Anatoly Maltsev
Encyclopedia
Anatoly Ivanovich Maltsev (Malcev) (Russian
: Анато́лий Ива́нович Ма́льцев 27 November N.S.
/14 November O.S.
1909 - 7 June 1967) was born in Misheronsky, near Moscow
, and died in Novosibirsk
, USSR
. He was a mathematician
noted for his work on the decidability of various algebraic group
s. Malcev algebra
s (generalisations of Lie algebra
s) are named after him.
to study Mathematics
. While he was there, he started teaching in a secondary school in Moscow. After graduating in 1931, he continued his teaching career and in 1932 was appointed as an assistant at the Ivanovo Pedagogical Institute
located in Ivanovo
, near Moscow.
Whilst teaching at Ivanovo, Maltsev made frequent trips to Moscow to discuss his research with Kolmogorov. Maltsev's first publications were on logic
and model theory
. Kolmogorov soon invited him to join his graduate programme at Moscow University, and, maintaining his post at Ivanovo, Maltsev effectively became Kolmogorov's student.
In 1937, Maltsev published a paper on the embeddability of a ring
in a field
. Two years later, he published a second paper where he gave necessary and sufficient conditions for a semigroup
to be embeddable in a group
.
Between 1939 and 1941, he studied for his doctorate at the Steklov Institute of the USSR Academy of Sciences, with a dissertation on the Structure of isomorphic representable infinite algebras and groups.
In 1944, Maltsev became a professor at the Ivanovo Pedagogical Institute where he continued to work on group theory
and linear groups in particular. He also studied Lie group
s and topological algebra
s.
In 1958, Maltsev became an Academician of the Soviet Academy of Sciences.
In 1960, Maltsev was appointed to a chair in mathematics at the Mathematics Institute at Novosibirsk and chaired the Algebra and Logic Department of Novosibirsk State University
. He founded the Siberian section of the Mathematics Institute of the Academy of Sciences, the Siberian Mathematical Society and the journal "Algebra i Logika". Maltsev also founded the "Algebra and Logic Seminar" attended by his students Igor Lavrov, Larisa Maksimova, Dmitry Smirnov, Mikhail Taitslin, and A. Vinogradov, as well as by Yuri Ershov
and others. This seminar, in essence, started a new and extremely fruitful school in model theory
and decidability of elementary theories.
During the early 1960s, Maltsev worked on problems of decidability of elementary theories of various algebraic structures. He showed the undecidability of the elementary theory of finite group
s, of free nilpotent group
s, of free soluble group
s and many others. He also proved that the class of locally free algebras has a decidable theory.
Maltsev received many honours, including the Stalin Prize in 1946 and Lenin Prize
in 1964.
In 1962 he founded the mathematical journal Algebra i Logika
.
Russian language
Russian is a Slavic language used primarily in Russia, Belarus, Uzbekistan, Kazakhstan, Tajikistan and Kyrgyzstan. It is an unofficial but widely spoken language in Ukraine, Moldova, Latvia, Turkmenistan and Estonia and, to a lesser extent, the other countries that were once constituent republics...
: Анато́лий Ива́нович Ма́льцев 27 November N.S.
Old Style and New Style dates
Old Style and New Style are used in English language historical studies either to indicate that the start of the Julian year has been adjusted to start on 1 January even though documents written at the time use a different start of year ; or to indicate that a date conforms to the Julian...
/14 November O.S.
Old Style and New Style dates
Old Style and New Style are used in English language historical studies either to indicate that the start of the Julian year has been adjusted to start on 1 January even though documents written at the time use a different start of year ; or to indicate that a date conforms to the Julian...
1909 - 7 June 1967) was born in Misheronsky, near 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...
, and died in Novosibirsk
Novosibirsk
Novosibirsk is the third-largest city in Russia, after Moscow and Saint Petersburg, and the largest city of Siberia, with a population of 1,473,737 . It is the administrative center of Novosibirsk Oblast as well as of the Siberian Federal District...
, USSR
Soviet Union
The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....
. He was a mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
noted for his work on the decidability of various algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s. Malcev algebra
Malcev algebra
In mathematics, a Malcev algebra over a field is a nonassociative algebra that is antisymmetric, so thatxy = -yx\ and satisfies the Malcev identity = x + x + y.\...
s (generalisations of Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s) are named after him.
