Interview with Academician I. M. Gelfand

Kvant 1989, No. 1 p. 3, 4, 5, 6, 7, 9, 10, 12 Tag 91IX

Rough automatic draft, low quality: expect mistakes and awkward wording. It will be replaced by a better translation. Original →

INTERVIEW WITH ACADEMICIAN
I. M. GELFANDOM

Gelfand’s mathematical seminar, which has been meeting on Monday evenings at Moscow University for 45 years, is universally famous. Both first-year students and famous scientists study at the seminar. Several generations of outstanding mathematicians were educated by this seminar.

Israel Moiseevich is the founder of the All-Russian Correspondence Mathematical School and the chairman of its Scientific Council. The main task of this school is to help those schoolchildren who, unlike residents of Moscow, Leningrad and other large cities, are practically deprived of mathematical literature and contacts with scientists. Created 25 years ago, the All-Use Correspondence Mathematics School was the first school of its type in our country; Subsequently, other correspondence schools were opened following its example.

The meeting with Academician Gelfand was always planned as a general interview «Kvant». We prepared questions that we thought would be interesting to both our reader and our interlocutor. Israel Moiseevich, having looked through the list of questions, noted that the questions were very interesting, but he… did not consider himself competent to answer them.

You understand, I do not consider myself to have the right to express my opinion to your young reader. Let me tell you how we do mathematics at an age that coincides with the age of most of the readers of Quantum - from 13 to 17 years old. I’m not sure that I now remember all the tasks that I was doing during this time, but I remembered the tasks that I’m going to talk about very well. Here is the story of I.M. Gelfand.

One of Graham Greene’s novels is called “The loser takes all”). My mathematical biography was surprisingly happy, and for many years it was the fulfillment of this statement. How am I lucky? In short, firstly, that I did not study at the university (or any educational institution at all), and, secondly, that as a result of certain life difficulties that befell my family, I found myself in Moscow without parents, unemployed, at sixteen and a half years old.

I. M. Gelfand talks with the outstanding French mathematician J.-P. Serre after his speech at Gelfand’s seminar at Moscow State University. April 1984.

I’ll try to understand the meaning of the phrase “the loser takes all” with the help of a story by another English writer, Somerset Mose. The main character of the story, a certain church servant, fails: during the certification of church personnel, it turns out that he is illiterate, and he is kicked out of work. He starts working as a seller of sirah, then buys out a table kiosk, then several other kiosks… and makes a brilliant commercial career, becoming the richest man in his city, its zer. And so they interview him - like you are interviewing me now - and he tells the journalist that he is illiterate. The needy journalist will exclaim: “What heights would you have reached if you were literate!” The answer immediately follows: “I would be a church minister.”

So in February 1930, at the age of 16 and a half, having arrived in Moscow to visit distant relatives, I was partly unemployed, took on any temporary work, and mostly sat in the Lenin Library and “acquired” the knowledge that I had not received at school and at an independent vocational school. In the library I met university students and began attending seminars. At the age of 18 I was already teaching, and at 19 I went to graduate school, and my further mathematical biography followed an ordinary, regular course, here it fits into the mainstream that is inculcated for mathematicians.

But I didn’t want to tell the reader «Kvant» about this part of my life, but about the previous one - about how I studied mathematics between the ages of 13 and 17. I want to talk about this for two reasons. Firstly, in my deep conviction, in the majority of future professional mathematicians, mathematical talent manifests itself precisely in the period from 13 to 16 years**), although, of course, there are fluctuations, from earlier development to much later development (from 20 to 30 and even 40 years), which is observed in some strong mathematicians. Secondly, this period formed

*) Loser takes all
**) Israel Monseevich spoke about this in great detail in an interview published in Izvestia on December 3, 1987.

Val is my way of doing mathematics. The subject of research, of course, varies
was created, but the artistic image of mathematics that developed at that time
was the basis in the choice of tasks that have attracted me to this day.
Have a nice day. It seems to me that without understanding this motivation it is impossible
understand each algorithm of my ways of knowing and choosing topics
busy. In light of this motivation, they actually developed very differently.
therefore it is logical.

