# Ilya Nestorovich Vekua

### Born: 23 April 1907 in Shesheleti, Kutaisi Governorate, Russian Empire

(now Ochamchira District, Abkhazia, breakaway region of Georgia)

Died: 2 December 1977 in Tbilisi, Georgia, USSR

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**Ilya Vekua**'s parents were Nestor and Lisa Vehua, who were described in [2] as peasants:-

The village of Shesheleti where Ilya was brought up, about 12 km from the Black Sea, had no high school, so Vekua attended the secondary school in Zugdidi, about 25 km to the south. At the school in Zugdidi of all subjects he studied and enjoyed it was mathematics that captivated him. He loved its rigour and its logical internal structure. He was described as serious and thoughtful during his school days and, in 1925, he completed his schooling and entered the Faculty of Physics and Mathematics of Tbilisi State University. This university had opened in 1918 and one of the founders of the university was Andrei Razmadze who held a chair in the Physics and Mathematics Faculty. Razmadze was assisted by Nikoloz (Niko) Muskhelishvili who had joined the new university in 1920. Among the more junior members of the mathematics staff at the time that Vekua was a student we mention Archil Kharadze (1895-1976) and George Nikoladze (1888-1931).... who could have never fancied that their first-born child would become one of the prominent mathematicians of the20^{th}century, a great organiser, an outstanding scientist and a public figure.

Although the standard of teaching was high, the classes were very large and students got relatively little personal contact with their professors. There were no research seminars at this time and few opportunities for involving students in research topics. However, for the best students this problem was to some extent minimised by the students themselves who organised their own student physics and mathematics circle. They set up meetings in which the students gave talks on solving mathematical problems. They also invited their professors and visiting academics to give lectures to the circle. From his third year onwards, Vekua was elected each year as chairman of the student physics and mathematics circle. In 1929, while still an undergraduate, Vekua was employed as an Observer at the Georgian Geophysics Observatory. He retained a position with the Geophysics Observatory until 1933 but for the latter part of this time he changed from the role of Observer to that of Physicist at the Observatory's Karsani Magnetic Department (near Tbilisi).

Vekua graduated from the Tbilisi State University in 1930 and became a postgraduate student at the USSR Academy of Sciences in Leningrad. At the Academy of Sciences he came into contact with Muskhelishvili who had taught him at Tbilisi, but Muskhelishvili had not really known the young student while he was an undergraduate as he explained. Muskhelishvili wrote (see [2]):-

Advised by Aleksei Krylov, Niko Muskhelishvili and Vladimir Smirnov, Vekua undertook research on the equations of mathematical physics at the USSR Academy of Sciences in Leningrad for his Candidate's Degree (equivalent to a Ph.D.) [7]:-Strangely enough, I really got to know Vekua only in1930in Leningrad, where he had been sent on my recommendation along with a group of other young mathematicians to the graduate school in the Physics and Mathematics Institute. While there were very few mathematics professors in Tbilisi, we had to teach a large number of subjects, and since there were quite a lot of students it was hard to memorise all of them. Besides Vekua always behaved modestly and quietly. Some time after arriving in Leningrad(I often go there to give lectures and work with graduate students), I went to the Director of the Institute, the leading naval architect and outstanding specialist in applied mathematics Academician Aleksei Nikolaevich Krylov, to ask how the postgraduates we had sent were coping with the work. Frankly, I was somewhat surprised that Aleksei Krylov only talked about Vekua. I must say that the speciality of Aleksei Krylov was very far from that in which Vekua was engaged, but Aleksei Krylov was one of the smartest people I ever met, and he could spot talent in a far removed area. One day I participated in the seminar which was led by Professor Vladimir Ivanovich Smirnov(also a very prominent scientist), and found a report which had been written by Vekua. This report gave me a very strong impression of maturity of thought and clarity of presentation. It was then that I saw clearly that this is a talented young man. Our acquaintance began just after the workshop and developed into a close friendship. The difference in age, which at that time could pose a huge problem but is now almost obliterated, did not prevented us from the very beginning from being friends, and our scientific work, especially before Vekua moved to Moscow in1951, and then to Novosibirsk, always proceeded closely. In particular, the work of Ilya Vekua in the field of so-called 'singular integral equations' had a great influence on the direction of my own work, and I felt it my pleasant duty to emphasize that fact in the introduction to my monograph in this area.