Biography
At school, Maltsev demonstrated an aptitude for mathematics, and when he left school in 1927, he went to Moscow State UniversityMoscow 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...
to study Mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. While he was there, he started teaching in a secondary school in Moscow. After graduating in 1931, he continued his teaching career and in 1932 was appointed as an assistant at the Ivanovo Pedagogical Institute
Ivanovo State University
Ivanovo State University or IvSU is the oldest and one of the most famous universities of Ivanovo Oblast, located in Ivanovo, about 300 km east of Moscow, Russia. The University was founded in 1918 and now has one of the best resources, faculty and students among the classical universities...
located in Ivanovo
Ivanovo
Ivanovo is a city and the administrative center of Ivanovo Oblast, Russia. Population: Ivanovo has traditionally been called the textile capital of Russia. Since most textile workers are women, it has also been known as the "City of Brides"...
, near Moscow.
Whilst teaching at Ivanovo, Maltsev made frequent trips to Moscow to discuss his research with Kolmogorov. Maltsev's first publications were on logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
and model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
. Kolmogorov soon invited him to join his graduate programme at Moscow University, and, maintaining his post at Ivanovo, Maltsev effectively became Kolmogorov's student.
In 1937, Maltsev published a paper on the embeddability of a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
in a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
. Two years later, he published a second paper where he gave necessary and sufficient conditions for a semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
to be embeddable in a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
.
Between 1939 and 1941, he studied for his doctorate at the Steklov Institute of the USSR Academy of Sciences, with a dissertation on the Structure of isomorphic representable infinite algebras and groups.
In 1944, Maltsev became a professor at the Ivanovo Pedagogical Institute where he continued to work on 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...
and linear groups in particular. He also studied Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s and topological algebra
Topological algebra
In mathematics, a topological algebra A over a topological field K is a topological vector space together with a continuous multiplication\cdot :A\times A \longrightarrow A\longmapsto a\cdot bthat makes it an algebra over K...
s.
In 1958, Maltsev became an Academician of the Soviet Academy of Sciences.
In 1960, Maltsev was appointed to a chair in mathematics at the Mathematics Institute at Novosibirsk and chaired the Algebra and Logic Department of Novosibirsk State University
Novosibirsk State University
Novosibirsk State University was founded in May 1959 in the USSR by Soviet academicians Mikhail Alekseevich Lavrentiev, Sergei Lvovich Sobolev and Sergey Alekseyevich Khristianovich in a program of establishing a Siberian branch of the USSR Academy of Sciences...
. He founded the Siberian section of the Mathematics Institute of the Academy of Sciences, the Siberian Mathematical Society and the journal "Algebra i Logika". Maltsev also founded the "Algebra and Logic Seminar" attended by his students Igor Lavrov, Larisa Maksimova, Dmitry Smirnov, Mikhail Taitslin, and A. Vinogradov, as well as by Yuri Ershov
Yuri Ershov
Yuri L. Ershov or Yershov is an outstanding Soviet and Russian mathematician.Yuri Ershov was born in 1940 in Novosibirsk. In 1958 he entered the Tomsk State University and in 1963 graduated from the Mathematical Department of the Novosibirsk State University...
and others. This seminar, in essence, started a new and extremely fruitful school in model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
and decidability of elementary theories.
During the early 1960s, Maltsev worked on problems of decidability of elementary theories of various algebraic structures. He showed the undecidability of the elementary theory of finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
s, of free nilpotent group
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
s, of free soluble group
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
s and many others. He also proved that the class of locally free algebras has a decidable theory.
Maltsev received many honours, including the Stalin Prize in 1946 and Lenin Prize
Lenin Prize
The Lenin Prize was one of the most prestigious awards of the USSR, presented to individuals for accomplishments relating to science, literature, arts, architecture, and technology. It was created on June 23, 1925 and was awarded until 1934. During the period from 1935 to 1956, the Lenin Prize was...
in 1964.
In 1962 he founded the mathematical journal Algebra i Logika
Algebra i Logika
Algebra i Logika is a Russian mathematical journalfounded in 1962 by Anatoly Ivanovich Malcev, published by the Siberian Fund for Algebra and Logic at Novosibirsk State University. An English translation of the journal is published by Springer-Verlag as Algebra and Logic since 1968...
.