Gelfand’s seminar. For per-
second table from the left -
the growth of the seminar is famous
American mathematician
R. McPherson. 1981

The first thing I remember is from 5th-6th grade. I then realized that there was a task
questions in geometry that cannot be solved algebraically. I made a table
face the ratio of chord length to arc length every 5°. Only a lot of poses
I learned that there are trigonometric (not algebraic!) functions
and that in essence I then compiled trigonometric tables.

Around the same time, I worked through a problem in elementary algebra.
I didn’t have an algebra course, I didn’t know the theory, but sometimes I had to decide
difficult tasks, I use, for example, (unknown to me) abbreviated formulas -
multiplication. If I couldn’t decipher how to solve a given problem,
Well, I looked into the answer, learned from the formulation of the problem and the answers to restore
identify methods for solving them. In particular, I then understood and remembered completely
life, that a new area can be mastered by solving problems, and there is no shame in
look back, because when we solve any problem, we always
there is a hypothesis about the answer. Doing mathematics is generally similar to solving
problems in which something is known about the answer. These mathematical
bot differs from the training required, of course, to perform a competition
no work for entering a university.

At the beginning of 6th grade, I paid attention to geometry problems in which
a right-angled triangle often appeared with sides 3, 4, 5, and even
sides 5, 12, 13. I wanted to find all right triangles with
whole sides, and I found the promised formula for their sides, i.e. I found all Pi-
Fagorian triplets. (Of course, I didn’t know this term then.) Unfortunately,
I don’t remember now how I did it.

I studied mathematics during illnesses and holidays. I have now replaced
It’s amazing how much strong schoolchildren manage to do when they stay away due to illness
Houses. Therefore, after illness for several personal days, I
left it at home.

We go through geometry according to Kiselev, not yet spoiled by subsequent
processing. Some theorems were given in the form of a problem. I achieved my promise
notebook (in those years it was not so easy), wrote on each page

condition of the theorem and filled almost all the pages with proofs over the summer. This is how I learned to write mathematical papers.

I’ll miss a big period. I will only mention a book on Davydov’s algebra with ingenious methods for solving maximum and minimum problems using elementary methods (i.e., without the use of differential calculus). For example, find the maximum of $ab$ if $a+b$ is given; find a rectangle of maximum area for a given period; find the maximum of the product of non-negative numbers $a_1a_2...a_n$ , if their sum $a_1+a_2+...+a_n$ is given; squares are cut out of a square on a given side, and a box is made from the remaining part; How many squares should be cut to maximize the volume of the box?

I was greatly impressed by combinatorics and Newton’s biome, which I had been thinking about for a long time.

I lived in a small town with only one school. My mathematics teacher was a very kind, although seemingly stern man named Titarenko. He had a large Zaporozhye mustache. I have never met a better teacher, although I knew more than him, and he understood it. He loved me very much and encouraged me in every possible way. Encouraging is the most important thing for a teacher, isn’t it?

The supply of books on mathematics was significant. I saw in advertisements that there were books on higher mathematics, and I imagined higher mathematics as something very interesting. My parents could not order these books - there was no money. But here I was lucky again. At the age of 15, I was taken to Odessa to have appendicitis operated on. I told my parents that I would not go to hospital until they bought me books on higher mathematics. The parents agreed and bought Belyaev’s higher mathematics course for colleges in Ukrainian. However, there was only enough money for the first part, with a presentation of differential calculus and analytical geometry on the plane.

I was lucky that I didn’t start with an in-depth university course. It was a very basic book. The level of Belyaev’s book can be judged by the introduction to it. It said, in particular, that there are three types of functions: analytical, defined by formulas; empirical, given by tables; correlational. I learned what correlation functions are many years later from a student studying probability theory.

On the third day after the operation, I took up this book and read it, interspersed with reading the novels of E. Zola, for nine days (then after such operations it was 12 days). During this time I read it.

From the book I got two great ideas. Firstly, any geometric problem on a plane and in space can be written down by formulas. However, I suspected this before. I also learned about the existence of such wonderful funcurds, such as the ellipse.

The second idea - it revolutionized my worldview - was that there is a formula for calculating sinhus: sin $x=x-\frac{x^3}{3!}+\frac{x^5}{5!}$ -…

Before this, I believed that there are two mathematics - algebraic and geometric, and that geometric mathematics belonged to the “transcendent” for algebraic. Take, for example, the formula for circularity - there is a “geometric” number l. Or, let’s say, sinhus - it is determined purely geometrically.

