Viviane Ledoux

Guillaume Moroz

Evaluation of Chebyshev polynomials on intervals and application to root finding

Keywords: Clenshaw algorithm, Chebyshev polynomials, Root finding, Ball arithmetic, Interval arithmetic

In approximation theory, it is standard to approximate functions by polynomials expressed in the Chebyshev basis. Evaluating a polynomial f of degree n given in the Chebyshev basis can be done in O(n) arithmetic operations using the Clenshaw algorithm. Unfortunately, the evaluation of f on an interval I using the Clenshaw algorithm with interval arithmetic returns an interval of width exponential in n. We describe a variant of the Clenshaw algorithm based on ball arithmetic that returns an interval of width quadratic in n for an interval of small enough width. As an application, our variant of the Clenshaw algorithm can be used to design an efficient root finding algorithm.

Introduction

Clenshaw showed in 1955 that any polynomial given in the form

p(x) = n i=0 a i T i (x) (1)
can be evaluated on a value x with a single loop using the following functions defined by recurrence:

u k (x) =          0 if k = n + 1 a n if k = n 2xu k+1 (x) -u k+2 (x) + a k if 1 ≤ k < n xu 1 (x) -u 2 (x) + a 0 if k = 0 (2)
such that p(x) = u 0 (x).

Unfortunately, if we use Equation [START_REF] Boyd | Finding the zeros of a univariate equation: Proxy rootfinders, chebyshev interpolation, and the companion matrix[END_REF] with interval arithmetic directly, the result can be an interval of size exponentially larger than the input, as illustrated in Example 1.

Example 1. Let ε > 0 be a positive real number, and let x be the interval [ 1 2ε, 1 2 + ε] of width 2ε. Assuming that a n = 1, we can see that u n-1 is an interval of width 4ε. Then by recurrence, we observe that u n-k is an interval of width at least 4εF k where (F n ) n∈N denotes the Fibonacci sequence, even if all a i = 0 for i < n.

Note that the constant below the exponent is even higher when x is closer to 1. These numerical instabilities also appear with floating point arithmetic near 1 and -1 as analyzed in [START_REF] Gentleman | An error analysis of Goertzel's (Watt's) method for computing Fourier coefficients[END_REF].

To work around the numerical instabilities near 1 and -1, Reinsch suggested a variant of the Clenshaw algorithm [START_REF] Gentleman | An error analysis of Goertzel's (Watt's) method for computing Fourier coefficients[END_REF][START_REF] Oliver | An Error Analysis of the Modified Clenshaw Method for Evaluating Chebyshev and Fourier Series[END_REF]. Let d n (x) = a n and u n (x) = a n , and for k between 0 and n -1, define d k (x) and u k (x) by recurrence as follows:

d k (x) = 2(x -1)u k+1 (x) + d k+1 (x) + a k u k (x) = d k (x) + u k+1 (3)
Computing p(x) with this recurrence is numerically more stable near 1. However, this algorithm does not solve the problem of exponential growth illustrated in Example 1.

Our first main result is a generalization of Equation 3 for any value in the interval [-1, 1]. This leads to Algorithm 1 that returns intervals with tighter radii, as analyzed in Lemma 2. Our second main result is the use of classical backward error analysis to derive Algorithm 2 which gives an even better radii. Then in Section 3 we use the new evaluation algorithm to design a root solver for Chebyshev series, detailed in Algorithm 3.

Evaluation of Chebyshev polynomials on intervals

Forward error analysis

In this section we assume that we want to evaluate a Chebyshev polynomial on the interval I. Let a be the center of I and r be its radius. Furthermore, let γ and γ be the 2 conjugate complex roots of the equation:

X 2 -2aX + 1 = 0. (4)
In particular, using Vieta's formulas that relate the coefficients to the roots of a polynomial, γ satisfies γ + γ = 2a and γγ = 1. Let z n (x) = a n and u n (x) = a n , and for k between 0 and n -1, define z k (x) and u k (x) by recurrence as follows:

z k (x) = 2(x -a)u k+1 (x) + γz k+1 (x) + a k u k (x) = z k (x) + γu k+1 (x) (5)
Using Equation ( 4), we can check that the u k satisfies the recurrence relation

u k (x) = 2xu k+1 (x) -u k+2 (x) + a k , such that p(x) = xu 1 (x) -u 2 (x) + a 0 .
Let (e k ) and (f k ) be two sequences of positive real numbers. Let B R (a, r) and

B R (u k (a), e k ) represent the intervals [a -r, a + r] and [u k (a) -e k , u k (a) + e k ]. Let B C (z k (a), f k ) be the complex ball of center z k (a) and radius f k .
Our goal is to compute recurrence formulas on the e k and the f k such that:

z k (B R (a, r)) ⊂ B C (z k (a), f k ) u k (B R (a, r)) ⊂ B R (u k (a), e k ). (6)
Lemma 1. Let e n = 0 and f n = 0 and for n > k ≥ 1:

f k = 2r|u k+1 (a)| + 2re k+1 + f k+1 e k = min(e k+1 + f k , f k √ 1-a 2 ) if |a| < 1 else e k+1 + f k (7)
Then, (e k ) and (f k ) satisfy Equation [START_REF] Moroz | [END_REF].

Proof (sketch). For the inclusion z k (B R (a, r)) ⊂ B C (z k (a), f k ), note that γ has modulus 1, such that the radius of γz k+1 is the same as the radius of z k+1 when using ball arithmetics. The remaining terms bounding the radius of z k follow from the standard rules of interval arithmetics.

For the inclusion u k (B R (a, r)) ⊂ B C (u k (a), e k ), note that the error segment on u k is included in the Minkowski sum of a disk of radius f k and a segment of radius e k+1 , denoted by M . If θ is the angle of the segment with the horizontal, we have cos θ = a. We conclude that the intersection of M with a horizontal line is a segment of radius at most min(e

k+1 + f k , f k √ 1-a 2 ). Corollary 1. Let B R (u, e) = BallClenshawForward((a 0 , . . . , a n ), a, r) be the result of Algorithm 1, then p(B R (a, r)) ⊂ B R (u, e)
Moreover, the following lemma bounds the radius of the ball returned by Algorithm 1.

Lemma 2. Let B R (u, e) = BallClenshawForward((a 0 , . . . , a n ), a, r) be the result of Algorithm 1, and let M be an upper bound on

|u k (a)| for 1 ≤ k ≤ n. Assume that ε k < M r for 1 ≤ k ≤ n, then        e < 2M n 2 r if n < 1 2 √ 1-a 2 e < 9M n r √ 1-a 2 if 1 2 √ 1-a 2 ≤ n < √ 1-a 2 2r e < 2M (1 + 2r √ 1-a 2 ) n -1 if √ 1-a 2 2r < n Proof (sketch). We distinguish 2 cases. First if n < 1 2 √
1-a 2 , we focus on the relation e k ≤ e k+1 + f k + M r, and we prove by descending recurrence that e k ≤ 2M (nk) 2 r and f k ≤ 2M r(2(nk -1) + 1).

For the case

1 2 √ 1-a 2 ≤ n, we use the relation e k ≤ f k √ 1-a 2 +M r, that we substi- tute in the recurrence relation defining f k to get f k ≤ 2rM + 2r √ 1-a 2 f k+1 +f k+1 + M r √ 1 -a 2 .
We can check by recurrence that

f k ≤ 3 2 M √ 1 -a 2 (1 + 2r √ 1-a 2 ) n -1 , which allows us to conclude for the case √ 1-a 2 2r
≤ n. Finally, when

1 2 √ 1-a 2 ≤ n < √ 1-a 2
2r , we observe that

(1 + 2r √ 1-a 2 ) n -1 ≤ n exp(1) 2r
en ← 0 for k in n -1, n -2, . . . , 1 do f k ← 2r|u k+1 | + 2re k+1 + f k+1 e k ← min(e k+1 + f k , f k √ 1-a 2 ) + ε k f0 ← r|u1| + 2re1 + f1 e0 ← min(e1 + f0, f 0 √ 1-a 2 ) + ε0 return B R (u0, e0)

Backward error analysis

In the literature, we can find an error analysis of the Clenshaw algorithm [START_REF] Elliott | Error analysis of an algorithm for summing certain finite series[END_REF]. The main idea is to add the errors appearing at each step of the Clenshaw algorithm to the input coefficients. Thus the approximate result correspond to the exact result of an approximate input. Finally, the error bound is obtained as the evaluation of a Chebyshev polynomial. This error analysis can be used directly to derive an algorithm to evaluate a polynomial in the Chebyshev basis on an interval in Algorithm 2. Lemma 3. Let e n = 0 and for n > k ≥ 1:

e k = 2r|u k+1 (a)| + e k+1 (8)
and

e 0 = r|u 1 (a)| + e 1 . Then (e k ) satisfies u k (B R (a, r)) ⊂ B R (u k (a), e k ).
Proof (sketch). In the case where the computations are performed without errors, D. Elliott [3, Equation (4.9)] showed that for γ = xx we have:

p(x) -p(x) = 2γ n i=0 u i (x)T i (x) -γu 1 (x)
In the case where x = a and x ∈ B R (a, r) we have γ ≤ r and

|T (x)| ≤ 1 which implies e k ≤ r|u 1 (a)| + n i=2 2r|u i (a)|.

Application to root finding

For classical polynomials, numerous solvers exist in the literature, such as those described in [START_REF] Kobel | Computing real roots of real polynomials ... and now for real![END_REF] for example. For polynomials in the Chebyshev basis, several approaches exist that reduce the problem to polynomial complex root finding [START_REF] Boyd | Computing zeros on a real interval through chebyshev expansion and polynomial rootfinding[END_REF], or complex eigenvalue computations [START_REF] Boyd | Finding the zeros of a univariate equation: Proxy rootfinders, chebyshev interpolation, and the companion matrix[END_REF] among other.

In this section, we experiment a direct subdivision algorithm based on interval evaluation, detailed in Algorithm 3. This algorithm is implemented and publicly available in the software clenshaw [START_REF] Moroz | [END_REF].

We applied this approach to Chebyshev polynomials whose coefficients are independently and identically distributed with the normal distribution with mean 0 and variance 1.

As illustrated in Figure 1 our code performs significantly better than the classical companion matrix approach. In particular, we could solve polynomials of degree 90000 in the Chebyshev basis in less than 5 seconds and polynomials of degree 5000 in 0.043 seconds on a quad-core Intel(R) i7-8650U cpu at 1.9GHz. For comparison, the standard numpy function chebroots took more than 65 seconds for polynomials of degree 5000. Moreover, using least square fitting on the ten last values, we observe that our approach has an experimental complexity closer to Θ(n 1.67 ), whereas the companion matrix approach has a complexity closer to Θ(n 2.39 ).