He published two papers in 1933, both joint works with A K Rukhadze and both in Russian, namely (i)Using the methods of the complex variable function theory, he investigated a number of problems of static and dynamic elasticity. During that period he wrote his major works devoted to the theory of distribution of elastic waves in an infinite layer with parallel plane boundaries. These studies formed the basis of his candidate's thesis ...

*Problem of torsion of a circular cylinder reinforced with a longitudinal circular rod*and (ii)

*Torsion and bending by transverse force of a bar composed of two elastic materials bounded by confocal ellipses*. He returned to Tbilisi in 1933 and was appointed as a Junior researcher in the Faculty of Physics and Mathematics of Tbilisi State University. He also became the scientific secretary of the Mathematical Institute of the Georgian Branch of the USSR Academy of Sciences and we now look briefly at how this was set up.

On 8 October 1933 a research institute of mathematics, physics and mechanics was set up run by Tbilisi State University. Its director was Niko Muskhelishvili. On 1 October 1935, at the initiative of Muskhelishvili and Vekua, the mathematics and mechanics section of the above-mentioned institute was transformed into a mathematical research institute under the auspices of the Georgian Branch of the USSR Academy of Sciences. This Institute was incorporated into the Georgian Academy of Sciences after it was founded in February 1941.

In 1937 Vekua defended his Candidate's thesis "Propagation of elastic waves in an infinite layer" and, immediately afterwards, was promoted to assistant professor. In 1939 he defended his doctoral thesis (equivalent in standard to a D.Sc. or habilitation) "A complex representation of solutions of elliptic differential equations and its application to boundary value problems". The representations which Vekua introduced in this thesis, and published in papers which appeared in 1941 and 1942, were named 'Vekua representations' by Muskhelishvili. There have been a number of papers studying Vekua representations and in recent times they have been generalised.

The years of World War II were difficult although the German armies never reached Tbilisi. During these hard years, Vekua served first as the dean of the Faculty of Physics and Mathematics of Tbilisi State University and later as pro-rector. He had other roles too being head of the Geometry Section of the Faculty and, outside the university, he headed the Theoretical Mechanics section of the Transcaucasian Institute of Railway Transport. After the war, in 1947, he became head of a department of the Institute of Mathematics and of the Department of Mathematical and Natural Sciences of the Georgian Academy of Sciences. He held these roles until 1950.

In 1951 Vekua was invited by Sergei Alekseevich Khristianovich, a mechanical engineer who was pro-rector of Moscow State University, to come to Moscow and work in the scientific institutions and higher educational institutions in that city. He left Tbilisi, going to Moscow in the autumn of 1951. He was appointed head of a Laboratory of the Central Institute of Hydro-aerodynamics and also head of Theoretical Mechanics at the Moscow Physico-Technical Institute. In the following year he became a professor in the Department of Differential Equations of Lomonosov State University of Moscow. He also became a senior scientific worker at the Steklov Institute of Mathematics of the USSR Academy of Sciences in 1953.

The authors of [7] describe some of Vekua's mathematical achievements while he was working in Moscow:-

The Siberian Branch of the USSR Academy of Sciences was established in 1957 and Vekua was among those elected to the Presidium. Novosibirsk State University was constructed at Akademgorodok and when it opened in 1959 Vekua became its first rector. At the same time he became head of the Theoretical Department of the Siberian Institute of Aerodynamics. He held these positions from 1959 to 1965 when, at the request of the Georgian Academy of Sciences, he returned to Tbilisi to become Rector of Tbilisi State University. In 1972 he retired as Rector and was elected President of the Georgian Academy of Sciences. He held this position until his death in 1977.In Moscow I N Vekua wrote and published a large cycle of studies devoted to the so-called theory of generalized analytic functions. An attempt to construct the theory of such functions was made back in the191930^{th}century by the Italian mathematician Beltrami. In the earlys Carleman and Theoclorescu showed that a number of properties of analytic functions of one complex variable could be transferred to the solution of an elliptic system of two first order differential equations in the case of two real independent variables. I N Vekua created a general theory which at present forms the basis of the theory of generalized analytic functions. Using the theorems he had derived, Vekua obtained an analytic substantiation of M A Lavrentev's geometrical theory of quasiconformal mappings of plane domains, which has been recognized as one of the best achievements in the theory of functions over the last fifty years. l N Vekua's results in the theory of first order elliptic systems were included in his monograph 'Generalized Analytic Functions' which was awarded the Lenin Prize in I963

