Okay, first of all, I want to thank Eric for the kind invitation to visit this very beautiful

city and university.

And I apologize.

In my talk, I don't have machine learning.

I have a PD, but no machine learning.

I see that on many of the talks, I see many of the terms I know, like Allen-Corn equation,

Hamilton-Giapofi equation, I see the title in these talks.

So I'm looking forward to see the connection between this equation and machine learning.

But I apologize.

I don't have any machine learning in my talk.

Okay, so this is a recent work with my former postdoc, Li Min-Seng and Wen Yang.

So this is about Brzeece's first open question.

And this is in memory of Harn Brzeece, who died last year, unfortunately.

Okay, so I start with a differential geometry problem, which is a well-known problem in

differential geometry.

So if you have a compact closed Riemannian manifold, the question is, can we find a conformal

metric, which is a scalar multiple of your metric, so that the new metric has a constant

scalar capture?

So this problem can be rephrased as a nonlinear elliptical problem with critical exponent,

which is u to the n plus 2 over n minus 2.

And this is related to the Solvier-Fieberian.

So a critical problem for Solvier-Fieberian.

And this problem, we know this has been completely solved.

By Trudinger and Orban and Hsion after 1984, I assume the possible math.

Now, so what happens if we replace the manifold by Euclidean domain?

Because this is much easier.

Everything will be much easier.

So then the domain, then the equation, everything disappeared, because you don't have

scalar capture.

Scalability disappeared.

And so the equation just minus n plus plus u to the n plus 2 over minus 2.

And we put a dual-shaped boundary condition of the boundary.

Now, for this problem, there are no solutions if your domain is a star shape.

For example, if you're a unit ball, you don't have any solution for this problem.

So what are Brzez and Niemberg in the similar work in 1983?

So what they considered is the i-th and lower order term, which is lambda u.

Okay, so this problem, Brzez and Niemberg problem, has a few characteristics inside.

First, there's a critical exponent, which is u to the n plus 2, n minus 2,

which is a nonlinear problem.

But there's a lambda u plus lambda u.

This is a linear problem, and this is an eigenvalue problem.

So there's an interaction between the eigenvalue problem and the Yamaabi problem.

This is an interaction between the eigenvalue problem and the Yamaabi problem.

So first, for this problem, lambda should be positive,

because if lambda is less than or equal to zero, you don't have any solution.

And if you want a positive solution, lambda must be strictly less than lambda 1,

because you multiply both sides by the first eigenfunction,

and you see the lambda must be less than lambda 1.

And this is exactly what Brzez and Niemberg in the 1983 paper prove.

If lambda is for n greater than 4, if lambda is between 0 and lambda 1,