So welcome everyone to this seminar.

So we have Professor Erin Francesca from the Carnegie Mellon University in the USA.

And today she is speaking about genetic flows and phase transition in heterogeneous medium.

Please, Professor Farnesca, grab the floor.

Thank you.

And thank you for the invitation.

I'm delighted to be here.

I guess it's a good afternoon for most of you here where I meet you at 10 o'clock in

the morning.

So the work that I'm going to talk about actually has kind of like two parts to it, but of course

I integrated.

And the beginning of all these was carried with some of my former trainees, so Ricardo

Christofori, postdoc, and then my PhD students, Adrian Haggerty, Christina Popovici.

And then that resulted in papers that have been published.

And currently we are continuing this work with Rustam Shoxi and Jessica Lynn from McGill

and my postdoc, Raghav Venkataman at Carnegie Mellon.

And so that's work in progress.

There's still nothing out there yet in writing, that second part of the talk, but there will

be shortly we'll have some preprints out.

Okay, so the beginning of my talk, and some of you I'm sure it's very well known, but

forgive me to just recap so that we are all on the same page.

And basically this whole starts with the Van der Waals can heal your theory for phrase transitions.

And you can think of a model where you're going to have a mixture of two fluids, they

equilibrium phases lie at the bottom of a double well energy like the one I depicted

here.

Say for example, one minus square square.

It doesn't have to be like that, but say let's fix these for now.

And so you ask, okay, how can I minimize this energy?

Right?

And if you just ask like that, and probably with some fixed volume fractions that you

are forced to mix to have part of your domain in one with one material and part of the domain

with the other material.

So you fix the average of you if you wish in between the two wells.

And so, as I said, the problem is let's minimize this energy, right?

Subject to this constraint.

In what I'm going to talk about, omega is going to be bounded domain and is going to

be two or more.

Right here, as this problem was first posed in the beginning, you were scalar valued.

It does not have to be, but let's keep it scalar valued for this next couple of slides.

Again, fix the volume fraction and say W, okay, in the prototype, I gave you W with

wells at minus one and one.

Put it at wells at A and B, but otherwise W strip is positive.

Right?

Okay.

So if you just leave it like that, of course, we all know that you have a really many solutions.

And any U that just looks like A or B, right, like in here, suppose that I normalize and

put the measure of omega equal to one, I take the mass between A and B.

So any U that's like, and I write it as U sub E, which means it's basically the characteristic

function of a set E at times A and outside E, the value is B. And of course the volume

of E has to be adjusted so that the constraint is satisfied.