√ 1 -a 2 Algorithm 1

 121 which leads to the bound for the last case. Clenshaw evaluation algorithm, forward error function BallClenshawForward((a0, . . . , an), a, r) Computation of the centers u k un+1 ← 0 un ← an for k in n -1, n -2, . . . , 1 do u k ← 2au k+1 -u k+2 + a k ε k ← bound on the rounding error for u k u0 ← au1 -u2 + a0 ε0 ← bound on the rounding error for u0 Computation of the radii e k fn ← 0

Corollary 2 .Algorithm 2

 22 Let B R (u, e) = BallClenshawBackward((a 0 , . . . , a n ), a, r) be the result of Algorithm 2, and let M be an upper bound on |u k (a)| for 1 ≤ k ≤ n. Assume that ε k < M r for 1 ≤ k ≤ n, then e < 3M nr. Clenshaw evaluation algorithm, backward error function BallClenshawBackward((a0, . . . , an), a, r)Computation of the centersu k un+1 ← 0 un ← an for k in n -1, n -2, . . . , 1 do u k ← 2au k+1 -u k+2 + a k ε k ← boundon the rounding error for u k u0 ← au1 -u2 + a0 ε0 ← bound on the rounding error for u0 Computation of the radii e k en ← 0 for k in n -1, n -2, . . . , 1 do e k ← e k+1 + 2r|u k+1 | + ε k e0 ← e1 + r|u1| + ε0 return B R (u0, e0)

2Fig. 1 .

 1 Fig. 1. Time for isolating the roots of a random Chebyshev polynomial, on a quad-core Intel(R) i7-8650U cpu at 1.9GHz, with 16G of ram

Algorithm 3 Subdivision algorithm for root finding

Require: (a0, . . . , an) represents the Chebyshev polynomial approximating f (x) (b0, . . . , bn) represents the Chebyshev polynomial approximating df dx (x) Ensure: Res is a list of isolating intervals for the roots of f in [-1 , 1] function SubdivideClenshaw((a0, . . . , an), (b0, . . . , bn))

append the pair (B R (a, r), "plus") to P artition else if f + s < 0 then append the pair (B R (a, r), "minus") to P artition else if g -s > 0 or g + s < 0 then append the pair (B R (a, r), "monotonous") to P artition else B1, B2 ← subdivideB R (a, r) append B1, B2 to L Compute the sign of F at the boundaries B R (f, s) ← BallClenshaw ((a0, . . . , an), -1, 0) append the pair (B R (-1, 0), sign(B R (f, s))) to P artition B R (f, s) ← BallClenshaw ((a0, . . . , an), 1, 0) append the pair (B R (1, 0), sign(B R (f, s))) to P artition


Athanase Papadopoulos

Galina Sinkevich

A NOTE ABOUT MIKHA ÏL LAVRENTIEFF AND HIS WORLD OF ANALYSIS IN THE SOVIET UNION

Keywords: AMS classification: 30C20, 30C35, 30C70, 30C62, 30C75, 37F30, Mikhaïl A. Lavrentieff, theory of functions of a real variable, theory of functions of a complex variable, quasiconformal mapping, history of quasiconformal mappings, Nikolai N. Luzin, Pavel

We survey the life and work of Mikhaïl Lavrentieff, one of the main founders of the theory of quasiconformal mappings, with an emphasis on his training years at the famous Moscow school of theory of functions founded by Luzin. We also mention the major applications of quasiconformal mappings that Lavrentieff developed in the physical sciences. At the same time, we review several connected historical events, including the sad fate of Luzin during the Stalin period, the birth of the scientific Siberian center of Akademgorodok, and the role that Lavrentieff played in the organization of science and research in the Soviet Union. The final version of this paper will appear in Vol. VII of the Handbook of Teichmüller theory (European Mathematical Society Publishing house, 2020).

Introduction

This is a one-sided report on the life and work of Mikhaïl Alekseïevich Lavrentieff 1 (1900-1980), one of the main founders of the theory of quasiconformal mappings. Besides a few notes on Lavrentieff's work on the theory of functions of a complex variable, the report includes information on some aspects of his life and on some of the persons who influenced him. Lavrentieff had a densely packed academic life, often with heavy administrative duties, and he interacted with a diverse collection of people in the Soviet Union. He held many honorific positions, and some of them were with much responsibility: he was Vice-President of the Ukrainian Academy of Sciences (1945)(1946)(1947)(1948), Vice-President of the USSR Academy of Sciences, President of the Siberian Division of that Academy (1957)(1958)(1959)(1960)(1961)(1962)(1963)(1964)(1965)(1966)(1967)(1968)(1969)(1970)(1971)(1972)(1973)(1974)(1975), and Vice-President of the International Mathematical Union (1966)(1967)(1968)(1969)(1970). He played a preeminent role in the creation of Akademgorodok, the well-known science town in Siberia, which was founded in 1957. In more political settings, he was an elected deputy of the Supreme Soviet of the Ukraine SSR (1947)(1948)(1949)(1950)(1951) and of the Supreme Soviet of the USSR .

The present report will not do justice to Lavrentieff from the purely scientific point of view because his work encompasses much more than the topics we review here. In fact, Lavrentieff was one of the most preeminent 20th century scientists of the Soviet Union. He had a brilliant career, as a mathematician, as a physicist, and as a leader in science organisation. In mathematics, he worked on functions of one and several complex variables, on conformal and quasiconformal geometries, on partial differential equations, on nonlinear problems, and on the calculus of variations, but perhaps his most important achievement was the broad use he made of mathematics in the other sciences. With his profound intuition, Lavrentieff applied the mathematical tools he developed (in particular the theory of quasiconformal mappings) to solve difficult problems in physics and engineering. The topics included flows around air wings,2 shock waves, the motion of a plate under the surface of a liquid (in particular the motions of diving planes of submarines) and the resistance of buildings to the flow of ground water. He was the first to understand that matter (metal) behaves like an incompressible fluid under the effect of an explosion, an idea which led to several technological applications of explosives, like the construction of dams by blasting. He was responsible for the construction of several networks of soil projects which permitted the regulation of lakes and rivers, preventing natural catastrophes in several regions of the USSR. We shall report briefly on some of these achievements.

Beyond the mathematics and physics of Mikhaïl Lavrentieff, we have tried to give an idea of his creative style, of the incredible amount of energy that emanated from him, and on his ability to find connections between abstract mathematical theories and concrete physical situations.

The personal life of Lavrentieff and his mathematical training are tied to an important chapter of the history of mathematics in the Soviet Union: the Luzin school and the "Luzin affair," and I have included in this report a section on that subject.

Acknowledgements. Bill Abikoff read carefully a first version of this article and sent me a number of suggestions that led to a thorough revision. Yuri Neretin and Alexei Sossinsky read a second version and made several suggestions and corrections. Galina Sinkevich sent me very useful suggestions that amend some of my statements concerning the Moscow School of analysis in which Lavrentieff was trained and shed a different light on some facts that I relate. I asked her to write an appendix explaining her ideas. The appendix follows this article in the present volume. I would like to thank Abikoff, , Neretin, Sossinsky and Sinkevich for their care and invaluable help.

Family

The information we have on the history of the Lavrentieff family begins with the birth of Mikhaïl Lavrentieff's father. Our description of this event and of the few years that followed it is based on the collection of papers The age of Lavrentieff [START_REF] Dobretsov | The age of Lavrentieff, Novosibirsk, Siberian division of the Russian Academy of[END_REF] which appeared on the occasion of Lavrentieff's hundredth anniversary. 3 Mikhaïl's father, Alexey Lavrentievich Lavrentieff, was born in 1875, as an illegitimate child, in Paris, where his mother, who was Russian, was sent to give birth. His names were given to him in the Russian orthodox church of Paris by the priest who baptized him, whose own name was Lavrentii (Lawrence). Deriabina [19] tells us that Alexey Lavrentievich never knew his mother and was brought up in a foster family. His childhood was so difficult that he never wanted to talk about it. He married, at the age of 20, a woman who was 12 years older than he was, and their son Mikhaïl Lavrentieff was born at the turn of the twentieth century.

Alexey Lavrentievich studied mathematics and became a mathematics teacher at the technical school of Kazan, the capital of Tatarstan. In 1910, he passed the Master's exam in mechanics at Kazan University. This is a provincial university, but it is also the university where Lobachevsky studied and spent his whole career.

After obtaining his diploma, Alexey Lavrentieff was sent for two years of study in Göttingen and Paris, the two cities which were the world mathematical centers. In those times, it was common that a talented Russian young scientist go to France or to Germany, for a few years of training. Lavrentieff 's family accompanied him.

Mikhaïl Lavrentieff, who was ten years old in Göttingen, recalls in his memoirs [START_REF] Dobretsov | The age of Lavrentieff, Novosibirsk, Siberian division of the Russian Academy of[END_REF] that at school, he was frequently harassed by his fellow pupils who used to point their fingers at him, shouting: "Russian, Russian." Even the teachers were harsh; they hit him because of his lack of knowledge in German. His parents eventually stopped sending him to school; instead, each night his father read to him, in German, the tales of the Brothers Grimm. The Lavrentieffs socialized with the Russians living in Göttingen; accompanied by his wife, Nikolai Luzin was there, studying with Edmund Landau. The young Mikhaïl was fascinated by the conversations he heard at home concerning mathematical problems and new theories.

On his return from Paris, Alexey Lavrentieff was appointed to a position on the Faculty of Mathematics and Physics of Kazan University. He remained in contact with Luzin, who, after his stay in Göttingen, spent a few months in Paris and then returned to Moscow. In 1921, Luzin suggested that the Lavrentieff family move to Moscow. In 1922, after obtaining his undergraduate diploma, Mikhaïl became a graduate student at the Institute for Mechanics and Mathematics of Moscow State University and was supervised by Luzin. Egorov was teaching there and in 1923, he became director of the Institute; he noticed Lavrentieff's talent during an examination session and later assisted him and helped prepare him for his career as a researcher. During his studies, Lavrentieff attended courses on function theory by Luzin, and seminars on topology by P. S. Alexandrov, who was Luzin's student.

