Great to make that recording, right?

Yes.

So, happy New Year to everyone and welcome to the first seminar of this year.

Today we have Professor Stefan Steinerberger from the University of Washington, Seattle.

And today he will be telling us about the PD description of polynomials on that differentiation.

Please.

Okay, so thanks very much for the introduction.

I've not given this talk too many times, so please feel free to interrupt me.

I'll actually, maybe I'll try to make sure there's a chat that's open if we have chat.

I mean, just interrupt me, okay?

That works.

So, okay, so this is a very fun story.

It's very new, and most papers on this have been written in the last two months and not

by me.

So at this point I'm losing the overview a bit, but I think it's a very exciting area.

There's lots of opportunities to do interesting work.

And what I'll tell you is I'll tell you a little bit about the roots of polynomials,

not too much, nothing too crazy.

And then I'll tell you about a nonlinear PD.

And this PD has really nice properties.

So at some point we sort of simulated it a little and we started proving cool results

about polynomials, which was, you know, it's rare that you can use PDEs to prove results

about polynomials.

So that was sort of cool.

And then it turns out there's some connection to other fields of mathematics that maybe

I don't know too much about.

And I'll tell you what I know.

And I'll conclude with a PD in a complex plane that is also quite interesting and has many

beautiful properties.

Okay, so throughout this talk, PN is always a polynomial, and it would be a polynomial

of degree N. And I will assume that the roots are all distinct.

If a couple of roots happen to collide, many things still go through, but it's much easier

and might save you some.

And my interest was sort of started by the following theorem, which goes back to Gauss

and Lucas sort of independently.

It's not clear Gauss ever published it.

I think it's in his collected works in some sort of minor comment section.

But it says if you have roots, if you have a polynomial and you differentiate the polynomial,

then the roots of the derivative are contained in the convex hull of the original roots.

So I thought this was quite funny because this is true for both real and complex polynomials,

so with really complex roots if you want.

So you have your roots in the plane and there's a complex hull, and then the roots of the

derivative are all inside.

And it seems a bit mysterious until you sort of see the trick.

The trick is not too hard.

The trick is that you look at PN prime over PN, the roots are all simple and easy, so

this thing is going to have roots in the places where the derivative of PN has a root.

And you see this is essentially something like an electrostatic potential.

So if you think of the roots of the polynomial as electrons that are sort of fixed in the

plane, maybe complex valued electrons, but okay.