When I discovered that sinhus can be written algebraically in the form of a series, the barrier fell, mathematics became unified. And now I perceive the various branches of mathematics together with mathematical physics as a single whole.

Of course, I was convinced that extreme problems are solved automatically, and although they lose their charm, you have in your hands a powerful apparatus for solving them.

While studying differential calculus, I learned that there is also interral calculus related to areas and volumes. But I didn’t know what it consisted of: after all, I didn’t have the second volume of Belyaev’s course!

Here I want to remember one more task. When we studied the volumes of bodies of rotation at school the following fall, my classmate, the later famous mathematician D.P. Milman, drew my attention to the following problem: find the volume of a body formed by the rotation of a circle around a tangent

Presentation of an honorary doctorate degree from the University of Lyon to I. M. Gelfand. June 1984.

Noah. To solve this problem, I divided the circle into strips and calculated the sum of the differences in the volumes of the corresponding rotation cylinders. Thus I came to the need to find the sum

$\cos \varphi + \cos 2\varphi + \cos 3\varphi + \dots + \cos n\varphi.$

(*)

What happened next was, as always, a mixture of ingenuity and stupidity. I simplified the mismo of the elementary solution using general trigonometry, but used the formula $e^{i\varphi} = \cos \varphi + i \sin \varphi$ (this formula is called Euler’s formula, but I didn’t know that). I got this formula from the series for sine, cosine and $e^x$ that struck me deeply. Next, it remains to find the sum of the geometric progression $e^{i\varphi} + e^{2i\varphi} + \dots$ and get the sum (*) from it, which is what I did.

On this task, I got into the habit of thinking about the problem even after solving it. Namely: he moved the circle away from the straight line and realized that when rotated, a body is obtained that resembles a rubber circle, on which

*) $i$ — minimum unit ( $i^2 = -1$ ); about complex numbers can be read, for example, in «Kvant» (1983, No. 2, p. 16).

Adding these equalities, I got

$F(n+1) - F(1) = S_0 + \frac{S_1}{2!} + \frac{S_2}{3!} + \dots,$

where $S_0$ is the amount I’m interested in, and

$S_1 = f'(1) + f'(2) + \dots + f'(n),$

$S_2 = f''(1) + f''(2) + \dots + f''(n),$

Then I wrote down the system

$F(n+1) - F(1) = S_0 + \frac{S_1}{2!} + \frac{S_2}{3!} + \frac{S_3}{4!} + \dots,$

$f(n+1) - f(1) = S_1 + \frac{S_2}{2!} + \frac{S_3}{3!} + \dots,$

$f'(n+1) - f'(1) = S_2 + \frac{S_3}{2!} + \dots,$

i.e. an infinite system with an infinite number of unknowns $S_0, S_1, S_2$ , … I already mentioned that Kagan’s book had infinite determinants, so I was able to use the “Cramer’s rule” to find $S_0$ :

$S_0 = \frac{1}{1}$

I expanded the numerator of this “fraction” into the first column, obtaining

$S_0 = B_0(F(n+1) - F(1)) + B_1(f(n+1) - f(1)) + B_2(f'(n+1) - f'(1)) + \dots,$

(1)

where $B_0 = 1, B_1, B_2, \dots$ are numerical determinants of infinite order. The resulting expression is called the Euler-Maclaurin formula, but of course I didn’t know about it. To calculate it, it was necessary to find the coefficients $B_0, B_1, B_2, \dots$ .

To do this, I applied considerations that today are called “functional”. Namely, taking advantage of the fact that the coefficients $B_0, B_1, \dots$ does not depend on $f$ , I selected a function $f$ such that the left side of the system formed a geometric progression (which I knew how to sum). The function $f(x) = e^{\alpha x}$ is suitable for this, substituting it into formula (1), I got (do the calculations!)

$B_0 + \alpha B_1 + \alpha^2 B_2 + \dots = \frac{\alpha}{e^{\alpha}-1},$

i.e., he obtained the product of the function for the required numbers $B_0, B_1, B_2, \dots$ (they are called Bernoulli numbers, and the polynomial $S_0$ for $f(x) = x^p$ is called the Bernoulli polynomial).

