So welcome everyone. Today we have Professor Pierre-Emmanuel Jabin from the University of

Maryland.

Penn State actually.

Penn State, okay. Today he will be speaking about the main field limit for non-exchangeable

community agent system. Please, Professor Gamba, you have the floor.

Thank you. So I actually changed a little bit what I'm going to talk about. I hope it's

going to be okay. I wasn't sure how much you knew about main field limits and things like

that. So I decided to go to a still related, but another topic that is maybe a little bit

simpler. So that's why you see the title is a little bit different.

Should be fine.

And also, yeah, what I should have said first, of course, thank you so much for the invitation.

I just regret that, you know, as everything goes days, it has to be through Zoom and not

in person. But all right. So what I want to do here is first give a very broad overview

of what I mean, which kind of particle system I'm studying, what I mean by the main field

limit, what are some examples of models that we are interested in, what type of challenges

we have, and then give a relatively, probably relatively brief overview of the new method

that we are bringing.

So the first thing is, of course, particles means so many things that it doesn't mean

much, right? And classically, the term particles comes from physics and typically from plasmas,

you know, this sort of example was the canonical setting for studying the main field limit.

But even in physics, it actually ends up referring to a lot of other objects. And astrophysics

is another very canonical example. I mean, the first many particle system was probably

Newton's dynamics. So for point masses in gravitational interaction. And nowadays, this

is still commonly used because at some large scales, the universe is essentially flat.

And so you have this kind of point masses in gravitational interaction that are studied

for understanding the formation of galaxies or galaxy clusters, and whose either are,

you know, ordinary matter or sometimes particles of dark matter. But still in physics, and

an important example that I'll go back to is in fluids and point vortices. But of course,

nowadays, those models, you find them a little bit everywhere. So in biomechanics, hair soles,

biosciences in general, and any dynamical behavior of animals, microorganisms, for examples,

bacteria, social sciences, opinion dynamics is a very popular topic these days, economics,

and of course, the main field games have taken a huge importance as well. So there's a lot

of potential applications. And here, I'm typically going to focus on two of them, which are those

point vortices in fluids, and the dynamic of microorganism around the question of cable

taxis. So to give you a bit of a visual idea of the range that could be covered, right,

so this is an actual numerical simulation of gravitational interactions between point

masses. The color that you see is the density of mass, and the small yellow lines are the

directions of the velocity. And at the other end of the spectrum, right, both in terms

of size and in terms of field. So this is biological neurons, in which you also very

often find this kind of many articles or multi-agent, and they are often called multi-agent systems.

All right, so this is a piece of the brain of a rat. And what you see here is actually

the cytoskeleton of the neurons, but that lets you see relatively well the center of

the neurons and the axons and the way they connect to each other. So another important

question to discuss here is I'm going to write down the system of couples of these or couple

SDEs or stochastic differential equations. And the question is, how many of them should

we have? And just as there is a huge variation in the sizes and in the kind of settings,

well, you also have quite a bit of variation in the number of particles that you are considering.

And not very surprisingly, the higher numbers are found in physics, right, with something

like 10 to the 20, 10 to the 25 particles being relatively common. And in plasma, I

mean, think Avogadro number, for example, that will give you the order of magnitude