Andrey Andreyevich Markov (russian: Андре́й Андре́евич Ма́рков; St. Petersburg, September 22, 1903 –

Moscow Moscow ( , US chiefly ; rus, links=no, Москва, r=Moskva, p=mɐskˈva, a=Москва.ogg) is the capital and largest city of Russia. The city stands on the Moskva River in Central Russia, with a population estimated at 13.0 million ...

, October 11, 1979) was a

Soviet The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, the son of the Russian mathematician Andrey 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 In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...

and the foundations of mathematics. His name is in particular associated with Markov's principle and Markov's rule in mathematical logic, Markov's theorem in

knot theory In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...

and

Markov algorithm In theoretical computer science, a Markov algorithm is a string rewriting system that uses grammar-like rules to operate on strings of symbols. Markov algorithms have been shown to be Turing-complete, which means that they are suitable as a gener ...

theoretical computer science Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumsc ...

. An important result that he proved in 1947 was that the

word problem for semigroups A word is a basic element of language that carries an objective or practical meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of what a word is, there is no consen ...

was unsolvable; Emil Post obtained the same result independently at about the same time. In 1953 he became a member of the

Communist Party A communist party is a political party that seeks to realize the socio-economic goals of communism. The term ''communist party'' was popularized by the title of ''The Manifesto of the Communist Party'' (1848) by Karl Marx and Friedrich Engels. A ...

. In 1960, Markov 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 In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. Alan Turing proved in 1936 that a g ...

. This result has profound implications for the limitations of mathematical analysis. His doctoral students include Boris Kushner, Gennady Semenovich Makanin, and

Nikolai Aleksandrovich Shanin Nikolai Aleksandrovich Shanin (russian: Николай Александрович Шанин) (25 May 1919 Pskov – 17 September 2011) was a Russian mathematician who worked on topology and constructive mathematics. He introduced the delta-system ...

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* 1903 births 1979 deaths Mathematical logicians Topologists 20th-century Russian mathematicians Soviet mathematicians {{Mathematician-stub