So please Karina, you have the floor please.

Yeah, many many thanks for your interest in Marius and I'm really really honored to be here.

So I'm going to keep it introductory and I marked the slides which are a bit more technical with a little star.

So if those are a bit too hard for those who are kind of not in the field, then don't worry.

These are really sort of minor and they don't sort of correspond to the major flow of the argument.

Yeah, so what I want to do, so let me see if I can.

So what I want to tell you today is basically another look on stochastic partial differential equations by discretizing them and what does it make interesting from the modeling point of view.

So the reason why I like to study stochastic PD is from a discrete point of view and on this I mean discrete in space is like two folds, like one of them is that it has its sense and applications.

For example, what you can see here in the picture on the right. This is a granular medium.

And you have naturally sort of one grain of sort of minimum size or you can say one pixelated image or something like this to go below this scale.

It says my internet connection is stable so I hope it's going to work still.

So yeah, so there's this minimum scale it's not not reasonable to go below it and that means that naturally we are in a discrete setting.

And that could happen like not only in granular floors but it's also sort of happens when you have a model coming from, from sort of like computational neuroscience or so where you have these myelated cybers and then I'll show them later.

And then you have to look at systems which come for models which come from particle systems, because then sort of like each particle has a representation as the bacterium or a population or something like one member of the population.

So there's also like naturally like one smallest entity.

And the other reason why I like to study those models is because it makes a difference from the phenomenological point of view. So I can see different phenomena happening in the discrete setting, then I can see from the continuous setting.

And that's what I really like.

And what can happen, for example, I mean, is that the discretization in space will influence the dynamical behavior of the system and something else and if you might see is that you might have special solutions in either discrete or a continuum case, but we don't have

that special solutions in the other case, right. So somehow solutions exist or not exist depending on if you look at them in a discrete or in a continuum scale. And that's somewhat surprising because we usually would think about the discrete systems being some sort of an

approximation of the continuum system. But sometimes this doesn't work.

And one I think we will see today.

So this is something which I like to add to my PD model and this is noise.

Also casting PD.

And this is our causticity also brings a new phenomenon. And one phenomena that I want to talk about today is called meta stability.

And if you haven't heard about it, I'll make you a quick example, which is this picture here to the right. And this is a situation, you can try to do this at home it's a bit tricky but it can work and there's YouTube videos of people who do it.

So you take a bottle of water, it's best if the water is distilled so the one that you use sort of for ironing your clothes, because then it doesn't have any sort of disturbing particles in it.

So you take pure water, you put it in a bottle, and you put it in your freezer, and hopefully your freezer such that you can open it with a, you can open the door without sort of tearing, tearing the whole sort of, so that it doesn't wiggle too much basically.

So if you do that and you put that bottle in your freezer, and you wait for a certain amount of time, it can happen that the temperature of the water is slightly below zero, but the water is still liquid.

And when this case happens, and then you pour this distilled water, you see this instantaneous freezing event which happens here on which you can see here on the picture.

So what, what is going on there is that somehow there is a barrier. That's the sort of, that the system needs to overcome to transfer to a different state.

In this time from a liquid to a frozen state, but it could also be something else. And that extra energy barrier to overcome that, that creates this phenomenon of metastability.

So metastability means that the system is stable for a very very long time when one parameter is changing a bit, but there might be one instance, or there will be one instance where very quickly, it will go to the other state.

And if you look at that sort of in a more involved example, this is the picture on the left bottom.

I wrote it, these are energy levels of confirmation of some protein. So this is what we see in nature is that you have these proteins, and they have, they kind of fold up in different, like, in different ways.

And in one folding layer is incorporated the geometric structure. And this geometric structure tells you like how are the atoms arranged in this protein.

And each geometric structure, or sort of has a specific energy to it, right. So then, if the temperature changes in the system, then another structure, another geometric structure of those atoms might be preferable.

So that's why chemists model those geometric structure of our proteins with potentials which look like a multiball. And every local minimum of this multiball potential corresponds to one geometric configuration, as you can see in that picture here.

And then, then this is exactly situation where metastability occurs, namely that your geometric structure is quite stable.

But if the energy level changes due to outside temperature influences, for example, very rapidly, each metric configuration will change.

So it's stable for a very long time, and then the change will happen quite suddenly. And that change means that you're moving from one local minimum to the other local minimum in your energy, potential energy picture.

And that is obviously quite important and sort of serious mathematicians here at Van den Aden. They've tried to sort of use those facts also to characterize how these conformational changes are working in a mathematical computational manner.

And here's one picture here.

Where they try to characterize conformational change in some HEPs.

So you see this like pretty exciting when you start to add noise to your picture. And so in this talk we're going to try to do both. We're going to look at the discrete in space perspective, and we look at the noise perspective.

Can I ask you something, Karina? Absolutely. Is this related to Turing unstability, in some sense?

You mean Turing instability for training of neural networks? No, no, no. For just the reaction diffusion problems in which by changing, you know, the diffusivity parameters on the system you can make a steady state that are stable to become unstable and so on, right?

Okay, so I don't know exactly. So you will see my point of view popping up in a couple of slides.

So like, I can't, like probably you're going to figure out by yourself more quickly than I know because I've never seen the name of the phenomena.

Otherwise, like probably like we can have a chat about it. But I think, I think it's probably just in other words what you already know.

That's my guess, but I can't tell you for sure because words are always misleading.