From this period I remember two more tasks. The first one came from a problem in our algebra assignment: Express $x_1^2 + x_2^2$ and $x_1^3 + x_2^3$ in terms of the coefficients of a quadratic equation whose roots are $x_1$ and $x_2$ . A NATURAL GENERALIZATION IS THE PROBLEM: Express the sums $x_1^2 + \dots + x_n^2$ and $x_1^3 + \dots + x_n^3$ , where $x_1, \dots, x_n$ are the roots of the equation $x^n + a_1x^{n-1} + \dots + a_n = 0$ , via

ego coefficients. Here Bezout’s theorem helped me, which I learned from Davydov’s textbook. I went further and set myself a more general task: to express the sum of the $k$ -th powers of the roots of the $n$ -th degree algebraic equation through the coefficients of this equation. I solved this problem (the answer is called Newton’s formula).

The second problem I was solving then arose when I discovered that cos $ix$ is a real number, or

$\cos ix = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \dots$

I thought about this unexpected fact and proved the following promising theorem: every even real function takes real values on the minimum axis.

To prove this, it was necessary to clarify what a “function” is. I thought about what to call a function, and accepted the following definition: a function is the sum of a convergent infinite degree of a series. After this, the proof of the theorem is almost obvious.

This task, apparently, was the last thing I thought about before my arrival in Moscow. I solved it in the summer of 1929, and in the next six months, which were very difficult for my family and for me, I had no time for mathematics.

The further period of my studies in Moscow, unlike what was described, was not a “pure experiment”. In Moscow I was already exposed to many different influences, and my development no longer followed its own path. During this period, as I already said, I studied on my own in the library named after. V.I. Lenin, surviving on official earnings. However, time was even a control in Leninka. I met some university mathematics students. One of them told me that my interest in expressions of the form $f(n+1)-f(n)$ is associated with a whole science called finite difference theory, and suggested that I need to read Nörlund’s book “Differenzenkalkül” on this topic. The book was in German, but I could understand it with the help of a dictionary.

I started going to university seminars, and then I found myself under heavy psychological pressure. I found that my style of doing math just wasn’t working. New things have appeared in mathematics - new requirements regarding the rigor of proofs, great interest in the theory of functions of a real variable (TFV). (I note that today both this level of strictness and the TFDP are considered old-fashioned and outdated, but then…)

Then I realized that it is very important that a function does not have to be wrong, that a continuous function does not have to be differentiable, that a differentiable function does not have to be twice differentiable, etc.; that even if a function has arbitrary ones of all orders, then its Taylor series does not have to converge and that, even if it converges, its sum does not have to coincide with the value of the function! If this coincidence takes place, then such a function is called analytic, and this class of functions, the adherents of the TFDP argued, is so narrow that it is outside the main intervals of mathematics. And I only considered such functions!

Under the pressure of this point of view, I read a “modern, rigorous” textbook on analysis by Vallée Poussin, similar to those textbooks that are used today at the Moscow State University Mechanics, but better than them. Therefore, I have the understanding of a sympathetic first-year mechanite who is allowed to the beauties of mathematical analysis only after a one-year probationary period, a test of “rigorous justification” of the analysis.

But even here I was lucky. I began to read the wonderful textbook by I. I. Pri-

the clock makes the box (see figure). What size squares should be cut to maximize the volume of the box?

It is clear that you cannot cut out very small squares, since the box will turn out to be too small. You cannot cut out very large squares, as then the box will be too narrow. How much should you cut?

Note: box volume $V = x(a - 2x)^2$ . Woo-
dem look for maximum $4V = (4x)(a - 2x)(a - 2x)$ ,
i.e., the maximum of the product of three numbers whose sum is 2a.

Zadchacha. Calculate the sums of $\cos x + \cos 2x + \dots + \cos nx$ and $\sin x + \sin 2x + \dots + \sin nx$ .
While calculating the volumes and surface areas of a rotating body, I.M. Gelfand discovered Guldin’s remarkable theorems:

the volume of a body of rotation of a flat figure around an axis that does not intersect it is equal to the product of the area of this figure and the length of the circle described by the center of the gravity;

The surface area of the body of rotation is equal to the product of the length of the boundary of the figure and the length of the circle described by the center of gravity of this boundary.

