Andrey Markov (Soviet mathematician)
Encyclopedia
Andrey Andreyevich Markov Jr. (September 22, 1903 - October 11, 1979) was a Soviet mathematician
, the son of the great Russian mathematician Andrey Andreyevich Markov Sr
, and one of the key founders of the Russian school of constructive mathematics and logic. He made outstanding contributions to various areas of mathematics, including differential equation
s, topology
, mathematical logic
and the foundations of mathematics
.
His name is in particular associated with Markov's principle
and with Markov's rule. An important result which he proved in 1947 was that the word problem for semigroups was unsolvable; Emil Post obtained the same result independently at about the same time. In 1953 He became a member of the Communist Party
.
In 1960, Andrey Markov Jr. also obtained fundamental results showing that the classification of four-dimensional manifolds is undecidable (no general algorithm exists for
distinguishing two arbitrary manifolds with four or more dimensions). This is because four dimensional manifolds have sufficient flexibility to allow us to embed any algorithm within their structure, so that classification of all four-manifolds would imply a solution to Turing's halting problem
. This result
has profound implications for the limitations of mathematical analysis.
[1] Markov A A 1960 Proceedings of the International Congress of Mathematicians, Edinburgh 1958 edited by J. Todd Cambridge University Press, Cambridge p 300
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....
, the son of the great Russian mathematician Andrey Andreyevich Markov Sr
Andrey Markov
Andrey Andreyevich Markov was a Russian mathematician. He is best known for his work on theory of stochastic processes...
, and one of the key founders of the Russian school of constructive mathematics and logic. He made outstanding contributions to various areas of mathematics, including differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s, topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
and the foundations of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
.
His name is in particular associated with Markov's principle
Markov's principle
Markov's principle, named after Andrey Markov Jr, is a classical tautology that is not intuitionistically valid but that may be justified by constructive means.There are many equivalent formulations of Markov's principle.- Statements of the principle :...
and with Markov's rule. An important result which he proved in 1947 was that the word problem for semigroups was unsolvable; Emil Post obtained the same result independently at about the same time. In 1953 He became a member of the Communist Party
Communist Party of the Soviet Union
The Communist Party of the Soviet Union was the only legal, ruling political party in the Soviet Union and one of the largest communist organizations in the world...
.
In 1960, Andrey Markov Jr. also obtained fundamental results showing that the classification of four-dimensional manifolds is undecidable (no general algorithm exists for
distinguishing two arbitrary manifolds with four or more dimensions). This is because four dimensional manifolds have sufficient flexibility to allow us to embed any algorithm within their structure, so that classification of all four-manifolds would imply a solution to Turing's halting problem
Halting problem
In computability theory, the halting problem can be stated as follows: Given a description of a computer program, decide whether the program finishes running or continues to run forever...
. This result
has profound implications for the limitations of mathematical analysis.
[1] Markov A A 1960 Proceedings of the International Congress of Mathematicians, Edinburgh 1958 edited by J. Todd Cambridge University Press, Cambridge p 300