This is actually super interesting to me,
because it's one of the most famous proofs of all time.
It's the proof for Fermat's last theorem, which Andrew Wiles famously solved.
Now I don't wanna explain that,
what I want to explain is what's at the heart of that proof.
It's actually not what Wiles spent seven years on, it's
it's actually something else.
It's the Taniyama-Shimura-Weil conjecture which is now called modularity.
So we're gonna say is Fermat's last theorem is true
and has to be just as obvious as the fact that one doesn't equal two.
In this case, Fermat's theorem reduces down to something called modularity,
it's the modularity theorem.
So it's not 1 doesn't equal 2 here, it's something else.
So it's all about two things,
about elliptic curves and this is specifically for
whole numbers and fractions are actually modular forms in disguise,
so what does that mean?
And that's how I feel even as a mathematician it's like, what?
So what I want to do, just trying to get you to appreciate what this means;
just an appreciation
not the full thing, just an appreciation of what's going on.
And what I'm gonna do is I'm just going to use much simpler equations,
right, what do you want to start with?
So starting with modular forms, a modular form, now the way you need to think about
this is it's something to do with symmetry, but there's different forms of symmetry.
And the one I want to show you today is called translational symmetry,
and what it is, is this. Imagine you have an infinite plane
this is being nicely provided by brown paper.
And imagine we actually have here so I'm just gonna
just going to kind of just put arrows to represent that I've chopped this up.
Imagine now cutting this up into strips, infinite strips but the important thing is is that the
strips all the same width, and so now something has the symmetry has a
translational symmetry 'cos this goes on forever,
like left and right as you can imagine this kind of situation.
What has the symmetry which you can basically take a strip of this, and cut and paste but
what's important is we want to know what functions you can cut and paste, cut and paste.
yeah, a graph, but what it is we're looking for an equation, we want something to generate it,
because say if you had like a picture or funny picture in here,
it might have a really complicated equation. Maybe it doesn't even have an equation at all.
I can think about a pattern I mean this doesn't even have regular pattern, my shirt
I usually like pretty regular patterned shirts
But the point is I want to show you something which actually has a very regular pattern and it's
something I'm hoping you've seen before it's one of these things like basically this is
a sine wave so it goes up and down up and down up and down now this actually has a long
complicated equation but the important thing is it's like even though it is this
equation that you crank out potentially you could cut and paste, cut and paste, cut and paste
so that's the idea, a modular form has that except hyperbolic geometry and
it's really complicated but if you get the ... just appreciate the fact that it's a
cut & paste thing, you take one section of it you cut & paste, cut & paste, that's the idea not all
functions are like that. How about a function like this is my attempt to circle
Brady: (laughing) Simon: thanks man
so a circle has an equation. I'm actually going to write the equation down just because it's fun
basically the circle is another equation except it doesn't have translational symmetry
think about it just exists in one section so what this relationship is
saying ellipitic curves (so whole numbers and fractions) are modular forms in disguise
it's actually saying is that you can fit a modular form into elliptic curves, you can fit a
translational symmetry equation into the equation of the circle. That's what it's saying.
this is magical stuff. So how does this work? how can we make this work
well the way you have to think about it is that imagine now we had a point on the circle
alright? now we start following it around yeah? So let's just have fun, maybe we could do it really quickly
maybe go slowly maybe go fast-slow-fast-slow but if we chose to do it at a constant rate
going round and round and round imagine what that'd look like if this was actually
moving through space so if you think about it if I'm doing the circle imagine
now if I go like this, so now I'm gonna start moving at the same rate as I'm going up
and so what's actually going on is I'm actually making a coil
Brady: It's like a slinky Simon: It's like a slinky
that's right so now just imagine right, what we're doing is we're actually creating a
coil now again if it was a slinky, slinkies you can make them bunched up and ... but imagine
we've got a slinky that's just perfectly evenly spaced so now that has an
equation; it has a helix equation, it's got an equation in 3D but what happens if you look
down the barrel of that helix? What would you see if you looked straight
down and there was no perspective problems like if you look straight down
the barrel what would you see? Brady: A circle
Simon: that's right, so what's actually going on there is that the helix can fit inside the equation of the circle
and the only reason this works is because 3D space and 2D space are both flat when you
dealing with curvy spaces this is really not an option and these are curvy
spaces the actual elliptic curves belong to projective space, modular forms belong
to hyperbolic space, the complex numbers, I mean they are really super curvy, but in this
example it's really nice because what we're doing is we're actually fitting
three-dimensional numbers into two-dimensional numbers. Not really
three-dimensional, they exist in three dimensions you gotta think they have a different
relationship going on. So what the coil has, the coil has translational symmetry
So this is the link between modular forms and elliptic curves. It says
that this really strange function over here can fit inside this really dull
function over here and when this happened no one believed it no one believed you
could stick, you know, you can stick this coil into the circle. The reason why no one
believed it was because no one could visualize it so nicely like we can, they're just
completely different spaces. That's the connection that needed to happen to
actually make Fermat's last theorem work. It's ... this is again, this is not the real
thing because if you think about it if you have a whole number or a fraction as
its cycling around ok as long as it's a fraction there'll b a point where it will come
back on top of itself. Like there is a point where fractions because there's a number
you can multiply fractions by, you'll get a whole number. So what that means is, in turning
around a circle if you start with some sort of fraction you'll end up back at
the same spot if you have an irrational number which by definition is something
that can't be a fraction you started at some point and went around by root two
we never get back to the .. to the actual original position so again this is the feeling here
of the proof the fact these these numbers matter the type of number
matters. This is not how the proof is worked! I again am trying to give you a
feeling for what's going on the fact that the reason why Fermat's last
theorem involves numbers is because these numbers the way the numbers are
actually they work together
impact whether you can put a curve into one of these and so then, how they linked it in
here is the fact that they said: Fermat's last theorem involves whole numbers therefore the circle
equation I'm making should permit one of these things except they proved all
these wonderful people that you've seen on numberphile who you should be seeing
if you haven't seen yet all the wonderful people that actually proved this
they showed that that created a contradiction so basically even though
you're playing by the rules and using whole numbers it doesn't permit one of
these things so again it's the one, not equaling two.
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