at which these waves move is the speed of light. And so people said, "Wow, the speed of light comes out of those equations ." And that had to be, I think, very persuasive. And of course, electromagnetism also plays a really significant role in chemistry because after all,atoms are held together by electromagnetic forces. There's more to how atoms work. There is all the quantum mechanics stuff.
And I know that this might not be what you signed up for. Equations are not usually what are focused on in popular level talks about physics, but don't worry about it.This is going to be OK.
So you can have a bunch of equations , but without some data to actually back it up, you don't really know if those equations are particularly predictive.And on the flip side of the pile of data, without any equations or model to kind of explain what the general thing
My friend Anne-Marie Slaughter, a wonderful scholar at Princeton and former State Department official under Barack Obama, speaks of every profession's having to adapt to these new equations of power and influence."The leader," she says, "must become a catalyst.
So, I can fix that by adding a fourth equation or a fourth satellite. Four equations , four unknowns, that makes the system solvable. Okay, but we've made one key assumption, that when I'm calculating my position, I know where the satellite is.
50 years and wrote his laws of electromagnetism, and they're really fascinating. If you look at the laws of electromagnetism, they are-- they're differential equations or inter- integral equations . But basically, what they say is on one side, you have a bunch of terms that have electricity in them, and then you have equals on the other side, a magnetism thing.
This is the language in which modern physics is written, equations . And equations means that you have something that is absolutely precise. This is not just a statement in words that the more you push on something, the faster it will accelerate.
to say to the question of the ultimate intrinsic nature of the stuff that satisfies the rules and equations over and above saying that it satisfies those rules and equations . They know that physics just isn't in the business of answering the question, and this is Hawking's words again, "the question, what is it that breathes fire into the equations and makes a universe for them to describe?"
You look again, you get something else. Treat equations on the page the same way. So that's sort of the idea.
And that's good enough. Linear equations and plugging in the answers and seeing that it worked out.
but also to maybe clarify some conceptual difficulties that even I had as a researcher, feeling so comfortable writing the equations now of a Feynman path integral without really stepping back to think of, OK, what's going on intuitively here? So this analogy with jazz improvisation kind of helped.
So this is amazing. No equations . So it's like bracketing my life from serious scientist to fuzzy guy.
future and I was able to really try to -- I was free to find the, the real beauty inside these equations , and inside the math. So, the first thing - oops. I was talking too loud. So the first project that
So, you might think three satellites is all you need. We have three equations , one for each satellite, and three unknowns that are my position, x, y, and z. So, on the left side, I have the geometric distance between me and each of the three satellites.
The mass also follows Newton's second law, F = ma. If you combine these equations , you get a = -kx/m. which links acceleration and displacement.
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. And so in mathematical physics, we care a lot about whether certain equations in wave equations are stable or not, whether they can create these singularities.
There's a famous unsolved problem called the Navier-Stokes regularity problem. So the Navier-Stokes equations that govern a fluid flow or incompressible fluids like water. The question asks, if you start with a smooth velocity field of water, can it ever concentrate so much that the velocity becomes infinite at some point?
- Right, yeah. So the key phenomenon that my technique exploits is what's called supercriticality. So in partial differential equations , often these equations are like a tug of war between different forces. So in Navier-Stokes, there's the dissipation force coming from viscosity, and it's very well understood, it's linear, it calms things down.
And we have a lot of technology to handle critical and also subcritical equations and prove regularity. But for supercritical equations , it was not clear what was going on. And I did a lot of work.
the wind speed is this. For supercritical equations , the fine-scale information is really important. - If we can just linger on the Navier-Stokes equations a little bit.
So what I realized is that if you could pull the same thing off for the actual equations . So if the equations of water supported computation. So you can imagine kind of a steampunk, but it's really water punk type of thing where so modern computers are electronic.
into, you know, what could be responsible for this. And we actually found that just like we use so-called Maxwell's equations in electromagnetic stuff for everything we do in electromagnetics, we have Einstein's equations in general relativity for, you know, black holes and all that kind of stuff. It turns out if you could engineer those, you would
best explains the pattern of lightness along the outer boundary. Here are the equations . I won't go through them.
But the equations are so powerful, we cannot solve them analytically.
There are no equations in the book but we do have to talk about maths.
is m times a In the x direction, et cetera. There's really three equations embedded in this simple set of symbols. That's going to be something, that idea that we can embed multiple relationships into a single equation, it's going to be very, very useful very, very soon.
called the speed of light. It was Maxwell's equations that really showed us, once and for all, that what light is, is a wave in the electrical and magnetic fields. But the speed of light had to play some special role.
He can't write equations down.
There's no equations in the whole of the book, actually.
And he and his equations gave rise to the TCP/IP protocol in the internet we use today.
So physicists love equations for good reasons.
They obey their own equations of motion.
Einstein's equations predicted it in 1916.
Einstein's equations give us rotating black holes, where there's a gateway to a parallel universe.
I have, for example-- showing the scribblings as I promised, actually showing how a theorist might write some equations about a particular issue that's being discussed. So that's one thing I can talk about.
As a professor in the university, do you do teaching? Whether it be the equations , or diagrams associated with the equations , or even metaphorical sketches about what the interior of a black hole is,
As a professor in the university, do you do teaching? Lots of concepts and equations in physics seem to lend themselves quite naturally to visual ways of explaining.
But what the equations have done-- and what the researchers we've been working with have done-- is they've taken all of this information,
obey the same equations that are affecting the bridge.
We know all the equations of movement, and motion.
problems and equations or something down there uh but it's actually very simple
about like the equations like you know um nun's equations and you think about if you push something with a certain
These two simple equations that seem to reveal the truth of the universe explain the universe is expanding.
The second big ingredient in the proof of Fermat's Last Theorem we covered in this book, that's the theory of elliptic curves. Which are certain cubic equations . And then, in our most recent book, The one we came to talk about is the third ingredient that Andrew needed.
First this is equations .
These very equations which I really think beautiful.
And I saw this picture of this guy. And I saw these equations . And I thought this was some kind of hidden code.
he actually altered the equations so that it would take away the idea that the universe was expanding.
We can write the equations down.
You get lots of equations .