The authors of [7] write:-

In the book Vekua sates that:-Vekua was a warm-hearted person with high civic qualities. His endurance, courage and self-control evoked; delight and admiration. Though suffering from an incurable grave illness, the scientist persistently continued his research, resulting in the development of a new version of the mathematical theory of elastic shells. This version was included in his monograph "Theory of Shells: General Methods of Construction" which was published posthumously in Moscow in1982. The monograph was awarded the State Prize of the USSR and its English translation was published by Pitman Publishers in1985.

Leonid P Lebedev writes in a review:-... the main purpose of shell theory consists precisely in constructing approximate solutions of relevant prioblems of the three-dimensional theory of elasticity.

We have mentioned a couple of Vekua's books but he wrote a number of others. There isThis monograph summarizes the author's many years of work on shell theory. In the first two chapters, which make up half of the book, the author constructs different variants of the linear theory of elastic shells by reducing three-dimensional problems to the corresponding two-dimensional ones. ... The book makes much use of the apparatus of tensor mathematics, surface theory, and the theory of generalized analytic functions. At the end of the book is a list of the author's works on shell theory.

*New Methods for Solving Elliptic Equation*s (Russian) (1948) and, a book mentioned above,

*Generalized analytic functions*(Russian) (1959). Lionel Cooper writes in a review of this book:-

An English translation with titleBy generalized analytic functions are meant functions of two real variables which obey elliptic differential equations analogous to the Cauchy-Riemann equations. ... It is well and clearly written ...

*Generalized analytic functions*was published in 1962 and a German translation appeared in the following year.

Vekua published

*Fundamentals of tensor analysis*(Russian) in 1967. Manindra Chandra Chaki writes in a review:-

His next book wasThe book provides a good introduction to tensor analysis and gives some of its applications to the theory of surfaces. ... The treatment of the subject is clear and rigorous.

*Foundations of tensor analysis and the theory of covariants*(Russian) (1978). Wlodzimierz Wrona writes in a review:-

As to Vekua's personality, we quote from his friend Lipman Bers [5]:-The book contains a systematic presentation of tensor algebra and tensor analysis with their applications to the theory of surfaces and to the theory of shells. It also gives a solid mathematical preparation for the study of the theory of elasticity. Many results of the author's own research related to special coordinate systems having applications in the general theory of elasticity, and to the theory of covariants, are included.

Vekua received many honours. He was elected Corresponding Member of the Academy of Sciences of the Georgian SSR in 1944 and, two years later, a Corresponding Member of the USSR Academy of Sciences and also a Full Member of the Georgian Academy of Sciences. He received the Gold Star of the Hero of Socialist Labour and five orders of Lenin. He also received much international recognition being elected to the Academy of Sciences of the German Democratic Republic, the German Academy of Scientists Leopoldina, the Sicilian Academy of Sciences, the Polish Society of Theoretical and Applied Mechanics, and the Danish Centre of Applied Mathematics and Mechanics. He was awarded honorary degrees from Halle University and Jena University.... there was nothing official or rigid in his personality. Ilya Nestorovich was a large man with a booming voice, enormously friendly and filled with joie de vivre. His courtliness was somewhat old fashioned, and when you saw him kissing ladies' hands, or officiating as "tamada"(toastmaster)at a Georgian style party, or enjoying food and good wine or a good story, you forgot all about his scientific and administrative prominence. But his benevolence went beyond mere social friendliness. He was willing to help when others would not, and one could talk to him, freely and openly, about many things. His judgements were shrewd, original, and unhampered. ... Vekua's deeply rooted optimism was not based on denying or ignoring the tragic aspects of life or of history. Once he and I watched a Broadway performance of Bolt's Man For All Seasons. Vekua may not have known the details of Sir Thomas More's death, and the dilemma of an honourable and reasonable man confronting a tyrant moved him deeply. I recalled that evening when I read in an article, written for his70([4])^{th}birthdayabout the "strength, courage and self-control Ilya Nestorovich showed under complicated circumstances."

**Article by:** *J J O'Connor* and *E F Robertson*

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