prime
number theorem which predicts how many
primes that up to a million up to a
trillion. It's not an obvious question
and basically what he did was that he
computed I mean mostly um by himself but
also hired human computers um people who
whose professional job it was to do
arithmetic um to compute the first
100,000 tribes or something and made
tables and made a prediction um that was
an early example of experimental
mathematics
um but until very recently it was not um
yeah I mean theoretical mathematics was
just much more successful I mean because
doing complicated mathematical
computations is uh was just not not
feasible until very recently. Uh and
even nowadays, you know, even though we
have powerful computers, only some
mathematical things can be um explored
numerically. There's something called
the comatorial explosion. If you want us
to study, for example, Zodius the you
want to study all possible subsets of
the numbers 1 to a,000. There's only
1,000 numbers. How bad could it be? It
turns out the number of different
subsets of of 1 to a,000 is 2 to the^
1,000 which is way bigger than than that
any computer can currently can can in
fact anybody ever will ever um
enumerate. Um so you have you have to be
um there are certain math problems that
very quickly become just intractable to
attack by direct brute force
computation. Uh chess is another um
famous example. The number of chess
positions uh we can't get a computer to
fully explore.
But now we have AI um um we have tools
to explore this space not with 100%
guarantees of success but with
experiment you know so like um we can
empirically solve chess now for example
we have we have very very good AIs that
that can you know they don't explore
every single position in in the game
tree but they have found some very good
approximation um and people are using
actually these chess engines to make uh
to do experimental chess um that they're
revisiting old chess theories about, oh,
you know, when you this type of opening,
you know, this is a good, this is a good
type of move, this is not, and they can
use these chess engines to actually
refine in some case overturn um um
conventional wisdom about chess. And I
do hope that uh that mathematics will
will have a larger experimental
component in the future perhaps powered
by AI. We'll of course talk about that
but in the case of chess and there's a
similar thing in mathematics that I
don't believe it's providing a kind of
formal explanation of the different
positions. It's just saying which
position is better or not that you can
intuit it as a human being and then from
that we humans can construct a theory of
the matter. You've mentioned the Plato's
cave allegory. Mhm. So in case people
don't know, it's where people are
observing shadows of reality, not
reality itself, and they believe what
they're observing to be reality. Is that
in some sense what mathematicians and
maybe all humans are doing is um looking
at shadows
of reality? Is it possible for us to
truly access
reality? Well, there these three
onlogical things. there's actual
reality, there's our observations and
our our models. Um, and technically they
are distinct and I think they will
always be distinct. Um, but they can get
closer um over time. Um, you know, so um
and the process of getting closer often
means that you you have to discard your
initial intuitions. Um so um like
astronomy provides great examples you
know like you know like you an initial
model of the world is is flat because it
looks flat you know and um and that it's
and it's big you know and the rest of
the universe the skies is not you know
like the sun for example looks really
tiny um and so you start off with a
model which is actually really far from
reality um but it fits kind of the
observations that you have um you know
so you know so things look good you know
but but over time as you make more and
more observations bring it closer to to
reality Okay. Um the model gets dragged
along with it and so over time we had to
realize that the earth was round that it
spins. It goes around the solar system.
Solar system goes around the galaxy and
so on and so forth. And the guys
universe is expanding the expansion
itself expanding accelerating and in
fact very recently in this year. So this
uh even the acceleration of the universe
itself is this evidence that this
non-constant and uh the explanation
behind why that is it's catching up. Um
it's catching up. I mean it's still you
know the dark matter or dark energy this
this kind of thing. We have we have a
model that sort of explains that fits
the data really well. It just has a few
parameters that um you have to specify.
Um but so you know people say that's
fudge factors you know with with enough
fudge factors you can explain anything.
Um but uh the mathematical point of the
model is that um you want to have fewer
parameters in your model than data
points in your observational set. So if
you have a model with 10 parameters that
explains 10 10 observations that is a
completely useless model. It's what's
called overfitted. But like if you have
a model with you know two parameters and
it explain