# Andrei Andreevich Bolibrukh

### Quick Info

Moscow, USSR (now Russia)

Moscow, Russia

**Andrei Bolibrukh**was a Russian mathematician known for his work on ordinary differential equations and especially Hilbert's 21st problem.

### Biography

**Andrei Andreevich Bolibrukh**'s father, Andrei Vlas'evich Bolibrukh, was a soldier reaching the rank of major by the end of World War II. His war service saw him suffer five wounds and receive seven awards. He continued to serve in the army, reaching the rank of lieutenant-general. Andrei Andreevich's mother, Tat'yana Ivanovna Pimanikhina, showed a talent for mathematics and entered college before marrying at nineteen and devoting herself to raising her son and his sister Tat'yana. Andrei Vlas'evich was posted to quite a number of different places as Andrei Andreevich was growing up and, as a consequence, the boy attended a variety of different schools in different parts of the country - mainly in Moscow, Tallin, and Kaliningrad. When he was six years old, the family were living in a forest in Belorussia and Andrei's father decided to prepare him to begin his schooling. Bolibrukh explained what happened (see [1]):-

When I was six years old my father decided to prepare me for school. I could read, write and add numbers. My father decided to teach me mathematics. In particular he tried to explain to me how to calculate the area of a circle!! And I remember very well that I could not understand him. He told me, "The area of the circle equals pi r squared. Do you understand?" No, I answered, you say to write numbers, but what is an area? ... now I understand that he was not a mathematician and he could not say that area is a functional on the set of measurable figures with the following properties, etc. ...Although Andrei saw comparatively little of his father, who was busy working, he remembered happy holidays making trips by car or train to interesting places in the Baltic states. They also made trips around Kaliningrad and in the Crimea where he spent time fishing. His first interests were in literature and he loved the school theatre where between age eleven and fourteen he won prizes for recitation. A good mathematics teacher gave him an increasing interest in that subject from the age of twelve. By age fourteen he was making deep mathematical studies. In 1965 he was a prize winner in the All-USSR Mathematics Olympiad organised for school children. Following this he went to a boarding school which was specially set up to educate scientifically talented children. This school, the Physics-Mathematics Boarding School no. 45, was attached to Leningrad State University and was staffed by very fine teachers. Bolibrukh writes of his memories of his teachers, classmates and the excellent organisation of the school in [10]. Mathematics was not his only love, however, and while at this school he developed a love of modern art, poetry and continued his love of the theatre. He graduated from the school in 1967, having won the gold medal, and entered the Mechanics and Mathematics Department of Moscow State University. This university was, he wrote [1]:-

... maybe the best place to study mathematics throughout all the world.Volodya Leksin writes in [13] about undergraduate years with Bolibrukh:-

Andrei and I were fellow students in the Department of Mechanics and Mathematics from 1967 to 1972. From our third year there (that is, from Autumn 1969) we were in one group containing mainly students working in higher geometry and topology. Our research supervisor was Mikhail Mikhailovich Postnikov. In those years he gave splendid lectures for the compulsory course in linear algebra, and when in the Autumn of 1968 he offered a specialised seminar "Algebraic topology and its applications" (which started as a study group and then turned into a research group ...), many students joined this seminar. ... Andrei kept a low profile among his extrovert and noisy contemporaries. At first I attributed this to his very courteous and considerate manner. But after a little contact with him during our study I realised that he was a man of very strong inner self-discipline, responsibility and precision. For example, he would write down in his notebook under the relevant date any obligation that he undertook (to the majority of the students this was unusual and surprising) and would fulfil it to the letter.Bolibrukh graduated with his diploma (equivalent to a Master's Degree) in 1972 having written a thesis on calculations of the cobordism of manifolds with certain relations on the characteristic classes of their tangent bundles. He then began working towards his Candidate's Degree (equivalent to a Ph.D.) with Postnikov as his advisor. Postnikov's research students joined L V Keldysh's seminar and [13]:-

... we began to organize our workshops systematically in various parts of the Moscow countryside, combining topology, football or skiing depending on the season, anecdotes, serious conversations about life, and so on.Alexei Chernavskii recalls in [13] that Bolibrukh was not quite as successful as one might have expected:-

Andrei was an active and lively participant. He put his heart into everything that he tried to do, and he acted - it is hard to think of a better word - with precision. [He gave the] impression of a very lively energetic and pleasant young man, who sometimes suddenly seemed rather more mature than a first glance would have suggested. I have no doubt that Andrei was linguistically gifted: he easily solved various word puzzles ... [but his] mathematical work was not being very successful, although his knowledge and understanding were no less than that of the other students. I can say now that the world of pure topology was not his world, it contained something dissonant with his very independent personality.Bolibrukh had begun to take an interest in mathematical topics outside the area of the research he was undertaking for his Candidate's degree, attending an informal seminar organised by A V Chernavskii to study the multi-dimensional Riemann-Hilbert problem. As a consequence his Ph.D. studies changed direction, influenced by Chernavskii, and he completed research for his thesis

*On the Fundamental Matrix of a Pfaffian System of Fuchsian Type*in 1975. He began working at the Moscow Institute of Physics and Technology but, rather surprisingly, it was two years after completing research for his thesis that Bolibrukh defended it at Moscow State University and was awarded his Candidate's Degree.

