When he was seven years old, in 1871, tragedy struck the family when his father died. After this sad event which also left the family in some financial difficulty, his mother left Kirillov, taking the young boy and his sister to live in St Petersburg. There, beginning in 1873, he attended a private school run by the teacher F F Bychkova. His performance at the school was remarkable and he completed his studies in 1881, three years early, winning the gold medal. In the same year, he began his studies in the Faculty of Physics and Mathematics at the University of St Petersburg. Now Grave was a very talented as a musician and he would have liked to have studied music at the conservatory in parallel with his mathematical studies at the university. However, this would have required a level of financial support that his family were unable to give so he had to give up the idea of formally studying music.
At university, he studied under Pafnuty Lvovich Chebyshev and his pupils Aleksandr Nikolaevich Korkin, Egor Ivanovich Zolotarev and Andrei Andreyevich Markov. Grave spoke highly of Chebyshev's teaching :-
Chebyshev was a wonderful lecturer. His courses were very short. As soon as the bell sounded, he immediately dropped the chalk, and, limping, left the auditorium. On the other hand he was always punctual and not late for classes. Particularly interesting were his digressions when he told us about what he had spoken outside the country or about the response of Hermite or others. Then the whole auditorium strained not to miss a word.He was much influenced by Chebyshev's advice that:-
... we should not be engaged in what is interesting and curious, but rather what is important and necessary.Grave graduated with his first degree in 1885, having written a dissertation On minimal surfaces, which was published in the Notes of the Physics and Mathematics Association of St Petersburg University Students. He had already begun research while an undergraduate and wanted to remain at the university to train to become a professor. However, the family was poor and, in order to support his mother and sister, he had to earn money by giving private lessons and even doing odd jobs. He obtained his masters degree in 1889 (equivalent to a Ph.D.) for his thesis On the Integration of Partial Differential Equations of the First Order (Russian) and, in the autumn of that year, began teaching at the University of St Petersburg. For his Master's Degree, Grave had undertaken work on Jacobi's methods for the three body problem, a topic which had been suggested by Korkin. The methods he used in this work were based on those of Jacobi and Korkin. At St Petersburg he taught analytic geometry, algebra, calculus I and a special course on the theory of surfaces. While undertaking research at the University, in 1890 he began teaching higher mathematics at the Institute of Railway Engineers and in 1892 at the Women's University in St Petersburg where he taught until 1896. He also began publishing books based on his lecture courses, for example: Course of analytical geometry (1893); and A course in differential calculus (1895).
His doctorate (essentially equivalent to the habilitation) involved a study of map projections, again a topic proposed by Korkin, the degree being awarded in 1896 for the thesis On the Main Problems of the Mathematical Theory of Construction of Geographical Maps (Russian). The work, on equal area plane projections of the sphere, built on ideas of Euler, Lagrange and Chebyshev :-
In it Grave presented a comprehensive study of equal-area plane projections of a sphere, with meridians and parallels being represented on the plane by straight lines and circumferences respectively.However, his health was poor at this time and he was advised by his doctors to move to a place with a milder climate than St Petersburg, preferably in the south. Grave took their advice and became professor at Kharkov in 1897 and, from 1899, he was appointed professor at the Saint Vladimir University of Kiev, where he remained for the rest of his life. We note that this university was founded in 1834 and renamed the Taras Shevchenko University of Kiev in 1939. He moved with his family (his mother accompanied him) to Kiev in 1901 and he was appointed as a full professor at Kiev in 1902.
Grave is considered as the founder of the Kiev school of algebra which was to become the centre for algebra in the USSR. At Kiev Grave studied algebra and number theory. In particular he worked on Galois theory, ideals and equations of the fifth degree. He was a outstanding speaker, and a highly gifted teacher. He could explain deep mathematical ideas in a remarkably clear and simple way, and this talent led to a large number of students attending his lectures. He was able to give his students a passion for the mathematical sciences both from the clarity of his presentations and from his own obvious love for the subject. Grave made a major contribution to the development and improvement of teaching mathematical sciences in the University of Kiev. One of the innovations that he introduced was a compulsory number theory course, and he introduced a new style for seminars designed to train students. Among the courses he taught at Kiev we mention: "Group theory"; "Elementary course in the theory of numbers"; "Elements of the theory of elliptic functions"; "Fundamentals of analytical geometry"; "Mathematics of insurance "; and "The elements of algebra".
