The invitation to Erika and Jasmine from a very, very large city, Egypt, they are here

and so I know them from time to time. So the reason why I keep my, most of them, whatever

they touch, they go, so I'm quite certain that this will be a very successful initiative.

So it's always, it's really my first time in Erwangen, I mean the city we were in 52 years

ago, that is trying to force this airway problem and as Jasmine mentioned, the airway is not

a city. Jackie mentioned the city of Max and Emynote. So Emynote certainly had a lot to

contribute to science, but certainly in a strange way to topology and water peyote and

education. So the relationship between Russia and Germany probably cannot be described with

a differential equation that has a smooth solution, but there was a period of time after

Checherian science, the peace treaty of Livorno in 1922 where there was quite a bit of interaction.

But what's most important for us is the intellectual interaction and in the 20s, so at the end of

the 20s, a little bit before she immigrated to Bryn Mawr, Emynote visited Moscow and lectured

there. And so there was a guy, a very young guy at the lecture who listened to her lecture and he

couldn't go home after the lecture by himself just because he was blind. But then he was taken to his

room in the dorm and he announced that he knows the solution of the problem proposed by Emynote at

her lecture. So Komogorov was very impressed by this and went to find this guy in his dorm. And indeed

he did and there was a solution. The solution was correct. This is how Puntragon was created. The

name of the at that time student was Lev Davidovich Puntragon and this is how we have

Komogorov theory and topology initiated. He was one of the fathers together with Dines Khov.

So now what I'll talk about is very little in common with what other talks were. I know nothing about

PDEs except that I personally know very well Idrits Titi, a dear friend of mine for many, many years.

But so the problem that I'll discuss is a problem for birational geometry, a very classical one,

which I'll formulate soon. But the approach to it will be different and will be based on ideas from

theoretical physics. So we'll introduce a new invariant based on homological mirror symmetry, which

I think not very smartly were called atoms, but it wasn't my choice. And then I'll give some examples

and if there is time I'll discuss new directions. So let me actually try to give a quick course in

birational geometry. So birational geometry starts when the fusion of rational functions

over briboracy is a dimension N, it's morphed into the fusion of rational functions at the variables.

But let's try to do it in a little bit more detail. So if you have a conic, so you can put a point on it

and then you project the lines, you see that there is one-to-one correspondence between this conic complex coordinates and

the line. So this goes back to Archimedes. So if you try to do this with the equation of the grid 3,

so you see that that cannot be quite done. And in fact, because this picture doesn't give you a way to do this,

doesn't mean that there is no other way to do this. But this is where the German school comes. And then Gauss and then Riemann,

by introducing the genus of Riemann surface as an abstraction to rationality, show that this cannot be done.

So now, close to 100 years later, people looked at a cubic equation, but with one more variable.

And as a product of the work of Italian school and also of Zaritsky, who was actually Ukrainian originally,

was shown that this actually is variational. Namely, it has a filter fraction of two independent variables.

Now, this is around 1930. And then people added one more variable. And there was a lot of work by Fano, Weill, and so on.

And then only in 1972, Griffiths, using highly non-gebrach methods,

coming from partially topology, partially from differential equations, since the heat equation has a solution theta functions.

The singularities of these theta functions were used by Andriot and Mayer theory. So this has shown that, indeed, if you go to these four variables, this is not variational.

So now the point of the talk today will be to add one more variable and try to answer this question.

But classical methods will not work. I mean, many mathematicians have looked at it, certainly in France, in Germany, Russia.

But since the audience is mainly people in differential equations, let me just give you an answer. What is going to follow is analysis of behavior and the solutions of the following differential equation.

So the asymptotics in the behavior of this differential equation under blow-off, the asymptotics of the behavior of this differential equation under blow-off,

is going to be the main thing that we will be studying. But of course, in order to make this connection, I need to go to several different subjects, including mirror symmetry.

So let me just briefly tell you what the modes of mirror symmetry are.

So the modes of mirror symmetry, as formulated by Kuntzevich, is a correspondence of categories.

And let me try to explain this with the example of P2. So P2 is a variety which all of you will learn as undergraduates.

So basically, it gets a lot of different line bundles and then complexes of line bundles and so on. And you put them together in something which is called a direct category of coherence and so on.