Certainly his career was moving less rapidly than one might have expected. Dmitrii Viktorovich Anosov writes in [7] about the following years:-

I am not sure that even his closest friends really knew about his inner growth during about 10 years after defending his PhD dissertation; all the more so since at that time he devoted a lot of attention to completely different things - for example, tribology, that is, the study of the influence of various types of friction on the work of mechanical systems. This was a useful activity connected with the applied orientation of the Moscow Institute of Physics and Technology and requiring a certain qualification, but it had nothing to do with linear differential equations in the complex domain. I met him on several occasions during that time and must confess that he did not then make a strong impression on me.During these years Andrei and his wife Nina started a family and his son, also called Andrei, and daughter Ekaterina were born.

What appeared to be turning into a rather ordinary career for Bolibrukh changed dramatically in 1989 when he solved Hilbert's 21st problem, also known as the Riemann-Hilbert problem. This problem, the 21st in Hilbert's famous list of problems which he gave in 1900 to set the mathematical scene for the 20th century, is as follows:-

Does there exist a Fuchsian system of linear ordinary differential equations in the complex domain having prescribed singularities and monodromy group?Hilbert believed that the question had a positive solution and the problem appeared settled in 1908 when Josip Plemelj proved this by giving a reduction of the problem to a known result. However, in around 1980, it was discovered, by Golubeva, Leksin and others, that there was an error in Plemelj's reduction. It was still believed that what was required was a correction in Plemelj's method to give the expected positive answer but Bolibrukh produced a major surprise when he proved in 1989 that certain prescribed conditions on the singularities led to a negative solution. This brilliant work constituted Bolibrukh's doctoral thesis (equivalent to a D.Sc. or habilitation), awarded in 1991, and suddenly he went from being a rather ordinary mathematician to being a real star. We should make it clear that it was not simply that Bolibrukh had struck luck with one result. Rather from this time on he produced results of the highest quality. Quite why someone should blossom in this way at an unusually advanced age for a leading mathematician, it is hard to understand.

Honours were quickly given to Bolibrukh for his extraordinary work. He was an invited lecturer at the International Congress of Mathematicians held in Zürich in 1994 giving the lecture

*The Riemann-Hilbert problem and Fuchsian differential equations on the Riemann sphere*. From 1990 his main work was done in the Steklov Mathematical Institute and he became deputy director of the Institute in 1996. In 1994 he was elected a corresponding member of the Russian Academy of Sciences, becoming a full member three years later. The Russian Academy of Sciences awarded him their Lyapunov Prize in 1995 and in 2001 he received the State Prize of the Russian Federation. Moscow State University honoured him in 1999 with the title of Honoured Professor.

Honours were certainly not confined to Russia. He was invited to spend 1995-97 as a visiting professor at the University of Nice, and a similar invitation saw him spend 1998-2002 a visiting professor at the University of Strasbourg. He became a member of the Executive Committee of the International Union of Mathematicians in 2002. He was an invited speaker at numerous international conferences, often as a plenary speaker. Over a period of five years he gave talks at no less than 31 international conferences and, even more remarkably, gave a talk on a different topic at each one. Sadly, however, his health deteriorated and he was diagnosed with cancer. He underwent treatment in hospital over many months but when his health appeared to improve somewhat he left hospital and attended a conference to celebrate the 100th anniversary of the birth Andrei Nikolaevich Kolmogorov [7]:-

... he was optimistic, mixed in good spirits with the participants, made plans for his future research (negotiations about medical treatment in France were nearing completion), gave his talk brilliantly as usual, and immediately afterwards attended the unveiling of a memorial bust of L S Pontryagin on Lenin Avenue.Sadly the improvement was short-lived and he was hospitalised again. However, while in hospital he completed writing his last research paper, wrote the memoirs [11] and continued advising his students at the Steklov Mathematical Institute. A year after he died, a conference in his honour was organized in Strasbourg, France.

Let us record other aspects of Bolibrukh's career. M Butuzov, I V'yugin, R Gontsov and V Poberezhnyi write in [13] about Bolibrukh's teaching:-

All of us, at different times, got to know Andrei during our second year at the Department of Mechanics and Mathematics. In those years he gave the specialized lecture course on the analytic theory of differential equations, vector bundles and the Riemann-Hilbert problem. Andrei was a wonderful lecturer. His style of presentation was precise and rigorous but accessible, and he would highlight the most important and beautiful points. He prepared his lectures thoroughly, and one of his courses was published as the book Fuchsian differential equations and holomorphic bundles, which is a good example of his style. After every lecture Andrei asked us, the youngest of his audience members, what we had not understood, and we stayed in the lecture room, sometimes for a whole hour, receiving not only answers to our questions but also improvised lectures on general topics. He lectured with great passion, becoming completely carried away by the material. Once during a lecture he even tore his shirt by accident. His engagement won over the audience and drew them to his subject. Therefore when we had to choose a research advisor we approached Andrei. He agreed, and we joined the group working on differential equations, and later, together with him, joined the newly formed dynamical systems group.Outside mathematics Bolibrukh's interests included tennis, literature, paintings, and the theatre.

