Igor Shafarevich

Igor Rostislavovich Shafarevich (И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry.

Outside mathematics, he wrote books and articles that criticised socialism and other books which were described as anti-semitic.

From his early years, Shafarevich made fundamental contributions to several parts of mathematics

including algebraic number theory, algebraic geometry and arithmetic algebraic geometry.

In particular, in algebraic number theory, the Shafarevich–Weil theorem extends the commutative reciprocity map to the case of Galois groups, which are central extensions of abelian groups by finite groups.

Shafarevich was the first mathematician to give a completely self-contained formula for the Hilbert pairing, thus initiating an important branch of the study of explicit formulas in number theory.

Another famous (and slightly incomplete) result is Shafarevich's theorem on solvable Galois groups, giving the realization of every finite solvable group as a Galois group over the rationals.

Another development is the Golod–Shafarevich theorem on towers of unramified extensions of number fields.

Shafarevich and his school greatly contributed to the study of algebraic geometry of surfaces.

Shafarevich came into conflict with the Soviet authorities in the early 1950s but was protected by Ivan Petrovsky, the Rector of Moscow University.

He belonged to a group of Pochvennichestvo-influenced dissidents who endorsed the Eastern Orthodox tradition.

He started a famous Moscow seminar on classification of algebraic surfaces that updated the treatment of birational geometry around 1960 and was largely responsible for the early introduction of the scheme theory approach to algebraic geometry in the Soviet school.

His investigation in arithmetic of elliptic curves led him, independently of John Tate, to the introduction of the group related to elliptic curves over number fields, the Tate–Shafarevich group (usually called 'Sha', and denoted as 'Ш', the first Cyrillic letter of his surname).

He contributed the Grothendieck–Ogg–Shafarevich formula and to the Néron–Ogg–Shafarevich criterion.

With former student Ilya Piatetski-Shapiro, he proved a version of the Torelli theorem for K3 surfaces.

He formulated the Shafarevich conjecture, which stated the finiteness of the set of Abelian varieties over a number field having fixed dimension and prescribed set of primes of bad reduction.

The conjecture was proved by Gerd Faltings as a partial step in his proof of the Mordell conjecture.

Shafarevich's students included Yuri Manin, Alexey Parshin, Igor Dolgachev, Evgeny Golod, Alexei Kostrikin, Suren Arakelov, G. V. Belyi, Victor Abrashkin, Andrey Todorov, Andrey N. Tyurin, and Victor Kolyvagin.

He was a member of the Serbian Academy of Sciences and Arts in the department of Mathematics, Physics and Earth Sciences.

In 1960, he was elected a Member of the German Academy of Sciences Leopoldina.

In the 1970s, Shafarevich, with Valery Chalidze, Grigori Podyapolski and Andrei Tverdokhlebov, became one of Andrei Sakharov's human rights investigators and so was dismissed from Moscow University.

Shafarevich opposed political interference in universities.

Shafarevich published a book, The Socialist Phenomenon (French edition 1975, English edition 1980), which was cited by Aleksandr Solzhenitsyn in his 1978 address to Harvard University.

Shafarevich's book The Socialist Phenomenon, which was published in the US by Harper & Row in 1980, analyzed numerous examples of socialism from ancient times to various medieval heresies and a variety of modern thinkers and socialist states.

From those examples, he claimed that all the basic principles of socialist ideology derive from the urge to suppress individualism.

The Socialist Phenomenon consists of three major parts:

Shafarevich argued that ancient socialism (such as Mesopotamia and Egypt) was not ideological, as an ideology socialism was a reaction to the emergence of individualism in the Axial Age.

He compared Thomas More's (Utopia) and Tommaso Campanella's (City of the Sun) visions with what is known about the Inca Empire and concluded that there are striking similarities.

He claimed that we become persons through our relationship with God and argued that socialism is essentially nihilistic and is unconsciously motivated by a death instinct.

He concluded that we have the choice of pursuing death or life.

Shafarevich adhered to Russian Orthodox Christianity and incorporated the neo-Platonic views of Eastern Orthodoxy into his understanding of the relation of mathematics and religion.

In his talk to the Göttingen Academy of Sciences upon receiving a prize, Shafarevich presented his view of the relationship between mathematics and religion.

He noted the multiple discoveries in mathematics, such as that of non-Euclidean geometry, to suggest that pure mathematics reflects an objective reality, not a set of conventional definitions or a formalism.

He claimed that the growth of mathematics itself is not directed or organic.

To have a unity and direction, mathematics needs a goal.

It can be practical applications or God as the source for the direction of development.

Shafarevich opted for the latter, as pure mathematics is not in itself driven by practical applications.

In 1981, he was elected as a foreign member of the Royal Society.

On 21 December 1991 he took part in the first congress of the Russian All-People's Union, headed by Sergei Baburin.

In 2017, Shafarevich was awarded the Leonhard Euler Gold Medal by the Russian Academy of Sciences.