The relation between Luzin and the Lavrentieff family continued. Mikhaïl Lavrentieff remained close to Luzin whose life was marred with tragic events; we shall outline some of them in the next section. 3 The collective book The age of Lavrentieff contains valuable information about Mikhaïl Lavrentieff's life and epoch. It is written by his former students and colleagues, and it also contains Lavrentieff's own memories. Let me also mention the paper "Lavrentieff" [19] by Angelina Deriabina. I am grateful to Valerii A. Galkin and Mikhaïl Lavrentiev Jr. for providing these two references.

Luzin

Nikolai Nikolaïevich Luzin (1883Luzin ( -1950) ) together with his mentor, Dmitri Fiodorovich Egorov (1869Egorov ( -1931)), 4 are the founders of the Moscow school of function theory. Luzin was one of the architects of "descriptive set theory," a field closely related to topology whose goal was to set up firm foundations for the topological and set-theoretical methods of the theory of functions of a real variable. This field emerged as one of the most active research topics in the first quarter of the twentieth century. The leaders in the field included Cantor (who, by the way, was Russian-born), Borel, Baire, Hadamard, Lebesgue, Denjoy, Painlevé, Hausdorff and Hilbert.

It is worth emphasizing here that some of the greatest geometers were reluctant to the search for functions with very unusual behavior that this modern theory of functions led to. We can quote here Poincaré who wrote in his book Science et méthode [83, § V], talking about this field:

Logic sometimes generates monsters. The last half-century saw the emergence of a horde of bizarre functions whose endeavor seems to be that of having the least possible resemblance with the honest functions that are useful. No more continuity, or continuity but no derivative, etc. [. . . ] In the old days, when a new function was invented, it had some practical aim; today, we invent them intentionally in order to destroy our father's reasonings, and this is all we can get from them. 5 We also recall Hermite's words in a well-known letter to Stieltjes, dated May 20, 1893 [5, p. 318]: "Analysis takes away with one hand what was given with the other. I move away with horror from that dreadful plague of continuous functions that have no derivative." 6 Borel, Baire and other French mathematicians among those we mentioned spent some effort putting some order in the multitude of new functions that were discovered and whose properties were shaking the classical theory of functions. They introduced the language of set theory in analysis and they tried a classification of functions in terms of set-theoretic properties and limit operations. Set theory was an emerging field which, with its pseudo-logical apparatus, gives the impression of a firm ground. One may recall, as an example of a new object introduced, the notion of "Baire function", obtained from continuous functions by a transfinite repetition of the operation of taking simple limits.

A foundational crisis had arisen in set theory, and more generally, in mathematics; topology and the theory of functions were affected by it. In particular, Luzin was concerned with the difficulties that arise from the use of transfinite constructions and the continuum hypothesis, from the discovery of hierarchies between infinities, from the formulation of the axiom of choice, 7 etc. considers that the Moscow school of theory of functions was born with Egorov's Comptes Rendus note Sur les suites des fonctions mesurables [START_REF] Egorov | Sur les suites de fonctions mesurables[END_REF]. 5 La logique parfois engendre des monstres. Depuis un demi-siècle on a vu surgir une foule de fonctions bizarres qui semblent s'efforcer de ressembler aussi peu que possible aux honnêtes fonctions qui servent à quelque chose. Plus de continuité, ou bien de la continuité, mais pas de dérivée, etc. [. . . ] Autrefois, quand on inventait une fonction nouvelle, c'était en vue de quelque but pratique ; aujourd'hui on les invente tout exprès pour mettre en défaut les raisonnements de nos pères et on n'en tirera jamais que cela. 6 L'Analyse retire d'une main ce quelle donne de l'autre. Je me retourne avec effroi et horreur de cette plaie lamentable des fonctions continues qui n'ont pas de dérivées. 7 I wrote "formulation" instead of "discovery" because before Zemelo formulated this principle as an axiom, it was used without any special notice in mathematics. Baire, Borel, Hadamard and Lebesgue are among the mathematicians who refused the use of the axiom of choice, considering it as counter-intuitive. According to Lebesgue, in order for a mathematical object to exist, one

The era was also marked by conflicting priority claims and other disputes, especially in France; for example, Lebesgue and Borel had noteworthy differences regarding the theories of measure and integration. There were other controversies concerning intuitionism, constructive vs. non-constructive mathematics, etc. The problem of the non-contradiction of the foundations of mathematics was seriously raised. In fact, in these years, mathematics was closely related with philosophy. This was not new; the relation between the two fields dates back to Greek Antiquity, but its appearance at the forefront of research was episodic. Constructions that use infinite processes were carefully revised, from the philosophical as well as from the mathematical viewpoint. It was during that period that Russell and Whitehead showed how to write an incoherent book on the foundations; see the comments by Grattan-Guinness in [START_REF] Grattan-Guinness | Algebras, Projective Geometry, Mathematical Logic, and Constructing the World: Intersections in the Philosophy of Mathematics of A. N. Whitehead[END_REF].

Religion was also present in the philosophico-mathematical debate, in France 8 and at a much higher level of acuteness in Russia, where a group of mathematicians who were profoundly religious raised questions such as to what extent infinitywhich, for them, is an attribute of God-can be used in human contexts, like the sciences.

Set theory combined with the theory of functions were concerned with the general classification of sets and their behavior under various operations and mappings. Open and closed sets were the basic sets of topology, but these sets are not stable under simple operations. G δ -and F δ -sets were introduced as countable unions (respectively intersections) of open (respectively closed) sets. Other new classes of sets arose: A-sets (A stands for "analytic"), 9 B-sets (B stands for Borel), 10 Baire first category sets and second category sets, 11 Luzin sets, 12 etc. Lavrentieff showed the topological invariance of Suslin sets and Borel sets. Lebesgue did not exclude the existence of sets that are neither finite nor infinite. Luzin introduced the notion of projective set, that is, a set obtained from a B-set in n-dimensional Euclidean space by operations of projection and taking complement. Many other classes of sets were given names. Some of these sets played an important role in the formulation of the bases of measure and integration theory. Notions of named sets (introduced by Lebesgue) and effective sets (sets whose definition does not use the axiom of choice) were also introduced.

At the time Luzin was studying mathematics, he traversed a spiritual crisis that shaked what he called his "materialistic worldview," an expression which, at that has to able to explicitly "name" a property that defines it in a unique way. See the interesting historical account made by Bourbaki on this matter in [9, p. 53ff].

8 Hermite and Appell are representative of the religious stream in France. 9 Analytic sets were later called Suslin sets. Suslin was a student of Luzin who died at the age of 25, in 1919, from typhus, an epidemic which followed the Russian Civil War. The existence of these sets started with a remark he made, saying that a continuous image of a Borel set is not necessarily a Borel set, correcting a mistake made by Lebesgue. He was led therefore to define a class of sets which became these "analytic sets" (see the 4-page note [START_REF] Ya | Sur une définition des ensembles mesurables B sans nombres transfinis[END_REF] which is the only work he published during his lifetime). Luzin, who gave Suslin the motivation to introduce these sets, continued the latter's work on these sets after his death. 10 Borel sets are subsets of an n-dimensional Euclidean space that are obtained from segments by repeated application of the operations of taking countable union and countable intersection.

11 A Baire first category set is a subset of a Baire topological space contained in a countable union of closed subsets which all have empty interior. A Baire second category set is a subset of a Baire topological space which is not a Baire first category set.

12 A Luzin set is a subset of the real numbers which is uncountable but whose intersection with every set of the first Baire category is countable. epoch, meant atheism." 13 The crisis lasted three years and, presumably, was eventually resolved with help of Pavel Florensky (1882Florensky ( -1937)), an extremely talented fellow student in mathematics at the University of Moscow who had strong philosophical and mystic inclinations. The relation between the two men was decisive, and it led to Luzin's recovery of a peace of mind and his return to mathematics.

A few words about Florensky are in order. Florensky entered the mathematics division of Moscow University in 1900, one year after Luzin-this was also the year Lavrentieff was born. Among Florensky's teachers were Nikolai Bugaev, an outstanding mathematician who studied under Weierstrass and Kummer in Berlin, and Liouville in Paris. Bugaev was also Egorov's mentor. He belonged to a group of mathematicians with strong religious beliefs, those who later were accused of being "reactionaries" and "enemies of the people". Bugaev was also the father of Boris Nikolaevich Bugaev, the writer and poet who published under the pseudonym Andrei Bely and who is sometimes considered as the founder the Russian symbolist movement. From the religious/intellectual point of view, one of the themes that was promoted by these mathematicians was that religion is not subject to mathematical logic. They developed instead a line of thought called "paraconsistent logic," where apparent inconsistencies and contradictions are allowed, and where the "principle of explosion" (ex contradictione quodlibet), saying that anything follows from a contradiction does not hold. Mathematically, this principle is the "reasoning by contradiction" which is at the basis of set theory, which, as we recalled, was a rapidly growing field.

It is not easy for a mathematician to accept that the logical system that he is used to is not the unique possible. The position these mathematicians held is in fact close to Pascal's point of view, that something else than "reason" is used when one talks about God ("La raison n'y peut rien déterminer [. . . ] il faut renoncer à la raison" (Pascal, Pensées). Leonid Sabaneeff, 14 in an article on Florensky, writes that the latter "often spoke of the 'many facets' of any true thought and of the compatibility of contradictions on the deepest level. He even asserted that every perceived law 'generates' its own negation [. . . ] He obviously regarded antinomy as the basic law of the universe, encompassing all others" [86, p. 314].

In 1903, Florensky wrote a paper titled On symbols of the infinite, which is an exposition, intended for a general reader, of the basic notions of set theory newly developed by Cantor. At the same time, the paper contains philosophical and theological thoughts. Through his friendship with Andrei Bely, Florensky had entered the Russian symbolist poets movement, and instead of submitting his paper to a mathematical journal, he published it, in 1904, in the newly founded periodical of Symbolic poetry, Novyj Put' (the New way). The same year, he graduated from the mathematics division of the University of Moscow with the highest grade, after submitting a thesis titled Singularities of algebraic curves. A concise overview of this thesis, with several interesting comments, is contained Demidov's article [START_REF] Demidov | On an early history of the Moscow school of theory of functions[END_REF].

