**Nikolai Vladimirovich Efimov** was a Russian mathematician. He worked on the geometry of surfaces of negative curvature.

- Nikolai Vladimirovich was born in Orenburg, a city in the south of Russia, near the Kazakhstan border, north of the Caspian Sea but grew up in Rostov-na-Donu, about 30 kilometres from the north east end of the Sea of Azov.
- Efimov graduated in 1931, having completed the four-year undergraduate course in three years.
- In 1934 Efimov graduated with a Candidate's degree (equivalent to a Ph.D.) and began work at Voronezh State University as a lecturer.
- The year 1934 when Efimov graduated, was highly significant for geometry research in Russia, for late in that year Stefan Cohn-Vossen emigrated there to escape from the Nazis in Germany.
- Under his influence a school of "geometry in the large" was set up in Moscow and in Leningrad and Efimov's meeting with Cohn-Vossen, who had worked closely with Hilbert, was a major influence on setting the direction of his future research.
- In 1940 Efimov was awarded a doctorate from Moscow State University after submitting the thesis The curving of surfaces with parabolic points.
- Dubnov had also been a student of Kagan and Efimov worked closely with him during these years.
- They advanced along a broad front from the Baltic to the Black Sea and, in August 1941, Efimov and 80 students from Voronezh State University enlisted as volunteers in the Russian army.
- Efimov worked at the Voronezh Aviation Institute in Tashkent until 1943 when he went to Moscow to head the Department of Higher Mathematics at the Moscow State Forest University (it was at that time called the Moscow Institute of Forestry).
- Although Efimov continued as head of the Department of Higher Mathematics at the Moscow State Forest University until 1961 he had several other academic roles within the city.
- As well as this contribution that Efimov made to the teaching programmes of Moscow State University, we must also give an indication of his own skills as a lecturer, his research contributions, and the impressive collection of books that he wrote.
- Efimov's result is a generalization in principle of Hilbert's famous theorem that a surface of constant negative curvature in three-dimensional space cannot be without singularities.
- Efimov's theorem on the qualitative theory of surfaces ranks alongside Hilbert's theorem.
- The theorem which P S Aleksandrov refers to in this quote was one that Efimov became interested in while working for his doctorate but it was not until around 1950 that he began to concentrate all his efforts on obtaining a solution.
- Only a few mathematicians have the determination shown by Efimov to spend over twelve years of their lives trying to solve a single problem but he did so and published his solution in 1963 in The impossibility in three-dimensional Euclidean space of a complete regular surface with a negative upper bound of the Gaussian curvature.
- We must not give the impression that Efimov did not publish in the years during which he battled with his big problem for nothing would be further from the truth.
- Efimov published many important books, some at undergraduate level, some at research monograph level.
- Efimov's book with the largest number of editions is Higher Geometry (Russian) (1945).
- Many important papers on the subject appeared in Russia in the following years, among them Efimov's review that forms the basis of the present book (which to an extent brings the theory back to Germany).
- Another teaching text was published by Efimov in 1963, namely Quadratic forms and matrices (Russian).
- Efimov received many honours for his contributions including the Lobachevsky Medal in 1951, the Lenin Prize in 1966 and two orders of the Red Banner of Labour.
- We end this biography by giving two quotes which tell us something of Efimov's character and his interests outside mathematics.
- Not only in his teaching, editing and administrative work, but in all his dealings with the people around him he was able to see what was important; and perhaps this feature of his character appeared particularly brightly in his mathematical research; Efimov's theorems occupy a worthy place in mathematics.
- Efimov was a remarkable person and a great mathematician.

Born 31 May 1910, Orenburg, Russia. Died 16 October 1982, Moscow, Russia.

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Origin Russia

**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive