Boris Yakovlevich Levin

(1906-1993)
B.Ya. Levin’s scientific school  B.Ya. Levin’s list of publications

Boris Yakovlevich Levin was born on December 22, 1906 in Odessa, the beautiful port city on the Black Sea, one of the main trade and cultural centers in the South of the Russian Empire. "I am an Odessit by social origin and nationality", he used to joke. His father was a clerk for a Black Sea steamer company, whose work often took him to various beach ports on long-term missions. B. Ya. (as he was called by his colleagues and friends, rather than more formal Boris Yakovlevich) spent his youth moving with his family from one port town to another. He kept his devotion to the sea all his life, was an excellent swimmer and longed for the Black Sea, while living far from it in Kharkov.

Being of nonproletarian social origin, B. Ya. had no right to higher education in post-revolutionary Russia after graduating from high school. For some time he worked as an insurance agent and newspaper dispatcher, and as a welder during the construction of oil pipe-lines in the North Caucasus. This gave him the right to enlist as a university student, and in 1928 he started his first year at the Department of Physics and Engineering of Rostov University, Russia. B.Ya. and a friend of his decided, before concentrating on physics, to widen and improve their knowledge of mathematics. It was the choice of destiny: once entering mathematics they never parted with it. Both became famous experts: Boris Levin in analysis and Nikolai Efimov in geometry. To a great extent they were influenced by Dmitrii Mordukhai-Boltovskoi, an interesting and original mathematician with wide interests who worked at the Rostov University at that time.

While a second year student B. Ya. obtained his first mathematical result when he solved a problem proposed by Mordukhai-Boltovskoi. He investigated the functional equation

where R(x)  is a given rational function. This equation generalizes the functional equation Ã(x+1)=xÃ(x) of the Euler Ã- function. B. Ya. proved that, apart from some exceptions, all solutions to this equation are hypertranscendental , just as Ã(x). He also explicitly described all exceptional cases. This theorem generalizes the famous Holder theorem.

In 1932 B. Ya. graduated from the university, and for the next three years worked on his thesis and taught mathematics at a technical university in Rostov. His close friendship and scientific collaboration with Naum Akhiezer (Kharkov) and Mark Krein (Odessa) started at that time. In 1936 B. Ya. submitted his Candidate of Science thesis "0n the growth of an entire function along a ray, and the distribution of its zeros with respect to their arguments" to Kharkov University, but was awarded with the highest degree of Doctor in Mathematics, which was an extremely rare event. In this thesis B. Ya. founded the general theory of entire functions of completely regular growth, whose creation he shared with Albert Pfluger.

In 1935 B. Ya. moved to Odessa and began teaching mathematics at the Odessa University of Marine Engineering. In due time he got a Chair of Mathematics of this university. Parallel to his teaching B. Ya. spent a lot of time and effort in advising his colleagues who worked on hydrodynamical problems of ships and mechanics of construction. In his later years he would say that teaching and communicating with engineers in a serious technical university is an important experience for a mathematician.

Starting from the middle 1930s a new school of functional analysis has been forming around Mark Krein in Odessa, and B. Ya., as he later used to say, experienced its strengthening influence. He became interested in almost periodic functions, quasi-analytic classes and related problems of completeness and approximation, algebraic problems of the theory of entire functions, and Sturm-Liouville operators. These remained the main fields of interest during his life.

In Odessa, the first students of B. Ya. have started their own research. Moshe Livshits and Vladimir Potapov, who became well-known specialists in functional analysis, were common students of Mark Krein and Boris Levin. Today, the family tree of B. Ya.'s mathematical children, grandchildren, and great grandchildren contains more than a hundred mathematicians.

During World War II, B. Ya. worked with his institute in Samarkand (Uzbekistan). His attempts to join active military service failed, since full Professors were exempt from the draft. After the war B. Ya. returned to Odessa. At that time a destruction of mathematics at Odessa University began. Mark Krein and his colleagues were not permitted to return to work at the university, and very soon an anti-Semitic campaign waged against Mark Krein, and B. Ya. forced the latter to leave Odessa. On invitation of Naum Akhiezer in 1949, B. Ya. moved to Kharkov. During several decades after the end of World War II, some other mathematicians moved from Odessa to Kharkov: Izrail Glazman, Mikhail Dolberg, Moshe Livshits, Vladimir Potapov. However, B. Ya. has kept close ties with Odessa, Mark Krein, and the mathematicians of Krein's circle for the whole life.

Despite all difficulties, the period from late forties to late sixties was the time of blossoming of the Kharkov mathematical school. At that time Naum Akhiezer, Boris Levin, Vladimir Marchenko, Aleksandr Povzner, and Aleksei Pogorelov worked in Kharkov, and their impact determined the image of Kharkov mathematics for many years.