Florensky turned down a graduate fellowship from Moscow University and entered the Moscow Theological Academy, from where he graduated in 1908 and that I have known up to now (material worldviews) absolutely do not satisfy me. I may be wrong, but I believe there is some kind of vicious circle in all of them, some fatal reluctance to accept the contingency of matter, some reluctance, which I find absolutely incomprehensible" ([24, p. 335], translated from [START_REF] Demidov | Correspondance entre N[END_REF]). 14 became a priest. At the same time, he became a teacher at the theological seminary. In 1914 he defended a doctoral thesis, with a dissertation titled The pillar and foundation of truth [START_REF] Florensky | The pillar and foundation of truth[END_REF], which was an expanded version of a pamphlet he wrote in 1908, the year he graduated from the mathematics and physics department, and which he had called On spiritual truth. The dissertation was published as a book which became a foundational text for Russian orthodox twentieth-century theology.

In some sense, this new approach to theology was a return to the ancient so-called "apophatic (or negative) theology" which finds its roots in the third-century monks of the desert of Egypt. It asserts that mystical experience does not follow the laws of logical reasoning, and that paradoxical situations, like the truth of an assertion and its negation, are acceptable in theology. It was important at that time that a specialist in the mathematics of Cantor and Peano declares that these logical theories are irrelevant to Christian theology. Florensky also wrote a treatise on religious painting [START_REF] Florensky | La perspective inversée, suivi de L'Iconostase et autres écrits sur l'art[END_REF], and his writings confirmed his belonging to the Russian symbolist artistic movement. During the Soviet period, his books were translated into French and published in Paris, and they were used there in the teaching of the Russian theological seminary which replaced the theological schools that were closed in the Soviet Union.

Egorov and Luzin corresponded with Florensky and read his books. Sabaneeff writes about Florensky: "It was clear to me that mathematics was his guide even in the area of mystic speculations: it helped him not only through the elementary language of numbers (as in the case of many earlier mystics) but by means of the whole panoply of the modern mathematical apparatus: analysis, the theory of sets, and all the latest theories on the boundary between physics and mathematics" [86, p. 321].

Florensky was arrested briefly in 1928, and again in 1933, where, this time, he was sentenced to ten years of corrective labor and deported. In 1934, he was sent to the Soloveckij concentration camp which was opened in 1923 in the Solovetsky Islands, in the White sea of Northern Russia. Historically, these islands were the location of an ancient Orthodox monastery complex and they were transformed into a Gulag, where half-a-million people perished. Solzhenitsyn described the Solovski camp as the "mother of the Gulag."

Florensky is sometimes referred to as the Gulag's Pascal and the Russian Leonardo. He was executed on December 8, 1937. For the relation between Luzin and Florensky, we refer the reader to the paper [START_REF] Ford | The influence of P. A[END_REF] by Charles Ford, which also contains excerpts of the correspondence between the two men, translated from the article [START_REF] Demidov | Correspondance entre N[END_REF]. This correspondence explains in part Luzin's religious crisis which started when he experienced the bloody events of the 1905 political troubles in Russia which were part of a failed revolution that was later referred to by Lenin as "the great rehearsal" for the 1917 October Revolution.

We now return to Luzin. In a tribute to him, published in 1974 [START_REF] Lavrentieff | Nikolai Nikolaevich Luzin (on the 90th Anniversary of His Birth)[END_REF], Lavrentieff writes that as a young boy, Luzin was found to be poor in mathematics, and a student tutor was brought to the house to help him. The tutor discovered that it was difficult for the young Nikolai to understand established theories, but that he was able to quickly solve difficult problems demanding originality, and often with unusual methods.

The University of Moscow was temporarily closed after the 1905 events, and Luzin went to Paris, having obtained, with the support of Egorov, a scholarship to study there. This allowed him to follow courses by Borel, Poincaré, Hadamard, and other preeminent French mathematicians. The period was for Luzin one of profound crisis. In a letter he sent to Florensky from Paris, dated May 1st, 1906, he writes ([24, p. 335], translated from [START_REF] Demidov | Correspondance entre N[END_REF])): "You found me a mere child at the university, knowing nothing. I don't know what happened, but I cannot be satisfied anymore with analytic functions and Taylor series . . . To be precise, it happened about a year ago [. . . ] To see the misery of people, to see the torment of life, to wend my way home from a mathematical meeting, to wend my way through the Alexandrovsky Garden [on the side of the Kremlin nearest to the university], where, shivering in the cold, some women stand waiting in vain for dinner purchased with horror-this is an unbearable sight. It is unbearable, having seen this, to study calmly (in fact, to enjoy) science . . . I barely remember what happened to me. I could not work in science, and I seem to have begun to lose my mind as a result of the impossibility of living quietly and understanding where, where the truth is. Dmitri Fedorovich [Egorov], seeing me in such a state, sent me here to Paris. That is how I got here... I have been here about 5 months, but have only recently begun to study. I lacked the self-awareness . . . I now understand that "science," in essence, is metaphysical and based on nothing . . . At the moment my scholarly interests are in principles, symbolic logic, and set theory. But I cannot live by science alone."

Egorov arranged Luzin's second stay abroad, in 1910-1914, in Göttingen and Paris. It is during that stay in Göttingen that he first met the Lavrentieff family. In Paris, Luzin attended the courses of Picard on the theory of functions and of Bôcher on second order differential equations. He also met Borel, Lebesgue and Denjoy, and he participated actively in Hadamard's seminar; see [START_REF] Phillips | Nikolai Nicolaevich Luzin and the Moscow school of the theory of functions[END_REF].

In 1911 and 1912, Egorov and Luzin published a series of papers in the French Comptes Rendus which sealed the support of the French school of analysis to the Russian one [START_REF] Egorov | Sur les suites de fonctions mesurables[END_REF][START_REF] Luzin | Sur les propriétés des fonctions mesurables[END_REF][START_REF] Luzin | Sur un problème de M. Baire[END_REF]. In particular, Egorov's result in [START_REF] Egorov | Sur les suites de fonctions mesurables[END_REF] on sequences of measurable functions was a substantial improvement of previous works by Borel and Lebesgue. It says that an arbitrary sequence of measurable functions can be made uniformly convergent up to a set of arbitrarily small measure. Luzin's result in [START_REF] Luzin | Sur un problème de M. Baire[END_REF] is a response to a question raised by Baire.

On his return to Moscow, Luzin joined the faculty of Mathematics of the University of Moscow. He was appointed professor there in 1917, just before the revolution.

Lavrentieff's memoirs published in [START_REF] Dobretsov | The age of Lavrentieff, Novosibirsk, Siberian division of the Russian Academy of[END_REF] contain several pages on Luzin. 15 We read there that Luzin's Master's dissertation, Integral and trigonometric series, which he had prepared in Paris and which he defended in 1915, on his return to Moscow, differed substantially from regular dissertations. Along with concrete results in each section, it contained formulations of new problems and statements that were unproved, with only sketches of evidence. Furthermore, the dissertation contained sentences such as: "It seems to me", "I am sure that,", etc. This style did not fit into the classical tradition of mathematical writing, and the mathematicians at Saint Petersburg-the capital-who were in charge of examining the manuscript, were not convinced by its value. Lavrentieff reports that the academician V. A. Steklov made several ironic comments on the document, like: "It seems to him, but it does not seem to me", "the author is a Göttingen chatter," etc.

Egorov appreciated Luzin's monograph and understood that it was a significant contribution to science. He made the rare recommendation to the Academic Council of Moscow University that the text be accepted as a doctoral (instead of Master's) dissertation. The dissertation was accepted as such, the thesis defense took place soon after, and Luzin became a doctor of science. Lavrentieff wrote in his memoirs: "Today, we can see the great importance of the innovative style of the book, and it was especially valuable for the young mathematical generation. The book played a huge role in the formation of the Luzin school. The problems addressed in it by Luzin, and his hypothetical formulations, found a solution in the subsequent works 15 A translation of this part of Lavrentieff's memoirs is also contained in Lavrentieff's paper on Luzin [START_REF] Lavrentieff | Nikolai Nikolaevich Luzin (on the 90th Anniversary of His Birth)[END_REF].

of Luzin himself, as well as in the works of his students. The part 'it seems' and 'I am sure that' was not justified immediately, but it was justified 15 to 30 years later." Luzin's memoir contained several substantial results, one of them saying that any measurable function which takes finite values up to a set of measure zero can be represented by a trigonometric series which converges in the sense of Poisson and Riemann to the given function. In some sense, the memoir is a sequel to Riemann's Habilitation dissertation on trigonometric series Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe (On the representability of a function by a trigonometric series) [START_REF] Riemann | Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe[END_REF].

The period 1917-1921, which saw the October revolution and the civil war was a period of economic devastation in Russia. Luzin, with his students Menchov, Suslin and Khinchin, left Moscow and worked as a teacher at the newly-founded Polytechnic Institute of Ivanovo-Voznesensk, about 250 km North-East of Moscow. He came back to Moscow in 1922.

Lavrentieff writes in [START_REF] Lavrentieff | Nikolai Nikolaevich Luzin (on the 90th Anniversary of His Birth)[END_REF] that about the years 1922-1926, Luzin introduced to the mathematical division of Moscow university a new style of unprepared lectures, far from the standard "good lecture," and that the result was a brilliant, deep and fascinating way of teaching, encouraging the development of originality and independent thinking. He also reports that the essential characteristic of the Luzin school (which became known under the name Luzitania) was "the fostering of independent thought, the capacity to crack problems, to find new methods and to pose new problems". Lavrentieff reports that Luzin prepared his lectures only in outline and that he was often late to his class. He adds that "nearly all his students would arrive on time and discuss problems while waiting in the corridor." Luzin believed that having a strict schedule was not compatible with doing mathematics.

In 1927, Luzin was elected corresponding member of the USSR Academy of Sciences. The following year, he was elected full member, but, surprisingly, in the Department of Philosophy. In 1928, he was Vice-President of the ICM (Bologna). He presented there a paper titled Sur les voies de la théorie des ensembles (On the paths of set theory) in which he discussed the problems that appeared in the theory of functions as a consequence of the questions that were raised in set theory regarding infinity and the definition of a real number.

In 1930, Luzin published an important monograph on set theory, Leçons sur les ensembles analytiques et leurs applications [START_REF] Luzin | Leçons sur les ensembles analytiques et leurs applications, Collection de Monographies sur la théorie des fonctions[END_REF], with an appendix by Sierpiński and a preface by Lebesgue. The book was published in Paris, in the series Monographies sur la théorie des fonctions founded by Emile Borel. In a review of this book in l'Enseignement mathématique, Adolphe Bühl writes: "It seems that this wonderful volume proves that set theory has undergone tremendous progress which ties it permanently with all mathematical disciplines. The philosophical aspect which emerges from the pages written by Mr. Luzin is of the same nature as the one which emerges from group theory or from some extremely general exposition of geometry."

