Thank you very much. Thank you for the introduction and very welcome invitation to this one world

seminar. I'm honored to be here, a little worried because if I say something more exotic,

it's hard to tell from the faces of the audience if I have to stop or to change pace. So what

I'm going to talk about is some gamma convergence applications to larger data, say, if you want.

And in particular, to continue limits of where you have some kind of interfacial energies

arising on graphs. So the type of problem we have in mind is the typical Romanov calculus

of variation. So we have a set and we have to subdivide the set into subsets, like two

subsets so that some energy is minimized. And we think of this as an interfacial energies

between the two sets. So it's the starting problem of calculus of variation, DDoS problems.

But now we have a different setting. So we have a graph. So instead of having a region

of the plane or the space, we have a graph, which is a set of vertices and a set of edges.

So not all points are connected in the graphs, but we have sets of edges. And the idea is

to minimize some energies defined on edges. The easiest energy is just the number of edges.

So the picture we have in mind, we picture our graph into colors and we minimize the

number of edges of the graph, which connect points with different colors. As the number

of nodes of the graph diverges, then we want to keep trace of what happens to these kinds

of problems and see what is the limit behavior. So the plan of the seminar is the following.

So first, we embed these kind of problems in a variational setting, and we talk about

the gamma convergence very briefly. Then we will see what happens for some class of lattice

energies. So when the graphs are parts of a lattice, and in particular, when they are

sparse. So embed the lattice in some lattice, embed the graph in some lattice of some dimension,

and we suppose that the number of edges is much smaller than the total possible number

of bonds, so pairs, of the lattice. Then we will see that we have a description in this

case, but somehow we see the arising of some issues due to topology. And there will be

a short intermission where we will see what kind of problem arises, also for very simple

sparse graphs. And then we will go to dense graphs. So when the number of edges is of

the same order of the total number of possible bonds. So what is variational analysis? So

we transform our problem, which is defined on sets and problem defined on functions.

So we introduce a parameter, a parameter which labels the families we want to take into account.

So the easiest case when you have two families. So we will stick to this case, and also because

in this two parameter setting, we can choose the parameter very symmetric, being 1 and

minus 1. So a subdivision or subsets of the graph are function defined on the graph with

values minus 1, 1. If you are one family, you value 1, the other minus 1. They're called

spin functions. We don't impose any cardinality on the number of nodes of vertices in which

you take value 1 or minus 1, because this can be imposed a posteriori. Then we define

the edge energies in terms of these spin functions. And this is a way to define them. So to every

edge we associate a number, positive or non-negative number, aij, which is the strength of the

edges, of the edge. And if you are not in the set of edges, then we set aij equal to

0. So the energy can be written as a sum of the role pairs of vertices. And we have a

normalization factor because here ui minus uj is equal to 0 or 2 in modules. So in the

case when aij are equal to 1 on the edges, this is just the number of edges. So this

is another way to write the same thing. And then we want to pass the limit and see what

happens to the minimal card problems. But this is done by defining a limit energy, not

looking at the problem themselves, not looking at the solutions, but looking at the behavior

of the energies. And to define a limit energy, which is gamma limit, and in such a way that

this gives the minimal card problems for the limit are approximate minimal card problem

for the converging graph, for the diverging graphs. It's important here to have some compactness

properties of minimizing sequence that suggest which is the limit energy. And this is different

in the case of sparse or dense graphs. So we start with what had been done in the last

20 years, a lot of work has been done in connection with image segmentation and fracture mechanics,