Alexei Skorobogatov on Yuri Manin

The following obituary of Yuri Manin by Alexei Skorobogatov is in the London Mathematical Society Newsletter 507 (July 2003), 41-42.

Alexei Skorobogatov writes:

Yuri Manin, the only child of Ivan Gavrilovich Manin and Revekka Zinovievna Miller, was born on the 16 February 1937 in Simferopol (Crimea). His father, a lecturer in geography, was a son of an illiterate Russian peasant. His mother, a postgraduate student of Russian literature, was a daughter of a playwright and journalist of Jewish origin from Hughesovka (modern Donetsk in the Ukraine).

Yuri Manin's early personal history is an intense family tragedy caused by the catastrophe of the Second World War. In the fifth year of his life, in the face of the Wehrmacht's rapid advance, the Manin family together with the grandparents Zinovy and Hannah Miller fled to the North Caucasus and then further still, across the Caspian Sea, to Chardzhou in the Soviet Central Asia (modern Turkmenistan).

Refugees in their own country, they had no place to stay and no means of existence. During the year of 1942, Yuri's grandmother fell ill and died; soon thereafter his grandfather committed suicide. Later in the year Yuri's father volunteered to fight in the Red Army and perished in the war. After the liberation of Crimea, Yuri and his mother returned to Simferopol only to find the apartment where Yuri lived before the war occupied by another family. Yuri and Revekka became effectively homeless, though they were allowed to move in to Yuri's aunt's communal apartment. The diffcult post-war childhood was further complicated when Yuri's mother lost her job in the Soviet state anti-Semitic campaign of 1948.

While at school, Yuri read I M Vinogradov's book Elements of Number Theory and sent the author a letter with a generalisation of the formula for the number of integer points in a circle. Yuri's life radically changed when he became an undergraduate in the Department of Mechanics and Mathematics of the Moscow State University in 1953, the year of Stalin's death and the inauguration of the grandiose Main Building of the University. Igor Shafarevich approached Yuri and offered to become his supervisor, and - already in 1956 - Yuri's first published paper On cubic congruences to a prime modulus appeared in Izvestia, a leading Soviet mathematical journal. The paper gives an elementary proof of Hasse's bound for the number of points on an elliptic curve over the finite field with pp elements. Manin's precocious talent manifested itself in a series of papers on algebraic curves, many of which are nothing short of miraculous. The high point of this period is Manin's proof of the functional analogue of Mordell's conjecture in characteristic zero (1963). This ingenious proof is based on the discovery of a major new method involving the algebraisation of the theory of Picard-Fuchs differential equations and what Grothendieck later called the Gauss-Manin connection, by now a standard tool when dealing with cohomology of families of varieties over a base. In the words of Robert Coleman, "this work is testimony to the power and depth of Manin's intuition." Another foundational result of this period is Manin's classification of commutative formal groups over fields of finite characteristic. For these achievements he was awarded the Lenin prize in 1967, one of the most prestigious awards of the Soviet Union usually given to composers, ballet dancers and rocket scientists.

In 1960, Yuri Manin became a researcher at the Steklov Mathematical Institute of the Academy of Sciences of the USSR. He took part in the study group organised by Shafarevich with the aim of giving a modern treatment to the results and methods of the Italian school of algebraic geometry. The outcome was the celebrated volume Algebraic Surfaces. Manin's contribution to the project concerned rational and ruled surfaces; thus began his long series of papers on the geometry, combinatorics, and arithmetic of geometrically rational surfaces summarised in his book Cubic Forms (1972). A closely related work is Manin's joint paper with Vasily Iskovskikh which proves birational rigidity of quartic three-folds, thus giving a negative answer to Lüroth's problem.

In May and June of 1967, Yuri Manin participated in Alexandre Grothendieck's Séminaire de Géométrie Algébrique (SGA 6) at IHES, Bures-sur-Yvette. A result of this visit, Manin's paper on motives was the first ever publication on this subject. Grothendieck liked it and recommended it to David Mumford as "a nice foundational paper."

When I became Manin's student in 1980, he was not allowed to do any undergraduate teaching, presumably because that would have brought him too close to the 'ideological frontline'. This may have been a blessing in disguise. Besides, he was not restricted in the choice of his 'special courses', seminars and study groups aimed at a smaller circle of his own students. The weekly Manin seminar had something of a cult status. The seminar was about much more than talks: it was a meeting place where mathematical ideas were passed around. In the breaks, participants walked in pairs in circular corridors of the Main Building of the university discussing mathematics. Many joint papers started in this way.

Manin's graduate students include a large number of highly successful mathematicians. He has had about 60 PhD graduates, and has influenced the development of a great many more young mathematicians, including people who became famous in their own right, such as the Fields medallists Drinfeld and Kontsevich, and many others.

Perhaps the most characteristic feature of Manin's mathematical genius was his uncanny ability to see a far-reaching theoretical potential in mathematical facts and observations that could be perceived by others as specific and isolated, or merely as part of a familiar picture. Manin generously shared his insights; his papers are scattered with open questions and suggestions. Two major examples of his amazing insight are the Brauer-Manin obstruction (1970) and the Manin-Batyrev conjectures (1989-91) that essentially created the area of arithmetic geometry concerned with rational points on higher-dimensional varieties. Many other of Manin's research interests have generated flourishing schools of contemporary mathematics. He made fundamental contributions to instantons, Yang-Mills fields, supergeometry, quantum groups, quantum cohomology, operads, and much more.

Manin's interests took him beyond mathematics: to the human sciences, the literary and cultural milieu. He was a close friend of Vladimir Vysotsky, the singer/songwriter and actor known to every single person in the USSR, and the brothers Arkady and Boris Strugatsky, the best-known science fiction writers of the country.

Perhaps Manin's trademark was his unique ability to see mathematics as a whole, and to form a profound vision of mathematics and its connections with other areas of science, language and the arts. This combination of cultural breadth with a supreme mastery of research in a way that is seldom seen in contemporary mathematics contributed to his immense popularity among students.

In 1992-93 Manin spent a year in MIT. In 1993 he moved to the Max Planck Institute for Mathematics in Bonn to take up the position of a director. From 2002 to 2011 he was also a professor at Northwestern University. He became director emeritus in 2005, without any slowing down in his research.

Yuri Manin was a foreign member of several national academies, and has received a long string of prizes and honours: the Brouwer Medal in 1987, the Frederic Esser Nemmers Prize in 1994, the Schock prize of the Swedish Academy in 1999, the King Faisal International prize in 2002, the Georg Cantor medal of the German Mathematical Society in 2002, Order Pour le Mérite for Science and Art in 2007, the Great Cross of Merit with Star in 2008, and the János Bolyai International Mathematical prize of the Hungarian Academy of Sciences in 2010. He was an invited speaker to the International Congress of Mathematicians on no fewer than five occasions.

I would like to finish with a quote from T S Eliot's Four Quartets that Manin used to describe his attitude to mathematics and, perhaps, to life itself: "For us, there is only the trying. The rest is not our business."

Last Updated September 2023