The so-called "Luzin affair" erupted in July 1936. It constitutes one of darkest episodes of the history of Soviet mathematics, if not the darkest one. It is a mixture of a generation dispute and personal interests with a conflict on political, national, religious and ideological beliefs and it ended with a disaster. From the political point of view, the moment looked relatively calm ; this was the period which preceded Stalin's Great Terror or Purges, which started in May-July 1937 and ended in September-November 1938. These purges rippled almost all opposition to Stalin's power. In particular, the scientific community and other independent thinkers were liquidated. About one million people were either executed or sent to concentration camps. 16 In the summer of 1936, a violent political campaign started against Luzin, whose purpose was to expel him from the Academy of Sciences. the campaign was initiated with two articles in Pravda, Answer to Academician N. Luzin (July 2, 1936) and On enemies is Soviet masks (July 3, 1936). These articles were anonymous. 17 In 1936, a Commission of the Presidium of the Academy of Sciences of the USSR concerning N.Luzin was created. It was headed by the Vice-president of the Academy, G. Krzhizhanovsky, and its members were the Academicians A. Fersman, S. Bernshtein, I. Vinogradov, O. Schmidt, A. Bakh, and the Corresponding members L. Shnirelman, S. Sobolev, P. Alexandrov, and A. Khinchin. Additionally, the Academician Nikolay Gorbunov was a member of the Commission, as a political representative in the Academia. Kolmogorov, and Lyusternik, were not members of the Commission, but were invited to attend the meetings. Pontryagin, visited some meetings, but his name is absent from the stenograph. He pronounced a heavy speech in Moscow University against both Luzin and Alexandrov. Most of the members of the Commission, including the invited members, were Luzin's former students or student's students. Gelfond (a student of Hinchin and Stepanov, who were members of the Luzitania) participated to several political attacks that took place at the meetings of the Commission. These meetings were also attended by Kol'man.

The academic accusation claimed that Luzin, by the end of the 1920s, showed a lack of interest in other mathematical fields than the one he was working in, and in the mathematical schools that were emerging in the Soviet Union. It is probably true that Luzin considered problems that are unrelated to function theory and topology as secondary, although his ideas had an undeniable influence on all the schools that were appearing, some of them founded by his students, or his student's students: Kolmogorov and Khinchin in probability, Pontryagin and Alexandrov in modern topology (to which Kolmogorov contributed as well), Gelfond and Shnirelman in number theory, Lyusternik and Shnirelman in functional analysis, etc. Luzin's accusation also included claims made by some of his students saying that he was putting pressure on (other) students to include his name in their publications, or to make them acknowledge his help in their work. He was also blamed for the fact that he published his papers in French rather than Russian journals, pretending that this was a sort of disloyalty to the Soviet power.

Shnirelman, Sobolev, Alexandrov, and Khinchin, Kolmogorov, Pontryagin and Lyusternik were actively against Luzin, whereas Bernshtein defended him, and 16 About the relatively calm period that preceded the Great Terror, we mention a passage that André Weil wrote in his Commentaries to Volume 1 of his Collected Works [90, p. 535]. Weil recalls an international conference on topology that was organized in Moscow, in September 1935, by P. S. Alexandrov and to which he participated. He says that the fact that the conference took place, with a selection of mathematicians from all over the world, seems to him retrospectively as a miracle, and that many mathematicians, including him, thought at that time that this was a sign of the beginning of a liberalization of the Soviet regime. He adds that the big trials that took place just after showed that this was only an illusion. (I am grateful to Yuri Neretin for this reference.) 17 According to Yuri Neretin (correspondence with the author), the first article, titled "Answer to Academician N. Luzin", was signed by Shulyapin, a school director. According to Alexei Sossinsky (correspondence with the author), experts agree that the author of all these articles was Ernest Kol'man, a Marxist philosopher and party bureaucrat who was well known for his activities in the Soviet science community as chief ideological watchdog. Kol'man had a long history (starting in the 1920s) of attacks against Egorov, Luzin and other scientists who did not share his views on science organization. At the time of the Luzin affair, he was the head of the Science Section of the Moscow Committee of the Communist party.

Krzhizhanovsky was at first undecided. According to the Stenograph, Vinogradov was completely silent during the meeetings. Shnirelman, two times and without success, proposed to submit the case to NKVD, the "People's Commissariat for Internal Affairs". 18 A number of scientists outside the Commission defended Luzin, in particular the applied mathematician A. Krylov, the mineralogist V. Vernadsky and the physicist and future Nobel Prize winner P. Kapitsa (the latter wrote a strongly worded letter attacking the Pravda article). Lavrentiev, P. Novikov and L. Keldysh refused to be members of the Commission and to come to the meetings, which apparently required a great deal of courage. Eventually, and surprisingly, Luzin was not condemned as being an "enemy of the people", as was planned by his detractors. But as a consequence of this affair, Luzin was rejected by an important part of the mathematical establishment and the case left a deep wound in the Russian mathematical community. 19 Luzin continued working in isolation. 20 Discussion of the Luzin case remained a taboo in the Soviet Union, until its re-examination several decades later. In 2012, the Russian Academy of Sciences reversed the decision that was taken against Luzin in 1936. The reconstruction of the minutes of the various trials that took place is reported on in the the book The case of Academician Nikolai Nikolaevich Luzin [START_REF] Demidov | The Moscow school of the theory of functions in the 1930s[END_REF] by Demidov and Lëvshin. The reader may refer to the additional information contained in the appendix to the present article, written by Galina Sinkevich. For more details on these events, including an account of the dramatic fates of Egorov, Luzin, Florensky and others, the reader may consult Demidov's article The Moscow school of the theory of functions in the 1930s [START_REF] Demidov | The Moscow school of the theory of functions in the 1930s[END_REF] which constitutes an excellent concise report (in English) about the school founded by Luzin. The Lavrentieff memorial book [START_REF] Dobretsov | The age of Lavrentieff, Novosibirsk, Siberian division of the Russian Academy of[END_REF] also contains a chapter dedicated to the Luzin affair. The book Naming infinity: A true story of religious mysticism and mathematical creativity by Graham and Kantor [START_REF] Graham | Naming infinity: A true story of religious mysticism and mathematical creativity[END_REF], despite many inaccuracies, contributed in the rising of interest among European readers in the Soviet history of mathematics.

Egorov, who had not been politically active, fell into disfavor with the Soviet authorities six years before Luzin. One of the reasons is that he had defended the Church against Marxist attacks. As early as 1903, he had protested a pogrom against Jews. In 1929, after having been President of the Moscow Mathematical Society for seven years, he was dismissed from his position at the University. The next year he was jailed for being a "religious sectarian." Little is known about this sad story and the information about the end of his life contains contradictions. Most probably, Egorov died after a hunger strike. According to Demidov, he died 18 This is the organisation which soon became responsible for mass extrajudicial executions of a large number of citizens of the USSR and which was the official administrator of the Gulag system of forced labour camps. 19 Let me quote here a passage from an email I received on April 29, 2019, from Alexei Sossinsky:

"When the Commission concluded its work, it was expected that it would declare that Luzin is 'an enemy of the people', which meant that he would be sent to the camps or even be given the death penalty. But it didn't! Until a few years ago, we did not know why this happened. The publication of certain documents now gives us the correct story, which involves Kapitsa's letter, the falling of Kol'man into disfavor, Krzhizhanovsky, Molotov, and Stalin, the final decision apparently being taken in a tête-a-tête between Krzhizhanovsky and Stalin." 20 I learned from Neretin that in 1937-39 Luzin made an ingenious work on some types of bendings of two-dimensional surfaces solving a problem which had been discussed during 50 years, see [START_REF] Luzin | Proof of one theorem in the theory of bendings[END_REF]. Unfortunately for him, the solution was negative and his work closed the topic and had no direct continuation. We refer the reader to the interesting review of this problem by Sabitov in [START_REF] Kh | The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research[END_REF]. The author writes there: "a truly dramatic end to this direction of geometry came with Luzin's paper [START_REF] Luzin | Proof of one theorem in the theory of bendings[END_REF]" . In 1946, Luzin published a 900 pages textbook on calculus for technological universities. In 1953 (the year Stalin died), 300 000 copies of the book were already sold.

in the Hospital of the Institute for continuous education of doctors (a branch of the Kazan University), [START_REF] Demidov | professor of Moscow State University, and the doctrine of the veneration of the name of God in Russia during the first third of the 20th century (Russian), Istor[END_REF].

Back to Lavrentieff: his mathematics

After he studied at the Physics and Mathematics Faculty of Kazan State University, Lavrentieff taught mathematics at the Moscow School of Engineering, from 1921 to 1929. During the years 1923-1926, he was affiliated to the Institute of Mathematics and Mechanics of Moscow State University as a postgraduate student, working on topology and the theory of functions, with Luzin as an advisor. In the period 1924-1925, he wrote four papers on the theory of functions [START_REF] Lavrentieff | Sur la recherche des ensembles homéomorphes[END_REF][START_REF] Lavrentieff | Sur la représentation des fonctions mesurables B par les séries transfinies de polynômes[END_REF][START_REF] Lavrentieff | Contribution à la théorie des ensembles homéomorphes[END_REF][START_REF] Lavrentieff | Sur les sous-classes de la classification de M. Baire[END_REF] and a paper on differential equations [START_REF] Lavrentieff | Sur une équation différentielle du premier ordre[END_REF]. The papers were published in French, Polish and German journals, and Lavrentieff became known internationally. The paper [START_REF] Lavrentieff | Sur une équation différentielle du premier ordre[END_REF] contains a celebrated example of a first-order differential equation of the form y = f (x, y), with f continuous, such that the uniqueness result on integral curves is violated everywhere: through every point in the domain of definition, at least two integral curves pass.