Zadchacha. At what distance from the center are the centers of the gravity of a semicircle and a semicircle?

To get acquainted with the determinants, let’s solve a system of linear equations:

$\begin{cases} a_1x + b_1y = c_1, \\ a_2x + b_2y = c_2. \end{cases}$

If $a_1b_2 - a_2b_1 \neq 0$ , its solution gives the formulas:

$x = \frac{c_1b_2 - c_2b_1}{a_1b_2 - a_2b_1}, y = \frac{a_1c_2 - a_2c_1}{a_1b_2 - a_2b_1}.$

The quantity $a_1b_2 - a_2b_1$ in the denominator is denoted by $\begin{matrix} a_1 & b_1 \\ a_2 & b_2 \end{matrix}$ and is called the determinant. Numerals also contain determiners $\begin{matrix} c_1 & b_1 \\ c_2 & b_2 \end{matrix}$ and $\begin{matrix} a_1 & c_1 \\ a_2 & c_2 \end{matrix}$ .

Similarly, when solving the system

$\begin{cases} a_1x + b_1y + c_1z = d_1, \\ a_2x + b_2y + c_2z = d_2, \\ a_3x + b_3y + c_3z = d_3 \end{cases}$

the denominator will contain third-order determinants. Determinant

$\begin{matrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{matrix}$

equalized $a_1(b_2c_3 - b_3c_2) + a_2(b_3c_1 - b_1c_3) + a_3(b_1c_2 - b_2c_1)$ . Note that there is a second-order determinant in parentheses. This notation of the determinant is called its decomposition in the first column.

It is possible to agree on which one-years of the variables $a_1, a_2, \dots, d_4$ the determinant of the fourth order consists of:

$\begin{matrix} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ a_3 & b_3 & c_3 & d_3 \\ a_4 & b_4 & c_4 & d_4 \end{matrix}.$

It is much more difficult to guess what the signs of these annuals are.

Square tables of numbers are called (square) matrices, and the corresponding determinants are called matrix determinants. It is interesting that now I.M. Gelfand is fascinated by the question: what should the determinant of a cubic matrix be equal to? When the answer is received, he will be found to tell about it on the pages «Kvant».

Zadchacha. Prove that

$1^2 + 2^2 + \dots + n^2 = n(n+1)(2n+1)/6$

And

$1^3 + 2^3 + \dots + n^3 = n^2(n+1)^2/4.$

It’s great that $1^3 + 2^3 + \dots + n^3 = (1 + 2 + \dots + n)^2$ . I wonder if this can be proven directly?

Bernoulille numbers make it possible to calculate the total sum $1^p + 2^p + \dots + n^p$ as a polynomial in $n$ of degree $p+1$ .

Problems. a) Express the sum of the squares of the roots of the equation $x^2 + px + q = 0$ in terms of its coefficients.

The same applies to the sum of cubes of the roots of this equation.

b) The same - for the sum of squares and cubes of the roots of the equation

$x^n + a_1x^{n-1} + \dots + a_n = 0.$

Hint: use Vieta’s formulas:

$a_1 = -(x_1 + x_2 + \dots + x_n), \quad a_2 = x_1x_2 + x_1x_3 + \dots + x_n-1x_n, \quad a_3 = -(x_1x_2x_3 + x_1x_2x_4 + \dots + x_n-2x_nx_1x_n),$ where $x_1, x_2, \dots, x_n$ are the roots of the equation.

Solving the general problem of how the sum of powers of the roots of an equation is expressed through its coefficients, I. M. Gelfand proved Newton’s formulas:

$p_k + a_1p_k-1 + a_2p_k-2 + \dots + a_k-1p_k = 0, \quad k \leq n, \quad p_k + a_1p_k-1 + a_2p_k-2 + \dots + a_np_k-n, \quad k > n.$

Here $p_k$ denotes the sum of the $i$ powers of the roots $x_i^1 + x_i^2 + \dots + x_i^n$ .

Newton’s formulas can be considered as a system of linear equations for $p_i$ with coefficients $a_1, a_2, \dots, a_n$ . If you solve this system, you will get the Variant formula, which directly expresses the sum of the degrees of the equations through its coefficients. In modern mathematics, this complexion has received far development.