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. Our thanks audible.com for supporting
this numberphile video. They've got something like a hundred and eighty thousand
spoken word titles in their collection and I often enjoy listening to well a
few of them on my daily dog walks here with the girls now if the beauty of
mathematics is something you like why don't you try Love and Math the Hidden
Heart of Reality it's by Edward Frenkel someone you've probably seen in quite a
few numberphile videos already. It's got all sorts of great stuff in it including quite
a lot about Fermat's last theorem something that's not a million miles
from Professor Frankel's own research and like so many books available in audio
form at Audible so go to audible.com / numberphile if you'd like to give it a
try when you signed on for their free one month trial you might choose a
different book from their extensive collection there are plenty of math ones
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supporting this video
I wish there were videos like this that assumed some kind of background in math (like an undergrad degree)
his curls are on point
Does n have to be an integer too in fermat's last theorem or can n be a complex/real/rational number?