In 1927, Lavrentieff was sent to Paris for six months as a science researcher. 21 He heard lectures on function theory by Goursat, Montel, Borel, Lebesgue and Julia, and he attended Hadamard's seminar. He was interested in conformal geometry. One should also note that Luzin was also interested in this field; together with Privalov, he proved in 1924 that if a disc is sent conformally onto a domain with rectifiable boundary, then the map induced on the boundary of the domains sends a set of measure zero to a set of measure zero. In 1925 Luzin and Privalov obtained several results on boundary values of conformal mappings known as the Luzin-Privalov theorems [START_REF] Luzin | Sur l'unicité et la multiplicité des fonctions analytiques[END_REF]. The results were later generalized by Lavrentieff. During his stay in Paris, Lavrentieff wrote two Comptes Rendus notes on the boundary correspondence induced by a conformal mapping [43,[START_REF] Lavrentieff | Sur un problème de M. P. Montel[END_REF]. The first one concerns the boundary correspondence of a conformal mapping between two simply connected regions, and more precisely the behavior under such a map of ratios the lengths of subarcs of the boundaries of the two regions. Among the results he obtained, we note the following two:

1.-Let D and D be two domains in the plane, bounded by two simple closed curves Γ and Γ with bounded curvature. Consider a conformal correspondence between D and D . Then the ratio of lengths of corresponding arcs of Γ and Γ is bounded.

2.-Let D and D be two domains in the plane, bounded by two simple closed curves Γ and Γ which have continuously varying tangents. Consider a conformal correspondence between D and D . If δ and δ denote the lengths of corresponding arcs on Γ and Γ , then we have

K1δ 1-> δ > K2δ 1+
where K1 and K2 are constants that depend only on .

The second paper, [START_REF] Lavrentieff | Sur un problème de M. P. Montel[END_REF], has a sequel carrying the same name, and which Lavrentieff published two years later [START_REF] Lavrentieff | Sur un problème de M. P. Montel[END_REF]. It concerns sequences of analytic functions. The basic question addressed there is the following: for a pointwise convergent sequence 21 D. E. Menchov, in an interview with A. P. Youchkevich whose French translation is published in 1985 [START_REF] Menchov | Impressions sur mon voyage à Paris en 1927[END_REF], gives some details on the visits to Paris made at that time by Luzin, Lavrentieff and himself. He reports that the three of them used to stay in a small hotel, Parisiana, situated 4 rue Tournefort, near the Panthéon. (The hotel does not exist anymore.) He recalls that Luzin and Denjoy were close friends, but that at that time Denjoy was not very active in research. He gives several details about Hadamard's seminar at the Collège de France. He also says that Montel and Lavrentieff were close to each other and had several common interests, and that Borel and Lebesgue were no more involved in research.

of analytic functions f n defined on an open subset Ω of the complex plane, find an open dense subset Ω 0 of Ω on which f n converges uniformly on compact subsets to an analytic function. The problem started with a question of Montel who showed that the limit f of f n on Ω may be continuous and not analytic [START_REF] Montel | Sur les séries de fonctions analytiques[END_REF], and later proved that f n converges uniformly on compact sets to f on an open dense subset Ω 0 of Ω. Montel was interested in the structure of the complement of Ω 0 , and considered the question of whether it is possible to have f analytic on Ω 0 [START_REF] Montel | Leçons sur les séries de polynômes à une variable complexe[END_REF]. Lavrentieff solved completely this problem. He later treated this question in a comprehensive manner in his monograph Sur les fonctions d'une variable complexe représentables par des séries de polynômes published in 1936 [START_REF] Lavrentieff | Sur les fonctions d'une variable complexe représentables par des séries de polynômes[END_REF], after he gave a necessary and sufficient condition for a set to be the set of nonuniform convergence points of a sequence of polynomials converging everywhere in a given domain. This book is in fact concerned with the extension to the complex plane of the Baire-Weierstrass theory of approximation of functions of a real variable by series of polynomials. Lavrentieff gave a complete characterization of closed connected sets on which arbitrary continuous functions can be approximated by polynomials. The monograph also contains extensions of the same theory to harmonic sequences, and results on the boundary correspondence of conformal mappings.

In 1928, Lavrentieff gave an address, as an invited speaker, at the ICM (analysis section), which was held in Bologna [START_REF] Lavrentieff | Sur une méthode geométrique dans la représentation conforme[END_REF]. The talk he gave there is reviewed in another chapter in the present volume [2]. He writes in his memoirs that during that conference, he met Leonida Tonelli and that the two men realized they had close interests.

In the paper [7], written in 1930 in collaboration with P. Bessonoff, Lavrentieff studied again boundary mappings of conformal representations. This subject occupied him during several years.

In a series of papers written in collaboration with his student Mstislav Vsevolodovich Keldysh, 22 Lavrentieff studied problems related to domains on which continuous functions may be approximated by entire functions or by polynomials, including the case of dimension three ; cf. [START_REF] Keldysh | Sur les suites des polynômes harmoniques[END_REF][START_REF] Keldysh | Sur les suites convergentes de polynômes harmoniques[END_REF][START_REF] Keldysh | Sur le problème de Dirichlet[END_REF].

At the end of 1927, Lavrentieff was appointed assistant professor at Moscow University and he began his teaching with a course on the theory of conformal mappings. In 1929, he received the title of professor at the Moscow Institute of Chemical Engineering and he was elected chairman of the department of mathematics there. In 1931, he was appointed professor at Moscow University.

During the decade that followed his return to Moscow, Lavrentieff published a series of papers on the theory of functions of a complex variable, including results on conformal mappings onto canonical domains and variational problems leading to extremal domains. His papers during that period contained also results on boundary correspondence of conformal representations, Bloch's constant, approximation of complex-valued functions by polynomials, distance geometry and measure theory, and many other topics. In his paper Sur une classe de représentations continues [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF] published in 1935, he developed the theory of "almost analytic" functions, a class of functions very close to that of quasiconformal mappings. (Later on, Lavrentieff called his functions quasiconformal.) Herbert Grötzsch introduced his own version of quasiconformal mappings in 1928. Whereas the latter's methods were purely geometric, Lavrentieff had a more analytic approach. In his 1946 paper [START_REF] Lavrentieff | The general problem of quasi-conformal mappings of plane regions[END_REF], a quasiconformal mapping is defined as a mapping which satisfies a partial differential 22 Mstislav Vsevolodovich Keldysh was the brother of Ludmila Keldysh, who was member of the Luzitania and who became the wife of Piotr Sergueïevitch Novikov. Sergei Novikov, the Fields medalist, is their son.

system of equations which is more general than the Cauchy-Riemann system. According to Migirenko, Lavrentieff began investigating "almost analytic" functions in 1927 [77, p. 2]. We refer the reader to Lavrentieff's 1935 paper [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF], translated in this volume, which concerns exclusively almost analytic functions, and to the commentary [2].

In the 1930s, Lavrentieff became gradually the leader of the Soviet school of theory of functions. At the same time, he was working in physics, and in particular on incompressible fluid dynamics and on shock waves. In some sense, he was revisiting the interplay that was so productive in the development of both mathematics and physics. It is good to recall that Riemann himself, the main founder of the theory of conformal mappings, was more involved in physics than in mathematics. His study of electric and magnetic fields was his motivation for his development of the theory of abelian integrals. His use of the Dirichlet Principle is just the statement that a harmonic function minimizes the energy in an electrical field. The electrical potential or voltage in a source-free domain satisfies Laplace's equation as does the basic statement of incompressible fluid flow-both are formulations of conservation laws. Shocks in fluid flows can arise in different ways (one may think of the sonic boom created by a plane crossing Mach 1). The relevant equation changes from elliptic to hyperbolic (wave) and the transition creates a shock.

In the period 1929-1935, Lavrentieff worked as a senior engineer at the Theoretical Department of the Russian Institute of aerodynamics. His work on conformal and quasiconformal mappings was of great help in that domain. In his memoirs [START_REF] Dobretsov | The age of Lavrentieff, Novosibirsk, Siberian division of the Russian Academy of[END_REF], he writes that in 1929, he was given the task of determining the velocity field of the fluids in a problem related to thin wings, and that he wanted in some way to "justify the mathematics." In a lapse of time of six months, he managed, on the basis of variational principles using conformal mappings, to find a number of estimates for the desired solution. These estimates allowed him to identify a class of functions, among which the solution had to be found. It turned out that the theory of conformal mappings, which was already used in aerodynamics, could not fully meet the needs of this field, because flight speeds have increased, with the possibility that they might exceed the speed of sound. It was also necessary to take into account the compressibility of air. Thus, the classical Cauchy-Riemann equations satisfied by conformal mappings were not sufficient any more, and the theory of conformal mappings needed to be extended to a wider class of functions, satisfying a certain nonlinear system of partial differential equations. This is how his theory of quasi-conformal mappings was born.

It is worth noting here that Lipman Bers's work on partial differential equations was also motivated by fluid dynamics. He introduced the notion of pseudo-analytic function by studying elliptic systems that arise in the study of subsonic flows; see the historical details in the articles by Abikoff and Sibner [1] and Nirenberg [START_REF]Lipman Bers and partial differential equations[END_REF].

In 1934, Lavrentieff was awarded a doctorate in engineering by Moscow State University and in the following year he obtained his doctorate in mathematical sciences, without having to defend a thesis. Mathematics and physics were for him one and the same field of research.

In 1934, Lavrentieff published a paper on two extremal problems [START_REF] Lavrentieff | Sur deux questions extrémales[END_REF], one of them mathematical and the other one on fluid mechanics. The first problem concerns the conformal representation of the annulus 1 < |z| < r in the complex plane onto a doubly connected region which is the complement of the union of the closed disc |w| ≤ 1 and a simple path contained in the complement of that disc and converging to infinity. The problem asks for the maximization or the minimization of a certain quantity involving the derivative of the conformal representation of the annulus. An extremal image domain (that is, a domain realizing the maximum or the minimum of the quantity involved) is found in both cases to be the complement of the union of the closed disc |w| ≤ 1 and a straight line going to infinity.

The second problem concerns a simple arc of class C 1 under the action of the flow of an incompressible fluid, and the extremal problem asks for the maximization of the sum of the pressure forces exerted on the arc.

Although the two problems have different characters, they are treated in the same paper because they use the same preliminary lemmas. It is also possible that Lavrentieff, in publishing the two results in the same paper, wanted to point out a parallel between a purely geometric problem and a problem in fluid dynamics.

The article [START_REF] Keldysh | Sur la représentation conforme des domaines limites par des courbes rectifiables[END_REF] (1937), written in collaboration with Keldysh, is again concerned with the boundary correspondence induced by the Riemann mapping. The main result gives a negative response to the following natural question, which bears the name Smirnov problem: 23 suppose that w = f (z) realizes the conformal representation of a simply connected domain D bounded by a rectifiable Jordan curve, and let z = φ(w) be the inverse function. Is it possible to represent the harmonic function log |φ (w)| defined on the unit disc |w| < 1 by means of the Poisson integral of its limit values on the circle |w| = 1?

