Sergei Adian
Encyclopedia
Sergei Ivanovich Adian, also Adjan is one of the most prominent Soviet and Russia
n mathematicians. He is a professor at the Moscow State University
. He is most famous for his work in group theory
, especially on the Burnside problem
.
. He grew up there in an Armenian
family. He studied at Yerevan
and Moscow
pedagogical institutes.
His advisor was Pyotr Novikov. He has been working at Moscow State University since 1965. Alexander Razborov
was one of his students.
the functional equation f(x + y) = f(x) + f(y) and having discontinuities is
dense in the plane. (Clearly, all continuous solutions of the equation are linear
functions.) This result was not published at the time. It is curious that about
25 years later the American mathematician Edwin Hewitt
from Seattle gave
preprints of some of his papers to Adian during a visit to MSU, one of which
was devoted to exactly the same result, which was published by Hewitt much
later.
By the beginning of 1955 Adian had managed to prove the undecidability of practically all non-trivial
invariant group properties, including the undecidability of being isomorphic to a
fixed group G, for any group G. These results made up his Ph.D. thesis and his first published work. This is one of the most
remarkable, beautiful, and general results in algorithmic group theory and is now known as the Adian–Rabin theorem.
What distinguishes the first published work by Adian, is its completeness. In spite of numerous attempts, nobody
has added anything fundamentally new to the results during the past 50 years. Adian’s result was immediately used by A. A. Markov in
his proof of the algorithmic unsolvability of the classical problem of deciding when topological manifolds are homeomorphic.
Before the work of Novikov and Adian an affirmative answer to the problem was known only for n 2 {2, 3, 4, 6} and the matrix groups.
However, this did not
hinder the belief in an affirmative answer for any period n. The only question was
to find the right methods for proving it. As later developments showed, this belief
was too naive. This just demonstrates that before their work nobody even came
close to imagining the nature of the free Burnside group, or the extent to which
subtle structures inevitably arose in any serious attempt to investigate it. In fact,
there were no methods for proving inequalities in groups given by identities of the
form X^n = 1.
An approach to solving the problem in the negative was first outlined by
P. S. Novikov in his note, which appeared in 1959. However, the concrete
realization of his ideas encountered serious difficulties, and in 1960, at the insistence
of Novikov and his wife Lyudmila Keldysh, Adian settled down to work on
the Burnside problem. Completing the project took intensive efforts from both
collaborators in the course of eight years, and in 1968 their famous paper
appeared, containing a negative solution of the problem for all odd periods
n > 4381, and hence for all multiples of those odd integers as well.
The solution of the Burnside problem was certainly one of the most outstanding
and deep mathematical results of the past century. At the same time, this result
is one of the hardest theorems: just the inductive step of a complicated induction
used in the proof took up a whole issue of volume 32 of Izvestiya, even lengthened
by 30 pages. In many respects the work was literally carried to its conclusion by
the exceptional persistence of Adian. In that regard it is worth recalling the words
of Novikov, who said that he had never met a mathematician more ‘penetrating’
than Adian.
In contrast to the Adian–Rabin theorem, the paper of Adian and Novikov in no way ‘closed’ the Burnside problem. Moreover,
over a long period of more than ten years Adian continued to improve and simplify
the method they had created and also to adapt the method for solving some other
fundamental problems in group theory.
By the beginning of 1980s, when other contributors
appeared who mastered the Novikov–Adian method, the theory already
represented a powerful method for constructing and investigating new groups (both
periodic and non-periodic) with interesting properties prescribed.
, A. L. Semenov and V. A. Uspensky.
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...
n mathematicians. He is a professor at the Moscow State University
Moscow State University
Lomonosov Moscow State University , previously known as Lomonosov University or MSU , is the largest university in Russia. Founded in 1755, it also claims to be one of the oldest university in Russia and to have the tallest educational building in the world. Its current rector is Viktor Sadovnichiy...
. He is most famous for his work in group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, especially on the Burnside problem
Burnside's problem
The Burnside problem, posed by William Burnside in 1902 and one of the oldest and most influential questions in group theory, asks whether a finitely generated group in which every element has finite order must necessarily be a finite group...
.
Biography
Adian was born near ElizavetpolGanja, Azerbaijan
Ganja is Azerbaijan's second-largest city with a population of around 313,300. It was named Yelizavetpol in the Russian Empire period. The city regained its original name—Ganja—from 1920–1935 during the first part of its incorporation into the Soviet Union. However, its name was changed again and...
. He grew up there in an Armenian
Armenians
Armenian people or Armenians are a nation and ethnic group native to the Armenian Highland.The largest concentration is in Armenia having a nearly-homogeneous population with 97.9% or 3,145,354 being ethnic Armenian....
family. He studied at Yerevan
Yerevan
Yerevan is the capital and largest city of Armenia and one of the world's oldest continuously-inhabited cities. Situated along the Hrazdan River, Yerevan is the administrative, cultural, and industrial center of the country...
and Moscow
Moscow
Moscow is the capital, the most populous city, and the most populous federal subject of Russia. The city is a major political, economic, cultural, scientific, religious, financial, educational, and transportation centre of Russia and the continent...
pedagogical institutes.