### References (show)

- B Yandell,
*The honors class: Hilbert's problems and their solvers*(A K Peters, 2002). - Andrei Andreevich Bolibrukh (December 30, 1950-November 11, 2003) (Russian),
*Sovrem. Mat. Prilozh. No.***10**(2003), 3. - Andrei Andreevich Bolibrukh (December 30, 1950-November 11, 2003),
*J. Math. Sci. (N. Y.)***129**(1) (2005), 3517. - D V Anosov, V I Arnol'd, V M Bukhshtaber, V A Golubeva, A A Gonchar, A B Zhizhchenko, Yu S Il'yashenko, V V Kozlov, S P Konovalov, L D Kudryavtsev, V P Leksin, O B Lupanov, A A Mal'tsev, E F Mishchenko, S P Novikov, Yu S Osipov, M M Postnikov, V A Sadovnichii, A G Sergeev, L D Faddeev and A V Chernavskii, Andrei Andreevich Bolibrukh (obituary) (Russian),
*Uspekhi Mat. Nauk***58**(6) (2003), 139-142. - D V Anosov, V I Arnol'd, V M Bukhshtaber, V A Golubeva, A A Gonchar, A B Zhizhchenko, Yu S Il'yashenko, V V Kozlov, S P Konovalov, L D Kudryavtsev, V P Leksin, O B Lupanov, A A Mal'tsev, E F Mishchenko, S P Novikov, Yu S Osipov, M M Postnikov, V A Sadovnichii, A G Sergeev, L D Faddeev, A V Chernavskii and Andrei Andreevich Bolibrukh (obituary),
*Russian Math. Surveys***58**(6) (2003), 1185-1189. - D V Anosov and V P Leksin, Andrei Andreevich Bolibrukh in life and science (Russian),
*Uspekhi Mat Nauk***59**(6)(360) (2004), 3-22. - D V Anosov and V P Leksin, Andrei Andreevich Bolibrukh in life and science,
*Russian Math. Surveys***59**(6) (2004), 1009-1028. - D V Anosov and E F Mishchenko, In Memory of Andrei Andreevich Bolibrukh (Russian), Dynamical systems and related problems of geometry, Collected papers. Dedicated to the memory of academician Andrei Andreevich Bolibrukh,
*Tr. Mat. Inst. Steklova***244**(Nauka, Moscow, 2004), 5. - D V Anosov and E F Mishchenko, In Memory of Andrei Andreevich Bolibrukh,
*Proc. Steklov Inst. Math.*(1) (244) (2004), 1. - A A Bolibrukh, Some memories of Boarding School no. 45,
*Amer. Math. Soc. Transl.*(2)**174**(1996), 1-5. - A A Bolibrukh, Reminiscences and reflections on times long past (Russian) (Moscow, 2003).
- A V Chernavskii, V P Leksin, M Butuzov, I V'yugin, R Gontsov, V Poberezhnyi, Yu S Il'yashenko, A.G. Sergeev and S P Konovalov, Reminiscences about Andrei Andreevich Bolibrukh (Russian),
*Uspekhi Mat. Nauk***59**(6)(360) (2004), 207-215. - A V Chernavskii, V P Leksin, M Butuzov, I V'yugin, R Gontsov, V Poberezhnyi, Yu S Il'yashenko, A.G. Sergeev and S P Konovalov, Reminiscences about Andrei Andreevich Bolibrukh,
*Russian Math. Surveys***59**(6) (2004), 1213-1224. - V P Leksin, The works of A A Bolibrukh on multidimensional regular and Fuchsian systems (Russian),
*Uspekhi Mat. Nauk***59**(6)(360) (2004), 151-160. - V P Leksin, The works of A A Bolibrukh on multidimensional regular and Fuchsian systems,
*Russian Math. Surveys***59**(6) (2004), 1155-1164. - C Mitschi and C Sabbah, Andrei Bolibroukh, un mathématicien, un ami,
*Gaz. Math. No.***100**(2004), 20-31. - C Mitschi and C Sabbah, Preface,
*Differential equations and quantum groups,**IRMA Lect. Math. Theor. Phys.***9**(Eur. Math. Soc., Zürich, 2007), v-vi. - C Sabbah, The work of Andrey Bolibrukh on isomonodromic deformations, in
*Differential equations and quantum groups*(Eur. Math. Soc., Zürich, 2007), 9-25.

### Additional Resources (show)

Other websites about Andrei Andreevich Bolibrukh:

### Honours (show)

Honours awarded to Andrei Andreevich Bolibrukh

Written by J J O'Connor and E F Robertson

Last Update March 2011

Last Update March 2011