From 1949, B. Ya. worked at Kharkov University. In addition to undergraduate courses of calculus, theory of functions of a complex variable and functional analysis, he taught advanced courses on entire functions, quasi-analytic classes, almost periodic functions, harmonic analysis and approximation theory, and Banach algebras. The lectures were distinguished by their originality, depth and elegance. B. Ya. used to include his own, yet-unpublished results as well as new original proofs of known theorems. He attracted a very wide audience of students of various levels and also research mathematicians. One can get some flavor of B.Ya. courses from a small book published in Moscow State University and based on one semester course given there in 1969. This book was translated by AMS in 1996. It is a great pity that notes of other courses are not available.

In 1956 he published his famous treatise "Distribution of zeros of entire functions", which greatly influenced several generations of analysts. It was translated into German and English and revised in 1980. Even now the book is the main source on the subject.

On the same year B. Ya. started his Thursday seminar at Kharkov University. For about 40 years it has been a school for Kharkov mathematicians working in analysis and has been a center of active mathematical research. The major part of seminar talks concerned complex analysis and its applications. Nevertheless, there was no restriction on the subject: there were talks on Banach spaces, spectral theory of operators, differential and integral equations, and probability theory. A meeting of the seminar usually lasted more than two hours, with a short break. In most cases detailed proofs were presented. Its active participants included Vladimir Azarin, Aleksandr Eremenko, Sergei Favorov, Gennadiy Feldman, Aleksandr Fryntov, Anatolii Grishin, Vladimir P. Gurarii, Illya Hachatryan, Mikhail Kadets, Victor Katsnelson, Vladimir Logvinenko, Yurii Lyubarskii, Yurii Lyubich, Vladimir Matsaev, Vitali Milman, M.V. Novitskii, Iossif Ostrovskii, Igor' Ovcharenko, Victor Petrenko, Lev Ronkin, Mikhail Sodin, Vadim Tkachenko, Alexander Ulanovskii and many others. B. Ya. has always been proud and delighted with achievements of the participants of his seminar.

In 1969, without interrupting his teaching at the university, B. Ya. organized and headed the Department of the Theory of Functions at the Institute for Low Temperature Physics and Engineering of the Academy of Sciences of Ukraine, where he gathered a group of his former students and young colleagues. He worked there till the last days of his life. A well-known western mathematician working in complex analysis once said: "It is a typical Soviet habit to make secrets from everything; evidently, "Low Temperature Physics" is just a code for function theory".

The name of the founder and first director of the institute Boris Verkin must be mentioned here. A specialist in experimental physics, he held mathematics in high esteem and gave a lot of support to its progress. "Mathematicians ennoble the institute," Verkin used to say. Due to his initiative, Naum Akhiezer, Izrail Glazman, Vladimir Marchenko, Anatolii Myshkis, Aleksei Pogorelov joined the institute in the early sixties and very soon the Mathematical Division of the institute became one of the leading mathematical centers in the former Soviet Union, with the wonderful creative atmosphere.

It is not our intention to give here a detailed description of mathematical activities of B. Ya. We only mention that he knew how to find unexpectedly simple ways leading to a solution of a problem which from the beginning seemed to be extremely complicated. After his talks and works one would be puzzled why other mathematicians who attacked the same problem did not have the same insight. The participants of his seminar remember that sometimes after somebody's "hard" talk B. Ya. proposed his simple and elegant solution. At the same time B. Ya. mastered the fine analytic techniques, which he successfully used if required.

The main part of results obtained by B. Ya. are related to the theory of entire functions. Being involved in the main problems of this theory, he found new and important connections with other areas of analysis. His results helped to transfer applications of the theory of entire functions to functional analysis and the spectral theory of differential operators to a deeper level. Often B. Ya. expressed the viewpoint that the theory of entire functions remains of importance due to its numerous applications.

Boris Levin lived a long life, full of mathematical quest and discoveries. He experienced many difficult periods, but despite all strokes of fate remained faithful to his highest moral principles which he defended openly and selflessly. He did not have, and did not try to seek favours from officialdom. Until mid-80s he was not allowed to travel abroad and had very scarce possibilities to contact foreign colleagues. In Kharkov he lived in a small and wet ground-floor apartment. Nevertheless, very often B. Ya. would invite his colleagues and students to his home. Several hours would be devoted to mathematics. Then supper time would arrive, and his wife Liya, a woman of great charm, kindness, and benevolence, joined the guests. After traditional strong tea which B. Ya. always made himself, there was the time for discussing politics and politicians, for storytelling and poetry, in which B. Ya. was the expert and connoisseur.

Outstanding mathematician, brilliant lecturer and storyteller, witty companion, B. Ya. radiated some kind of energy that attracted to him even people who were very far from mathematics. He was a person of the highest quality to the many people who knew him.