Lem, Tokarchuk, Krakow, mathematics

On September 3-7, 2019, the anniversary congress of the Polish Mathematical Society took place in Krakow. Anniversary, because the centenary of the founding of the Society. It existed in Galicia from the 1st years (without the adjective that the Polish-liberalism of the emperor FJ1919 had its limits), but as a nationwide organization it operated only from 1919. Major advances in Polish mathematics date back to the 1939s XNUMX-XNUMX. XNUMX at the Jan Casimir University in Lviv, but the convention could not take place there – and it’s not the best idea either.

The meeting was very festive, full of accompanying events (including a performance by Jacek Wojcicki at the castle in Niepolomice). The main lectures were delivered by 28 speakers. They were in Polish because the invited guests were Poles - not necessarily in the sense of citizenship, but recognizing themselves as Poles. Oh yes, only thirteen lecturers came from Polish scientific institutions, the remaining fifteen came from the USA (7), France (4), England (2), Germany (1) and Canada (1). Well, this is a well-known phenomenon in football leagues.

The best constantly perform abroad. It's a little sad, but freedom is freedom. Several Polish mathematicians have made overseas careers that are unattainable in Poland. Money plays a secondary role here, but I do not want to write on such topics. Maybe just two comments.

In Russia, and before that in the Soviet Union, this was and is at the most conscious level ... and somehow no one wants to emigrate there. In turn, in Germany, about a dozen candidates apply for a professorship at any university (colleagues from the University of Konstanz said that they had 120 applications in a year, 50 of which were very good, and 20 were excellent).

Few of the Jubilee Congress lectures can be summarized in our monthly journal. Headings such as "Limits of Sparse Graphs and Their Applications" or "Linear Structure and Geometry of Subspaces and Factor Spaces for High-Dimensional Normalized Spaces" will not tell the average reader anything. The second topic was introduced by my friend from the first courses, Nicole Tomchak.

A few years ago, she was nominated for the achievement presented in this lecture. Fields Medal is the equivalent for mathematicians. So far, only one woman has received this award. Also worth noting is the lecture Anna Marcinyak-Chohra (Heidelberg University) "The role of mechanistic mathematical models in medicine on the example of leukemia modeling".

entered medicine. At the University of Warsaw, a group led by Prof. Jerzy Tyurin.

The title of the lecture will be incomprehensible to Readers Veslava Niziol (z prestiżowej Higher Pedagogical School) “-adic theory of Hodge". Nevertheless, it is this lecture that I have decided to discuss here.

Geometry -adic worlds

It starts with simple little things. Do you remember, Reader, the method of written exchange? Definitely. Think back to the carefree years of elementary school. Divide 125051 by 23 (this is the action on the left). Do you know that it can be different (action on the right)?

This new method is interesting. I'm going from the end. We need to divide 125051 by 23. What do we need to multiply 23 by so that the last digit is 1? Searching in memory and we have :=7. The last digit of the result is 7. Multiply, subtract, we get 489. How do you multiply 23 to end up with 9? Of course, by 3. We get to the point where we determine all the numbers of the result. We find it impractical and more difficult than our usual method - but it's a matter of practice!

Things take a different turn when the brave man is not completely divided by the divisor. Let's do the division and see what happens.

On the left is a typical school track. On the right is "our strange ones".

We can check both results by multiplying. We understand the first: one third of the number 4675 is one thousand five hundred and fifty eight, and three in the period. The second one doesn't make sense: what is this number preceded by an infinite number of sixes and then 8225?

Let us leave the question of meaning for a moment. Let's Play. So let's divide 1 by 3 and then 1 by 7 which is one third and one seventh. We can easily get:

1:3=…6666667, 1/7=…(285714)3.

This last line means: block 285714 repeats indefinitely at the beginning, and finally there are three of them. For those who don't believe, here's a test:

Now let's add fractions:

Then we add the received strange numbers, and we get (check) the same strange number.

……95238095238095238095238010

We can check that this is equal to

The gist is yet to be seen, but the arithmetic is correct.

One more example.

The usual, albeit large, number 40081787109376 has an interesting property: its square also ends in 40081787109376. the number x40081787109376, which is ( x40081787109376)² also ends in x40081787109376.

Tip. We have 40081787109376²= 16065496 57881340081787109376, so the next digit is three to ten's complement, which is 7. Let's check: 740081787109376²= 5477210516110077400817 87109376.

The question of why this is so is a difficult one. It's easier: find similar endings for numbers ending in 5. Continuing the process of finding the next digits indefinitely, we will come to such "numbers" that ²=²= (and none of these numbers is equal to zero or one).

we understand well. The farther after the decimal point, the less important the number is. In engineering calculations, the first digit after the decimal point is important, as well as the second, but in many cases it can be assumed that the ratio of the circumference of a circle to its diameter is 3,14. Of course, more numbers need to be included in the aviation industry, but I don't think there will be more than ten.

The name appeared in the title of the article Stanislav Lem (1921-2006), as well as our new Nobel laureate. Lady Olga Tokarchuk I only mentioned this because screaming injusticeThe fact is that Stanislav Lem did not receive the Nobel Prize in Literature. But it's not in our corner.