His advisor was Pyotr Novikov. He has been working at Moscow State University since 1965. Alexander Razborov
Alexander Razborov
Aleksandr Aleksandrovich Razborov , sometimes known as Sasha Razborov, is a Soviet and Russian mathematician and computational theorist who won the Nevanlinna Prize in 1990 for introducing the "approximation method" in proving Boolean circuit lower bounds of some essential algorithmic problems, and...
was one of his students.
Scientific career
In his first work as a student in 1950, Adian proved that the graph of a function f(x) of a real variable satisfyingthe functional equation f(x + y) = f(x) + f(y) and having discontinuities is
dense in the plane. (Clearly, all continuous solutions of the equation are linear
functions.) This result was not published at the time. It is curious that about
25 years later the American mathematician Edwin Hewitt
Edwin Hewitt
Edwin Hewitt was an American mathematician known for his work in abstract harmonic analysis and for his discovery, in collaboration with Leonard Jimmie Savage, of the Hewitt–Savage zero-one law.He received his Ph.D...
from Seattle gave
preprints of some of his papers to Adian during a visit to MSU, one of which
was devoted to exactly the same result, which was published by Hewitt much
later.
By the beginning of 1955 Adian had managed to prove the undecidability of practically all non-trivial
invariant group properties, including the undecidability of being isomorphic to a
fixed group G, for any group G. These results made up his Ph.D. thesis and his first published work. This is one of the most
remarkable, beautiful, and general results in algorithmic group theory and is now known as the Adian–Rabin theorem.
What distinguishes the first published work by Adian, is its completeness. In spite of numerous attempts, nobody
has added anything fundamentally new to the results during the past 50 years. Adian’s result was immediately used by A. A. Markov in
his proof of the algorithmic unsolvability of the classical problem of deciding when topological manifolds are homeomorphic.
Burnside problem
About the Burnside problem:
Very much like Fermat’s Last Theorem in number theory, Burnside’s
problem has acted as a catalyst for research in group theory. The fascination
exerted by a problem with an extremely simple formulation which
then turns out to be extremely difficult has something irresistible about
it to the mind of the mathematician.
Before the work of Novikov and Adian an affirmative answer to the problem was known only for n 2 {2, 3, 4, 6} and the matrix groups.
However, this did not
hinder the belief in an affirmative answer for any period n. The only question was
to find the right methods for proving it. As later developments showed, this belief
was too naive. This just demonstrates that before their work nobody even came
close to imagining the nature of the free Burnside group, or the extent to which
subtle structures inevitably arose in any serious attempt to investigate it. In fact,
there were no methods for proving inequalities in groups given by identities of the
form X^n = 1.
An approach to solving the problem in the negative was first outlined by
P. S. Novikov in his note, which appeared in 1959. However, the concrete
realization of his ideas encountered serious difficulties, and in 1960, at the insistence
of Novikov and his wife Lyudmila Keldysh, Adian settled down to work on
the Burnside problem. Completing the project took intensive efforts from both
collaborators in the course of eight years, and in 1968 their famous paper
appeared, containing a negative solution of the problem for all odd periods
n > 4381, and hence for all multiples of those odd integers as well.
The solution of the Burnside problem was certainly one of the most outstanding
and deep mathematical results of the past century. At the same time, this result
is one of the hardest theorems: just the inductive step of a complicated induction
used in the proof took up a whole issue of volume 32 of Izvestiya, even lengthened
by 30 pages. In many respects the work was literally carried to its conclusion by
the exceptional persistence of Adian. In that regard it is worth recalling the words
of Novikov, who said that he had never met a mathematician more ‘penetrating’
than Adian.
In contrast to the Adian–Rabin theorem, the paper of Adian and Novikov in no way ‘closed’ the Burnside problem. Moreover,
over a long period of more than ten years Adian continued to improve and simplify
the method they had created and also to adapt the method for solving some other
fundamental problems in group theory.
By the beginning of 1980s, when other contributors
appeared who mastered the Novikov–Adian method, the theory already
represented a powerful method for constructing and investigating new groups (both
periodic and non-periodic) with interesting properties prescribed.
External links
On 75th birthday – an article by L. D. Beklemishev, I. G. Lysenok, S. P. Novikov, M. R. Pentus, A. A. RazborovAlexander Razborov
Aleksandr Aleksandrovich Razborov , sometimes known as Sasha Razborov, is a Soviet and Russian mathematician and computational theorist who won the Nevanlinna Prize in 1990 for introducing the "approximation method" in proving Boolean circuit lower bounds of some essential algorithmic problems, and...
, A. L. Semenov and V. A. Uspensky.