 And, as far as I know, there is still no proof of harmonic functions, no elementary proof that will explain you why this is true. However, there is a theorem of social logon of this published this year that tells that the Dirichlet conjecture is true. Moreover, it holds for all elliptic PDs. I think it started with smooth coefficients in the work. So, if you have an elliptic operator, say in the unit, twice the unit ball, and then there is a constant depending on the operator such that any solution that vanishes at the zero point satisfies that the length or area or the dimensional house of measure to be precise of the function is at least constant. This theorem will give you the law estimate in the yaw conjecture. This theorem implies that the law estimate in the yaw conjecture holds. We would not be able to prove this in this four lectures or even go close to it. It will be able to look at some ideas that are relevant and useful in the proof, but only in the simple versions of the ideas that are useful there. So, I think it's a good point to stop the first lecture here. I don't know if there are any questions or comments. What actually happens is that you will get other zeros here. You can make this one quite narrow, but it will result that there are other zeros coming from your harmonic function. I think that this is not true for one component of your zero set, but there are other zero sets. I don't think so. I don't know. I don't have any intuition. I think all the intuition ruins when you don't have elliptic coefficients for me. It's a very nice question, but I don't know. Is it known what's typical in any dimension? Do we have with probability one for any distribution of A j's any reasonable results? Beliefs or expectations? It's difficult to think because on the sphere and the torus we are in a special situation because our eigenfunctions have large multiplicity. And then you can talk about random eigenfunctions because of that. But on the sphere you know that. On this level you know that. On the sphere for each one. You don't know how many components. There are other questions about locally what is the structure there. But in the sphere everything is known due to Donal and Pfeffermann. There are other questions about random things that are different. I don't think there are nice estimates for the constant. I think there is existence for this one, but I'm not sure. This is true for any uniform elliptic operator or just with Lipschitz coefficient? At least Lipschitz. Time to speak it again.