The following content has been provided by the University of Erlangen-Nürnberg.

So we have all experienced a lot of very abstract terminology this week.

And of course when it's new to you, abstractions can be very difficult.

But then after a while when you get used to abstractions,

then they become part and parcel of your everyday life.

Then you start to imagine these things as being very real.

I want to start with a small anecdote, which is a true anecdote.

This is about the abstract painter Pablo Picasso.

As you know, he painted very abstract paintings.

After World War II, right at the liberation time of Spain,

an American soldier came into Picasso's atelier, his office.

Picasso was friendly. He showed him around.

He said, here's some of my work. Do you like it?

No, he didn't quite like it. It was too abstract.

You cannot see what it is.

He would like to see things that were much more real, close to life.

So Picasso decided to talk about something else.

He said, what about you? Do you have a girlfriend?

Oh, yeah, I have a girlfriend. Is she nice?

Oh, she's very nice. What does she look like?

Well, then in his wallet, he had a little photograph.

Oh, yeah, she's very nice. But isn't she very small?

Meaning that, too, was an abstraction.

Okay, so I'll start a somewhat different place than I did last time.

I'll start with a setting of M and G, Brummanian manifold.

It also could be a Lorentz manifold, or any sequence that you like.

So the metric tends to G.

And I want to introduce a special operator.

So I want to introduce the operator L, which is a second-order operator.

So that's the L plus operator plus N minus 2 over 4 N minus 1 times R.

So N here will be the dimension of the manifold, and R will be the scalar curvature.

This is sometimes called the Yamaabi operator.

So this is a famous operator in, well, of course, it's well-known to people in cosmology, relativity.

It was used a lot in connection with conformal geometry in the Riemannian setting.

And the point I want to make here is that suppose you make a change of the metric.

Suppose you change the metric in a conformal way.

So you take some function, which is just a smooth real function.

And if you make a conformal change, then you did not change any angles.

But you change other things.

So what happens is that this one gives, of course, it gives a new Yamaabi operator,

because you change the metric.

And as you know, the curvature involves two derivatives of the metric,

and the L plus operator also sets the metric and some derivatives and whatnot.

So you might ask, what does the new Yamaabi operator look like?

And it's actually quite simple.

So it turns out that it can be written in terms of the old Yamaabi operator.

And then like this, then there's an N minus 2 over 2 omega.

And then here there's an N plus 2 over 2 omega.

What I mean by this is that you, and here you can see that this gives us a certain conformal invariance

of the set of solutions of the kernel of L.