Thank you, Antonio, and thank you, Leon, and all of the organizers of this seminar.

This is great.

Also, thank you to Andrea.

It's, of course, an honor to be sharing kind of this platform with him.

And OK, so as Santonio said, so I'm going to talk about graph Laplacian's regularity

theory and these things.

And then the way that the talk is structured, so the title is this one.

It's about regularity theory.

But in fact, I'm only going to show regularity theory in the last 10 or 15 minutes.

But then the first part of the talk will be kind of like a path to get there.

So somehow I'm going to show you a bunch of results that they're going to be connected

one to the other.

And then they will culminate or they will finish with this regularity theory.

So please stay with me and let's see where this takes us.

OK, so then just kind of in more detail, the description of what I'm going to talk about,

I have this talk into these four acts.

So in the first one, I'm going to give you like a very brief motivation and show you

some of the very basic and let's say first results on studying this graph Laplacian's.

Then I'll tell you a little bit about asymptotic spectral consistency of this graph Laplacian's.

I'll make sure that the setup is clear by that point.

Then we're going to talk about quantitative convergence rates first in the L2 sense.

And then in the last part, I will talk about a stronger notion of convergence and also

finer properties of the eigenvectors of this graph Laplacian's.

And so this is more or less the outline of the talk.

And so OK, so then let's start by the beginning.

So OK, so then the context where this graph Laplacian's arise, at least for me, the motivation

I have for my work is in data analysis.

And in data analysis, we have data.

And then we want to learn from data.

And so there are at least two big frameworks or big paradigms of learning that are of interest.

There are more than two, but there are two that are very popular.

And those are the unsupervised learning and supervised learning.

So in unsupervised learning, the idea is that given the data set, you want to find a course

structure, find clusters.

And in the other setting, the supervised one, it's well, you have the data, but you also

have labels.

Typically, these labels are given to you by an expert, this one being a doctor, an engineer,

or an artist, for example.

And the idea is to find a regression function so that whenever you get a new data point,

you can label correctly.

So there are these two frameworks that are out there in learning.

Now the idea is that, of course, like you want to use structure from the data to try

to fulfill one of these tasks.

And there is a framework for learning that is known as graph-based learning, where the

idea is that you are going to inform your clusters or your regression function by basically

a graph that you have on your data set.

So the data set comes, we're going to imagine that it comes with a notion of similarity

between data points.

And then, of course, the idea is to use this similarity to fulfill those learning tasks.

So then, for example, if we have this data set of handwritten numbers, the way we have