In the paper [START_REF] Lavrentieff | Sur une classe de transformations quasi-conformes et sur les sillages gazeux[END_REF], written in 1938, Lavrentieff uses quasiconformal mappings to describe the stationary flow of a gas. The paper [START_REF] Lavrentieff | Sur un critère différentiel des transformations homéomorphes des domaines à trois dimensions[END_REF], written the same year, is the first paper where higher-dimensional quasiconformal mappings are studied. The definition given by Lavrentieff involves the ratio of the greatest axis to the smallest axis of ellipsoids that are images of infinitesimal spheres. Lavrentieff studied several properties of these mappings. In particular, he obtained a result saying that a locally injective quasiconformal mapping of three-dimensional space is injective. A proof in all dimensions was given 30 years later by V. A. Zorich [START_REF] Zorich | The theorem of M. A. Lavrent'ev on quasiconformal mappings in space[END_REF]. Higher-dimensional quasiconformal mappings are further studied by Lavrentieff in the papers [START_REF] Lavrentieff | On the theory of quasi-conformal mappings of three-dimensional domains[END_REF][START_REF] Lavrentieff | Sur la théorie des représentations quasi-conformes[END_REF] and others. In his paper [6] written with Belinskij, Results of Martio-Rickman-Väisälä and Gehring are used to prove normality of families of Q-quasiconformal local homeomorphisms in dimension ≥ 3. For a report on the current state of the study of higher dimensional quasiconformal mappings, the reader can refer to the article by Gaven Martin in Volume IV of the present Handbook [START_REF] Martin | The theory of quasiconformal mappings in higher dimensions, I[END_REF] In the paper [START_REF] Keldysh | Sur un problème de M. Carleman[END_REF] (1939), also written with Keldysh, Lavrentieff gives a characterization of Carleman sets, that is, closed subsets E of the complex plane such that for any continuous function f which is analytic in the interior of E and for any positive continuous function (r), r ≥ 0, there exists an entire function F (z) satisfying |f (z) -F (z)| ≤ (|z|). Such sets are used extensively in the theory of analytic approximation.

In 1947, Lavrentieff began to develop a new theory of non-linear quasiconformal mappings which he called "strongly elliptic," to be used in hydrodynamics; see the exposition in [77, p. 20] where the author writes that this turned out to be the widest generalization of the Cauchy-Riemann system of equations that preserves the basic geometric properties of conformal functions. He extended to this general class several properties of conformal mappings.

The paper [START_REF] Lavrentieff | Variational Methods for Boundary Value Problems for Systems of Elliptic Equations, translated from the Russian[END_REF] (1963) contains an exposition of a variational principle that Lavrentieff developed and which he called Lindelöf's principle (the principle is an 23 The name refers to Vladimir Ivanovich Smirnov (1887-1974), another preeminent Russian mathematician of the same generation as Luzin who experienced in his student years the same period of political unrest. Smirnov was born in Saint Petersburg, and he founded there the department of Theory of Functions of a Complex Variable. He spent his career at the university of his hometown whose name changed to Leningrad. For a concise biography of Smirnov, we refer the reader to the article by Apushkinskaya and Nazarov [4].

extension of a principle due to Lindelöf), for the solution of boundary value problems for the Laplace equation and for more general elliptic equations. The approach is again based on the use of conformal and quasiconformal mappings and their geometric properties. The paper contains applications to hydrodynamics, e.g. problems concerning free stream lines and gravity waves (like waves at the air-sea interface, where gravity force tries to restore equilibrium), and to subsonic compressible gas flows.

Lavrentieff's mathematical ideas continued to have relevance also in the West (see Courant [11] (1950) which contains a section on Lavrentieff's methods).

M. V. Keldysh, with whom Lavrentieff wrote several papers, became President of the Academy of Sciences of the USSR in the period 1961-1975, a period during which Lavrentieff was Vice-President of that academy. Bill Abikoff informed the author of these lines that he learned from Jewish mathematicians that at the time where they were experiencing discrimination, Laventieff, with his position of Vice-President, protected them heroically.

Siberia

Lavrentieff contributed in a substantial way to the transformation of the science landscape in the USSR when, together with the mathematician Sobolev and the physicist Christianovich, he convinced Khrushchev to approve the creation of the science city of Akademgorodok, a few kilometers from Novosibirsk. The place became the host of several research centers devoted to various branches of science, including mathematics, physics, chemistry, geology, economics, history and others. Novosibirsk State University and the headquarters of the Siberian Division of the Russian Academy were naturally hosted in Akademgorodok. For many years, Lavrentieff was the director of the Institute of Hydrodynamics there. He convinced many mathematicians and other scientists to move there from the European part of Russia. It is possible that among the scholars that settled in Novosibirsk, many found in Siberia some kind of intellectual freedom that was inexistent in other parts of the big cities of the Soviet Union where they used to work. In a few years, and due in large part to the immense energy of Lavrentieff, his courage and his talent as a science organiser, Siberia was transformed from a scientifically underdeveloped region into a new world center of science. Lavrentieff also implemented there a new way of selecting students and of training them, organizing olympiads in mathematics and physics and creating specialized boarding schools for talented students. As a matter of fact, Lavrentieff created the first physico-mathematical boarding school. (A year later, Kolmogorov created a second one.) One of Lavrentieff's favorite sentences was: "There can be no scientists without science students" [77, p. 7].

De Gaulle made a spectacular state visit to the Soviet Union, from June 20 to July 1st, 1966, which was part of his plan to enhance the ties between France and the Soviet Union, and to make his country less dependent on the United States. Akademgorodok was among the places he visited. He gave a famous speech there in which he expressed his admiration for this science city, which at that time was comprised of 20 research institutes, and for the people who were working there, including 6000 researchers. He communicated his hope that French mathematicians, physicists, chemists, biologists, philosophers, anthropologists and historians establish close relations and collaborations with their Soviet colleagues. He declared that he was touched by the cordial welcome he received "from those who live in this city, this strange and courageous land of Siberia," quoting Teilhard de Chardin: "a land where people give their life to know, rather than to obtain." 24 Jean Leray, in his memorial article on Lavrentieff, [START_REF] Leray | Sur Mikhaïl Alexievich Lavrentiev[END_REF] recounts that de Gaulle, who was very tall, during his visit to Akademgorodok, found himself surrounded by a group of men as tall as him; one of them was Lavrentieff.

Mechanics and engineering

In the period 1929-1969, Lavrentieff wrote several papers in which he applied mathematical methods, including the theory of conformal and quasiconformal mappings in dimension ≥ 2, to fluid mechanics and aerodynamics. The fields of application includes aircraft, hydrofoil and underwater wing theories, the study of vibrating plates of infinite extent, of turbine propellers, of the impact of a body on water and of jet flows of an ideal liquid. At the same time, he developed theories as varied as those of long waves, of dynamical stability of buckling elastic systems and of the movement of grass snakes and fish. He built mathematical models for discontinuous flow patterns and for flows by stream of finite breadth around obstacles. He studied problems concerning directional explosions that were used for dam constructions and he obtained results on the welding of metals by explosions utilized in the design of canals and funnels. 24 De Gaulle cites the following excerpt from Teilhard de Chardin: "Une Terre où les télescopes géants et les broyeurs d'atomes absorberont plus d'or et susciteront plus d'admiration spontanée que toutes les bombes et tous les canons. Un Terre où, non seulement pour l'armée groupée et subventionnée des chercheurs, mais pour l'homme de la rue, le problème du jour sera la conquête d'un secret et d'un pouvoir de plus arrachés aux corpuscules, aux astres ou à la matière organisée. Une Terre où, comme il arrive déjà, c'est pour savoir et être plutôt que pour avoir, qu'on donnera sa vie" [89, p. 281-282]. De Gaulle's discourse is contained in [START_REF] De Gaulle | Discours et messages[END_REF].

A spectacular result of Lavrentieff's ideas was the construction of the gigantic dam that protects Almaty, the largest city of Kazakhstan, 25 at the center of Eurasia. The dam was constructed by a series of 5 directed blasts which involved the displacement of a total of 5 million cubic meters of rock. The last explosion used 3600 tons of ammonium nitrate based explosive. Before the dam was constructed, the city was periodically devastated by mudflows from the Meteu Valley. The most severe one in the twentieth century occurred in 1921, it killed 500 people and destroyed a large part of the city.

Lavrentiev also worked on artillery and on the improvement of gunpowder, upgrading the production of bullets using ideas from fluid dynamics. He helped in the conception of the famous multiple weapon launcher called the Katyusha that was used by the Soviet army during World War II. It is conceivable that the applications of his ideas and discoveries in the military field played a non-negligible role in the Soviet Union's victory on Nazi Germany. His work on explosive material was extended after World War II. Migirenko writes that in the years 1951 to 1953, " [Lavrentieff] frequently became engaged in experiments which he literally conducted with his own hands, and he youthfully expressed delight with his success and disappointment when his expectations were not confirmed. With the help of young scientists he set charges under water and equipped fire hoses with grains of TNT or so-called unconditioned powder, in which he was extremely interested at the time. He could have simply burned powder according to the existing canons, but Mihail Alexeevič could not accept the possibility that 'property was being needlessly wasted'." [77, p. 5] One may safely assert that Lavrentieff's involvement in the military technology was crucial in the decision of the Soviet heads of the state to increase their support of science, creating new mathematical centers and hiring a large number of scientists.

Lavrentieff also studied metal welding by explosions, and he developed hydraulic jet pulse techniques with applications to the breaking down of hard rocks. He conceived a device which became the prototype of modern hydraulic jet guns, used for pulsed high-energy forming of metals and other works. He also studied the phenomena of high-velocity collision with various kinds of obstacles (metallic, porous, etc.), together with the question of detonation of high-speed explosives. He found new ways for using the underground hot water, with practical methods for dealing with tsunami waves and insuring the security of coal mines.

