How to pronounce “riemann” in English

Riemann gave him some examples, some list of possible thesis topics.

Riemann didn't know about Minkowski's spacetime, but he knew about other examples.

So Riemann set out to do that.

And Riemann is saying, if you give me those 16 numbers at every point in the space or the spacetime, you have completely determined the geometry.

And Gauss suggested that Riemann do what Riemann thought was the most boring one, on the foundations of geometry.

But what Riemann suggested is that you can also go the other way around.

So what Riemann says is, in a curved space time, if I give you a formula to calculate the length of very short straight line segments, that

So what Riemann says is, imagine doing that at every point in spacetime separately?

That's what the Riemann tensor provides you.

Plug it into the Riemann tensor, and it spits out how these lines will change in their relationship to each other as time goes on as you move down the path.

But the Riemann tensor is a 4 by 4 matrix of 4 by 4 matrices.

But the Riemann tensor isn't separate and independent from the metric.

And that Riemann tensor is responsible for spacetime curvature.

So this Riemann zeta function helps you understand how they're spaced out so I understand.

- It's the substitute for the Riemann hypothesis on average, and it turns out that that's what's actually necessary in a lot of these arithmetic applications.

But he did it without the generalized Riemann hypothesis.

you don't need the Riemann hypothesis.

And that task was undertaken by Bernhard Riemann in the 1850s.

The first is that they assumed that a generalization of the Riemann's hypothesis is true, which we don't know.

- It's one thing to just assume the Riemann hypothesis, and say, "Oh yeah, you can make some progress."

And you can use the metric to calculate the Riemann tensor.

There is a formula that lets you calculate the Riemann tensor once you know the metric.

So there's no more information in the Riemann tensor than there is in the metric.

The obvious strategy is to somehow cut the Riemann tensor down to size, to boil it down to a tensor with two indices

Similarly, Bernhard Riemann and all his ideas about geometry going back to the 1800s and so on and so forth.

So for example, the Riemann hypothesis is a great discovery that Riemann made, that, to understand prime numbers, we need to understand the zeros of the Riemman zeta function.

It can be written in terms of the Riemann zeta function.

on a physical embodiment of a machine to try and solve Riemann's Zeta function.

trying to do-- there's some mathematicians amongst you-- the Riemann zeta function might-- oh yes, we're getting one or two nods, so that's good.

- Now, the Annals gets a proof of the Riemann Hypothesis every other day, so they're like, okay, surely this isn't gonna work out,

They then showed that, if the generalized Riemann hypothesis is true, then the main term grows faster than the error term.

It's just pointing out that this very compact notation of the Riemann tensor conveys an enormous amount of information.

And you put them together and it will actually approximates to something that will solve Riemann's Zeta function.

Often when I'm asked about some of these big famous problems like Twin Primes or Riemann or something like that, one way of kind of avoiding the question

And that's going to be captured by something called the Riemann curvature tensor, R lambda rho mu nu, four different indices.

So to replace Newton's equation, somehow what we want is an equation that relates the Riemann tensor, representing

So I think where the limitations will be is-- for example, even just something like the Riemann hypothesis.

So we can see the evolution of things from these analog machines, like the Riemann's Zeta function machine into something that's more digital.

was in use, but this is a step up from the Riemann zeta function machine.

They're going to be attacks on the most fundamental and important questions of the day, whether it's the Riemann hypothesis, Navier-Stokes or P versus NP,

Because when a math AI spits out a proof in Lean of, let's say, the Riemann hypothesis, we're not going to need humans to go through every single line of the proof

- Well, I mean, in your undergraduate education you learn about the really hard, impossible problems like the Riemann Hypothesis, the Twin-Primes Conjecture.

You know, famous problem in mathematics is the Riemann hypothesis.