Lem often foresaw the future. He wondered what would happen when they became independent of humans. How many films on this topic have appeared lately! Lem quite accurately predicted and described the optical reader and the pharmacology of the future.

He knew mathematics, although sometimes he treated it as an ornament, not caring about the correctness of the calculations. For example, in the story "Trial", the Pirks pilot goes into orbit B68 with a rotation period of 4 hours and 29 minutes, and the instruction is 4 hours and 26 minutes. He remembers that they calculated with an error of 0,3 percent. He gives the data to the Calculator, and the calculator replies that everything is fine ... Well, no. Three tenths of a percent of 266 minutes is less than a minute. But does this error change anything? Maybe it was on purpose?

Why am I writing about this? Many mathematicians have also raised this question: imagine a community. They don't have our human mind. For us, 1609,12134 and 1609,23245 are very close numbers - good approximations to the English mile. However, computers may consider the numbers 468146123456123456 and 9999999123456123456 to be close. They have the same twelve-digit endings.

The more common digits at the end, the closer the numbers. And this leads to the so-called distance -adic. Let p be equal to 10 for a moment; why just “for a while”, I will explain ... now. The 10 point distance of the numbers written above is 

or one millionth - because these numbers have six common digits at the end. All integers differ from zero by one or less. I won't even write a template because it doesn't matter. The more identical numbers at the end, the closer the numbers (for a person, on the contrary, the initial numbers are considered). It is important that p be a prime number.

Then - they like zeros and ones, so they see everything in these patterns: 0100110001 1010101101010101011001010101010101111.

In the novel Glos Pana, Stanisław Lem hires scientists to try to read a message sent from the afterlife, coded zero-one of course. Does anyone write to us? Lem argues that "any message can be read if it is a message that someone wanted to tell us something." But is it? I will leave readers with this dilemma.

We live in XNUMXD space R³. Letter R recalls that the axes consist of real numbers, i.e. integers, negative and positive, zero, rational (i.e. fractions) and irrational, which readers met at school (), and numbers known as transcendental numbers, inaccessible in algebra (this is the number π, which has been connecting the diameter of a circle with its circumference for more than two thousand years).

What if the axes of our space were -adic numbers?

Jerzy Mioduszowski, a mathematician at the University of Silesia, argues that this could be so, and even that it could be so. We can (says Jerzy Mioduszewski) occupy the same place in space with such beings, without interfering and without seeing each other.

So, we have all the geometry of "their" world to explore. It is unlikely that “they” think the same way about us and also study our geometry, because ours is a borderline case of all “their” worlds. "Them", that is, all the hellish worlds, where they are prime numbers. In particular, = 2 and this fascinating world of zero-one ...

Here the reader of the article may become angry and even angry. "Is this the kind of nonsense that mathematicians do?" They fantasize about drinking vodka after dinner, with my (=taxpayer's) money. And disperse them into four winds, let them go to state farms ... oh, there are no more state farms!

Relax. they always had a penchant for such jokes. Let me just mention the sandwich theorem: if I have a cheese and ham sandwich, I can cut it in one cut to halve the bun, ham, and cheese. This is useless in practice. The point is that this is just a playful application of an interesting general theorem from functional analysis.

How serious is it to deal with -adic numbers and related geometry? Let me remind the reader that rational numbers (simplistically: fractions) lie densely on the line, but do not fill it closely.

Irrational numbers live in "holes". There are many, infinitely many of them, but you can also say that their infinity is greater than that of the simplest, in which we count: one, two, three, four ... and so on up to ∞. This is our human filling of "holes". We have inherited this mental structure from Pythagoreans. 

But what is interesting and important for a mathematician is that one cannot "fill" these holes with irrational and p-adic numbers (for all primes p). For those readers who understand this (and this was taught in every high school thirty years ago), the point is that every sequence that satisfies Cauchy's state, converges.

A space in which this is true is called complete ("nothing is missing"). I will remember the number 547721051611007740081787109376.

The sequence 0,5, 0,54, 0,547, 0,5477, 0,54772 and so on converges to a certain limit, which is approximately 0,5477210516110077400 81787109376.

However, from the point of view of 10-adic distance, the sequence of numbers 6, 76, 376, 9376, 109376, 7109376 and so on also converges to the "strange" number ... 547721051 611007740081787109376.

But even that may not be enough reason to give scientists public money. In general, we (mathematicians) defend ourselves by saying that it is impossible to predict what our research will be useful for. It is almost certain that everyone will be of some use and that only action on a broad front has a chance of success.

One of the greatest inventions, the X-ray machine, was created after radioactivity was accidentally discovered Bekkerela. If not for this case, many years of research would probably have been useless. "We are looking for a way to take an x-ray of the human body."

Finally, the most important thing. Everyone agrees that the ability to solve equations plays a role. And here our strange numbers are well protected. The corresponding theorem (I hate minkowski) says that some equations can be solved in rational numbers if and only if they have real roots and roots in every -adic body.

More or less this approach has been presented Andrew Wiles, which solved the most famous mathematical equation of the last three hundred years - I recommend readers to enter it into a search engine "Fermat's Last Theorem".