Now we mentioned that, when he had been in St Petersburg, Grave had been advised to move south for health reasons. In fact the move to Kiev did not solve his health problems and, particularly during hard winters, he suffered serious illnesses. For instance in the winter of 1904 he had a very bad cold which turned into an acute form of pulmonary tuberculosis. He was treated at different hospitals abroad, spending time on the shores of the Mediterranean Sea, spending some time on the French Riviera. During the time he spent on the French Riviera, he visited the local casino and became interested in probability while watching people playing roulette. This led Grave to think about applications of mathematics to insurance, and various actuarial problems. Back in Kiev he taught courses on these topics at the Commercial Institute and later wrote textbooks based on these courses.
The Revolution of 1917 had some major effects on the development of mathematics in Russia and the Ukraine. One effect was that mathematics in the Ukraine was required to be more practical and algebra did not fit into this applied mathematics and technology dominated scene. Grave had to discontinue his famous Kiev algebra seminar in the 1920s, give up teaching and research in algebra, and move to applied mathematics topics. It would not be before the 1950s, well after Grave's death, that Kiev would again play a major role in algebra research. The five years 1915-1920 were exceptionally difficult for Grave because, in addition to the political turmoil which tore the country apart, these were years of personal tragedy with the deaths of his children.
Grave chaired the Applied Mathematics Commission of the Academy of Sciences of the Ukraine in the 1920s. After Grave stopped work on algebra, he began to study mechanics and applied mathematics, but he never completely gave up algebra. From 1924 to 1932, he headed the Department of Mechanics at the University of Kiev. During the 1930s there were further changes to the Soviet educational system, and there was a fair amount of reorganisation. The Institute of Mathematics of the Academy of Sciences was founded in Kiev in 1934 and Grave served as the first director of the Institute from its foundation until his death in 1939. His work at Institute of Mathematics was in addition to his chair at Kiev University which he continued to hold.
Among the many books that Grave wrote were Theory of Finite Groups (Russian) (1908), A Primer in Number Theory (Russian) (1909-10), Arithmetical Theory of Algebraic Quantities (Russian) (1909-10), Elements of the theory of elliptic functions (Russian) (1910), Encyclopaedia of Mathematics. An Essay on Its Current State (Russian) (1912), Elements of Higher Algebra (Russian) (1914), Theory of pension funds (Russian) (1917), and A Course in Algebraic Analysis (Russian) (1932). He also studied the history of algebraic analysis and published two volumes of his Treatise on Algebraic Calculus (Russian) in 1938 and 1939. His ambitious intention was to write a 17-volume work on algebraic analysis but his death in 1939 meant that only two volumes were published. As an example of the style of his books we record that a reviewer of Elements of Higher Algebra writes:-
This book made a good impression on me, combining the clarity seen in French textbooks with the rigour seen in German textbooks.As well as this large number of good quality books, Grave published 180 scientific papers on a wide variety of pure and applied mathematical topics during his career. He also trained many mathematicians who went on to have leading roles in Soviet mathematics. For example, among his students we should mention: Naum Il'ich Akhiezer, Nikolai Nikolaevich Bogolyubov, Nikolai Grigorievich Chebotaryov, Boris Nikolaevich Delone, Alexander Markowich Ostrowski, Otto Yulyevich Schmidt, and Yurii Dmitrievich Sokolov.
Among the honours that were given to him was election to the Academy of Sciences of the Ukraine in 1919, election to the Shevchenko Scientific Society in 1923 and election to the Academy of Sciences of the USSR in 1929. In addition, he was awarded the Order of the Red Banner by the Soviet Government in 1935. He is remembered in his home town of Kirillov where he has been honoured by having a street named after him. He died in Kiev and was buried in the Lukyanivske cemetery.
Article by: J J O'Connor and E F Robertson
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