Experiences of transcendental mathematics or mathematical folklore

On March 16, at 19:00, an interactive lecture on the theme “Experiences of Transcendental Mathematics or Mathematical Folklore will be held in the framework of the project“ Science is not flour ”on the World Day of Pi. ".

Do not worry, if you have already forgotten what logarithm is and how to calculate the integral, you will not need this. The necessary knowledge for the lecture is common sense and elementary logic.

You can often hear that mathematics is unimaginably boring and too abstract. We will try to prove the opposite with numerous examples of mathematical folklore, and the starting point of our meeting will be the book by Eduard Fraenkel “Love and Mathematics. The heart of the hidden reality . The book of the famous scientist makes an attempt to dispel the myth that mathematics is a boring science. From the lecture you will learn why all the horses are of the same color, why the moon is made of cheese and how to catch a fly on the back of the moon.
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Your guide in complex mathematical labyrinths will be Vitaly Filippovsky - a mathematician, a graduate student at ITMO, a leading mathematician and programmer of Emoji Apps.

Below we offer you to read the excerpt “Exquisite Dance” from the book by Frenkel.

In the fall of 1990, I became a graduate student at Harvard. This was necessary in order to change the position of a visiting professor to something more permanent. Joseph Bernstein agreed to become my official supervisor. By that time, I had gained enough material for a PhD thesis, and Arthur Jaffe persuaded the dean of the faculty as an exception to allow me to reduce the period of postgraduate study (which usually takes 4 or 5 years, and in any case at least 2 years, according to the rules) years, so that I can defend myself in a year. Due to this, my “demotion” from professor to graduate student lasted quite a bit.

My PhD thesis was devoted to a new project that I have just completed. It all started with a discussion with Dringfeld of the Langlands program in the spring of that year. Here is an example of one of our conversations, decorated in the form of a script.

ACTION 1
SCENE 1
CABINET DRINFELDA IN HARVARD

Drinfeld paces the room along the wall on which the blackboard hangs.
Edward, sitting in a chair, takes notes (on the table next to him is a cup of tea).

Drinfeld
So the Shimura – Taniyama – Weil hypothesis opens up a connection between cubic equations and modular forms, but Langlands went even further. He predicted the existence of a more general correspondence, in which automorphic representations of a Lie group play the role of modular forms.

Edward
What is an automorphic representation?

Drinfeld (after a long pause)
The exact definition does not matter now. In any case, you can find it in the textbook. What is important for us is that this is a representation of the Lie group G, for example the SO (3) group of rotations of the sphere.

Edward
Good. And what are these automorphic representations associated with?

Drinfeld
This is the most interesting. Langlands predicted that they should
be related to Galois group representations in another Lie group.1

Edward
Clear. Do you mean that this Lie group is not the same G group?

Drinfeld
Not! This is another Lee group called the Langlands dual group for G. Drinfeld writes the symbol LG on the board.

Edward
The letter L in honor of Langlands?

Drinfeld (with a slight smile)
Initially, Langlands was driven by a desire to understand objects called L-functions, because he called this group L-group ...

Edward
That is, for each Lie group G there is another Lie group called LG, right?

Drinfeld
Yes. And it is present in accordance with Langlands, which schematically looks like this. Drinfeld draws a scheme on the board



Edward
I don't understand ... at least for now. But let me ask a simpler question: what would the dual Langlands group look like for SO (3), for example?

Drinfeld
It's pretty simple - the double covering of SO (3). Have you seen a trick with a cup?

Edward
Focus with a cup? Oh yes, I remember ...

SCENE 2
HOME PARTY OF HARWARD POSTGRADUATES

A dozen or so students, all in their twenties, talk, drink beer and wine. Edward talks with a graduate student.

Graduate student
Here is how it is done.
A graduate student takes a plastic cup of wine and puts it on the open palm of his right hand. Then she begins to rotate the palm, turning her hand as in a sequence of photos
(below). She makes one full turn (360 degrees), and her hand turns her elbow up. Still holding the glass vertically, it continues to rotate, and after
one more full turn is a surprise! - her hand and cup return to their original normal position.

Another graduate student
I heard that in the Philippines there is a traditional dance with wine in which they do this trick with both hands. He takes two glasses of beer and tries to turn both palms.
at the same time. But it’s impossible to keep track of hands, and he immediately spills beer from both. Everybody laughs.

SCENE 3
AGAIN CABINET DRINFELDA

Drinfeld
This focus illustrates the fact that there is a nontrivial closed path on the group SO (3), the double passage of which, however, gives us a trivial path.

Edward
Oh, I understand. The first full rotation of the cup rotates the hand at an unusual angle - this is an analogue of the non-trivial way to SO (3). He takes a cup of tea from the table and does the first part of the focus.

Edward
It would seem that the second turn should make you turn the hand even more, but instead the hand returns to its usual position. Edward completes the move.

Drinfeld
For sure.

Edward
But what does this have in common with the Langlands dual group?

Drinfeld
The Langlands dual group for SO (3) is the double covering of SO (3), so ...


Edward
So each element of the group SO (3) corresponds to two elements from the dual Langlands group.

Drinfeld
That is why in this new group there are no longer non-trivial closed paths.

Edward
That is, the transition to the dual Langlands group is a way to get rid of that dislocation?

Drinfeld
Right. At first glance it seems that the difference is minimal, but in reality the consequences are more than significant. This, for example, explains the difference in the behavior of building matter bricks, such as electrons and quarks, and particles carrying
interactions between them, such as photons. For more general types of Lie groups, the difference between the group itself and its dual Langlands group is even stronger. In fact, in many cases there is not even a visible link between the two dual groups.

Edward
Why did the dual group appear at all according to Langlands? Some kind of magic ...

Drinfeld
It is unknown.

The Langlands duality establishes a pairwise relationship between Lie groups: for each Lie group G, there is a dual group Lee Langlands LG, and a dual
LG is G.9 itself. The fact that the Langlands program links objects of two different types (one from number theory, and the second from harmonic analysis) is surprising in itself, but the fact that two dual groups, G and LG, are present in different parts of this correspondence - it is just incomprehensible to the mind!

We talked about the Langlands program that connects different continents in the world of mathematics. Let's continue the analogy: let it be Europe and North America and let there be a way
to compare each person in Europe with a person from North America, and vice versa. Moreover, suppose that this correspondence implies a perfect coincidence of various attributes, such as weight, height and age, with one exception: a woman is compared to every man, and vice versa. This situation is analogous to replacing a Lie group with its dual group,
according to the prediction of the Langlands program.

Indeed, this replacement is one of the most mysterious aspects of the Langlands program. We know several mechanisms that describe how dual groups appear, but we are
still do not understand why this is happening. This ignorance was one of the reasons why scientists are trying to extend the ideas of the Langlands program to other areas of mathematics (through the Weil Rosetta stone) and even to quantum physics, as we will learn in the next chapter. We are trying to find more examples of the phenomenon of dual Langlands groups in the hope that this will give us additional clues as to why they arise and what it means.

Let's focus our attention so far on the right column of the Rosetta Weil stone, which is dedicated to Riemann surfaces. As we established in the previous chapter, in the version of the correspondence of Langlands, relevant for this column, the characters are “automorphic pencils”. They play the role of automorphic functions (or automorphic representations) associated with the Lie group G. It turns out that these automorphic pencils “live” in a certain space attached to the Riemann surface X and the group G, which is called the moduli space of G-bundles on X. at the moment it does not matter what it is. 10 In the opposite part of the correspondence, as we saw in Chapter 9, the role of the Galois group is played by the fundamental group of a given Riemann surface. From the scheme above, it follows that the geometric correspondence of Langlands should schematically look like this:


This means that we should be able to compare the automorphic pencil to each representation of the fundamental group in LG. And Drinfeld had a radically new idea on how to do this.

ACTION 2
SCENE 1
CABINET DRINFELDA IN HARVARD

Drinfeld
So, we need to find a method for constructing these automorphic sheaves. And it seems to me that the representations of Kac-Moody algebras could help us.

Edward
Why?

Drinfeld
Now we are in the world of Riemann surfaces. Such a surface may have a boundary consisting of loops.

Drinfeld draws a picture on the board.



Drinfeld
Through loops, Riemann surfaces can be associated with loop groups and, therefore, with Kac-Moody algebras. And this connection gives us the opportunity to transform the views.
Kac – Moody algebras in sheaves on the moduli space of G-bundles on our Riemann surface. Let's not go into details yet. As I expect, schematically this
should look like this.

Drinfeld draws a scheme on the board.



Drinfeld
The second hand is clear to me. The main question is how to design the first arrow. Feigin told me about your work on the ideas of Kac-Moody algebras. I think it just needs to be applied here.

Edward
But then the representations of the Kac-Moody algebra for G must somehow be “known” about the LG Langlands dual group.

Drinfeld
Exactly.

Edward
But how is this possible?

Drinfeld
And this is a question you must answer.

A CURTAIN