Starting in the 1940s, Lavrentieff worked on solitary waves. These are waves whose envelope has a unique peak. They decay away from this peak and their shape and amplitude stay constant in time. They arise as ocean water waves and in optical fibers where intensity increases from dark to light. Such waves have been used in the modelisation of tsunamis. Solitons, which are waves of big amplitude and small spatial extension and which can travel for long distances without deformation, are special cases of solitary waves. From the mathematical point of view, solitary waves are solutions of certain non-linear partial differential equations (Korteweg-de Vries and nonlinear Schrödinger equations). Seen the range of interests of Lavrentieff, it is not surprising that he was implicated in this research. The theory of solitary waves is also a classical subject. Their existence under special assumptions was proved by Boussinesq (1871) [10], Lord Rayleigh (1876) [START_REF] Rayleigh | On waves[END_REF], Korteweg and de Vries (1895) and Levi-Civita (1911) [START_REF] Levi-Civita | Sulle onde progressive di tipo permanente[END_REF]. Lavrentieff was the first to give a mathematical proof of the existence of general solitary waves. Using approximation techniques by Stokes waves, he established their existence as limits of periodic waves with indefinitely increasing wave length; cf. [START_REF] Lavrentieff | On the theory of long waves[END_REF][START_REF] Lavrentieff | A contribution to the theory of long waves[END_REF].

Starting from 1956, Lavrentieff worked on the question of spiraling detonation of gases and on problems related to the formation of turbulent vortex rings in a viscous fluid, including the transfer of impurities under such phenomena. His results led to a complete re-organization of large earth-works in the Soviet Union.

In 1946, while he was Vice-President of the Ukrainian Academy of Science and Director of the Mathematics Institute of Kiev, Lavrentieff wrote to Stalin, urging him to accelerate the development of computers, emphasizing their utility for military purposes. Stalin responded positively and authorized him to create a new laboratory dedicated to computer science. In 1948, Lavrentieff was selected by the government to develop modern computing technology, and in 1951 he was appointed director of the Institute of Precision Mechanics and Computer Technology in Moscow. He remained director of this institute until 1953; see [77,26,[START_REF] Malinovsky | Pioneers of Soviet Computing[END_REF].

Lavrentieff writes in his memoirs [START_REF] Dobretsov | The age of Lavrentieff, Novosibirsk, Siberian division of the Russian Academy of[END_REF]: "I learned for myself personally, first, the experience of applying pure mathematics to important engineering problems, and secondly, a clear understanding that in solving such problems new ideas and approaches are born in mathematical theories themselves." He was responsible for projects using geothermal energy, in particular in the Eastern part of Russia bordering the volcanoes of Kamchatka. He was particularly concerned by the effect of extreme cold on constructions in Siberia, by the pollution of rivers and lakes (in particular, that of the Lake Baïkal, located in Southern Siberia and which is the largest freshwater reserve by volume in the world).

Lavrentieff received the most prestigious prizes awarded in the Soviet Union, among them the Lenin Prize (1962), 26 the Hammer and Sickle gold star (1967), which is the highest honor in Soviet Union for nonmilitary achievement, and the Lomonosov gold medal (1977). He was also a member of several foreign academies of sciences

In his article on Lavrentieff [START_REF] Leray | Sur Mikhaïl Alexievich Lavrentiev[END_REF], Leray described half a dozen works of the latter as Herculean works. In the same article, Leray compared him to Peter the Great, for the breadth of his achievements and his vision on the present and the future of Russian science.

The main street in Akademgorodok is naturally called Lavrentieff Prospekt.

mathematics. Hints, beautiful comparisons are something that revels and promises, is tantalizing, alluring and ineffectual. And in the end I stopped understanding what Florensky is. Or is it a precursor of the new, the petrel, or a capable person with subconscious hellish self-love, who, because of the desire to be the best of all, has retired here?" [7, p. 150].

In 1904-1907, Luzin experienced a spiritual crisis which was so severe that he wanted to stop practicing mathematics, and even wanted to take his own life. One reason for the crisis was the dramatic atmosphere that preceded the first revolution. His teacher Egorov hastily sent him from to France to prepare him for a professorship. At this time, Luzin was under the strong influence of Florensky. On the ambiguity of Florensky's personality and his negative charisma, L. Sabaneev wrote in his memoirs (1915): "Very black and very thin, for some reason he always looked down and slightly sideways, he did not like to show his eyes. He never smiled. It was a strange thing-he had many students, apparently, he taught them not only classical theological subjects common to the spiritual sciences, but also gave them esoteric knowledge and habits. Three of his students committed suicide-powerful vibes emanated from him, and I myself felt it, felt that not all fluids were good, there were very demonic among them. I don't remember exactly who, but speaking of him, one of the Russian 'neo-Christian' group called him 'a clever and cruel Lavra priest'. In any case, he was an absolutely extraordinary person, and I am very grateful to the fate that brought him together with me, although not for long" [11].

The relations between Luzin and Florensky were uneven and the influence of Florensky on the mathematical research of Luzin is problematic.

In 1915, Luzin presented his master's thesis Integral and trigonometric series, with so strong results that it was qualified as a doctor's thesis. Starting in 1917, Luzin began to teach at Moscow University; a group of talented students gathered around him. This circle was named Luzitania. It was a happy period of the relationship between a teacher and equally talented students. Years passed, the students themselves became leaders of scientific schools, and in the meantime, the Stalinist terror gained momentum in the Soviet Union. Trials began, the search for "enemies of the people" was going on. Under the blow was the intelligentsia, the old professorship. Much has been written about this in Russian literature. The Soviet government sought to subjugate the Academy of Sciences, to "domesticate" the old scientists, orienting them to the tasks of communist construction, re-educate Soviet scientists in the spirit of communist ideology. For this, any contradiction among scientists was used. The conflict between Luzin and his former disciples in 1936 served as a convenient pretext. About this tragic story, see above all [2].

Despite this fact, it is impossible to perceive the persecution of Luzin as a direct manifestation of Stalin's terror. It was part of a more general process. Stalin did not set himself the goal of punishing Luzin and did not even know about the beginning of the persecution. The reason was different: the happy period of Lusitania ended, the former disciples became independent scientific leaders, and rivalry began. A tangle of very complex contradictions both in the mathematical community and in society as a whole, multiplied by the difficult character of Luzin, who in his own way understood his role in the students' successes; denunciations written on Luzin by the pro-Communists E. Kolman and V. Molodshij; the clash of traditional ethics of old scientists with the new ethics of the younger scientific generation; the atmosphere of fear cultivated among the intelligentsia by the Stalinist search for "enemies of the people": all this inspired this persecution. The case was accompanied by violent attacks on Luzin in the newspapers. Nevertheless, it did not go beyond the academic environment, and a month later it was completed by the Decree of the Presidium of the Academy of Sciences of 08/05/1936 on Academician Luzin with the following wording: "Given the importance of N. N. Luzin as a major mathematician, and weighing the full force of social impact, what had revealed in such a broad, unanimous and fair condemnation of the behavior of N. N. Luzin, and based on the desire to give Luzin the opportunity to restructure his whole behavior and work, we consider it possible to limit the warning to N. N. Luzin that in the absence of a decisive change in his future behavior, the Presidium will have to urgently raise the question of expelling N. N. Luzin from the academic ranks" [START_REF] Alberge | On five papers by Herbert Grötzsch[END_REF].

Luzin escaped the terrible machine of Stalin's repression, which many well-known scientists got into, including Florensky who was shot in 1937. See also (in Russian) S. Novikov (Jr.) [10], a very deep study of Yu. Neretin [9], and (in English) A. E. Levin [5]. Luzin recovered with difficulty after the persecution. He was ill for a long time, then continued his scientific work, but he no longer made major discoveries.

Figure 1 .

 1 Figure 1. Lavrentieff welcoming de Gaulle at Novosibirsk airport, July 1st, 1966.

These were times where aircraft design was in a period of rapid development in the Soviet Union and elsewhere.

Demidov, in his paper On an early history of the Moscow school of theory of functions[START_REF] Demidov | On an early history of the Moscow school of theory of functions[END_REF],

In a letter sent to P. A. Florensky sent on May 1st, 1906, Luzin writes: "The worldviews

Appendix

Some clarifications on the history of Russian mathematics of the 20th century by Galina Sinkevich

In this note, I would like to bring a few additions that clarify some points in A. Papadopoulos's article related to the biographies of P. Florensky and N. Luzin. My additions may shed a different light on this complicated period of the history of mathematics in the Soviet Union.

According to B. Mlodzeevsky, it was characteristic for Moscow mathematicians of the turn of the 19th and 20th centuries, "to explain mathematics through the worldview, and worldview through the mathematics" [8, p.185]. The leading role in this orientation of the Moscow school belonged to N. V. Bugaev, whose theory of discontinuous functions was considered in a philosophical context and had a wide resonance among Moscow mathematicians [4], although his colleagues did not appreciate his mathematical results and lectures [12, p. 165-185].

In 1897, at the First Congress of Mathematicians in Zurich, Bugaev made a report On the influence of mathematics on the worldview [1]. According to him, world laws are explained in the language of the theory of continuous functions. This gave rise to determinism, while in the natural and human sciences many phenomena cannot be subordinate to the laws of mathematical analysis. Using continuity, only part of the world events can be explained. At the same time, a new science emerges from number theory, a theory of discontinuous functions, called arhythmology. In the realm of philosophy proper, arhythmology is refracted into monadology. The arhythmological point of view complements the analytical worldview. Combining both worldview approaches, analytic (using mathematical analysis) and arhythmological (using the theory of discontinuous functions), and adding probability theory in necessary cases, Bugaev obtains a scientific world view that allows us to explain mechanically world phenomena. Finally, whenever phenomena do not obey the correct laws, probability theory applies. From the cumulative application of all these departments of mathematics, a true scientific and philosophical outlook was formed.

N. Luzin and P. Florensky entered Moscow University with a difference of one year (1901 and 1900 respectively). They soon became friends. First a mathematics student and then a priest, Florensky was a very ambiguous figure. No doubt he was very talented. In his student years, he was fascinated by Bugaev's ideas. He then became interested in Cantor's set theory. Florensky was the first to publish a competent statement of Cantor's ideas in Russian (On the symbols of Infinity, 1904). But then his interests shifted towards theology. After graduating from university he entered as a student the Moscow Theological Academy. He needed mathematics only as a basis for building the philosophical foundation of the universe. His philosophical and religious works, The pillar and foundation of truth (1914), Reverse perspective (1919), Imaginaries (imaginary numbers) in geometry (1922) are widely known. In particular, in his last work, Florensky interprets the complex plane as a two-sided object for the rehabilitation of the Ptolemaic model of the Universe.

The statement made by Athanase Papadopoulos in his article that Florensky was extremely talented in mathematics is debatable. Luzin and his teacher D. Egorov estimated the mathematical success of the student Florensky low. In 1908, Luzin wrote to his wife about Florensky: "As soon as he showed his work in mathematics, again the old opinion began to stir in me: all his works have no value in the field of