Andrey Andreyevich Markov (; 22 September 1903,

Saint Petersburg Saint Petersburg, formerly known as Petrograd and later Leningrad, is the List of cities and towns in Russia by population, second-largest city in Russia after Moscow. It is situated on the Neva, River Neva, at the head of the Gulf of Finland ...

– 11 October 1979,

Moscow Moscow is the Capital city, capital and List of cities and towns in Russia by population, largest city of Russia, standing on the Moskva (river), Moskva River in Central Russia. It has a population estimated at over 13 million residents with ...

) was a

Soviet The Union of Soviet Socialist Republics. (USSR), commonly known as the Soviet Union, was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 until Dissolution of the Soviet ...

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, mathematical structure, structure, space, Mathematica ...

, the son of the Russian mathematician Andrey Markov Sr, and one of the key founders of the Russian school of

constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove th ...

and logic. He made outstanding contributions to various areas of mathematics, including differential equations,

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

and the

foundations of mathematics Foundations of mathematics are the mathematical logic, logical and mathematics, mathematical framework that allows the development of mathematics without generating consistency, self-contradictory theories, and to have reliable concepts of theo ...

. His name is in particular associated with

Markov's principle Markov's principle (also known as the Leningrad principle), named after Andrey Markov Jr, is a conditional existence statement for which there are many equivalent formulations, as discussed below. The principle is logically valid classically, but ...

and Markov's rule in mathematical logic, Markov's theorem in

knot theory In 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 joined so it cannot be und ...

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 gen ...

theoretical computer science Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Associati ...

. 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 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 consensus among linguists on its ...

was unsolvable;

Emil Leon Post Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory. Life Post was born in Augustów, Suwałki Gove ...

obtained the same result independently at about the same time. In 1953 he became a member of the Communist Party. In 1960, Markov obtained fundamental results showing that the classification of four-dimensional

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

s 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. Hence, classifying all four-manifolds would imply a solution to Turing's

halting problem In computability theory (computer science), 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 for ...

. Embedding implies failure to create a correspondence between algorithms and indexing (naturally uncountably infinite, but even larger) of the four-manifolds structure. Failure is in Cantor's sense. Indexing is in Godel's sense. This result has profound implications for the limitations of mathematical analysis. His doctoral students include Boris Kushner, Gennady Makanin, and Nikolai Shanin.

Awards and honors

* Medal "For Valiant Labour in the Great Patriotic War 1941–1945" (1945) *

Order of the Badge of Honour The Order of the Badge of Honour () was a civilian award of the Soviet Union. It was established on 25 November 1935, and was conferred on citizens of the USSR for outstanding achievements in sports, production, scientific research and socia ...

(1945) *

Medal "For the Defence of Leningrad" The Medal "For the Defence of Leningrad" () was a World War II campaign medal of the Soviet Union established on December 22, 1942 by decree of the Presidium of the Supreme Soviet of the USSR to recognise the valour and hard work of the Soviet ...

(1946) *

Order of Lenin The Order of Lenin (, ) was an award named after Vladimir Lenin, the leader of the October Revolution. It was established by the Central Executive Committee on 6 April 1930. The order was the highest civilian decoration bestowed by the Soviet ...

(1954) *

Order of the Red Banner of Labour The Order of the Red Banner of Labour () was an order of the Soviet Union established to honour great deeds and services to the Soviet state and society in the fields of production, science, culture, literature, the arts, education, sports ...

(1963)

Notes

External links

* 1903 births 1979 deaths 20th-century Russian mathematicians Mathematicians from Saint Petersburg Academic staff of Saint Petersburg State University Corresponding Members of the USSR Academy of Sciences Communist Party of the Soviet Union members Recipients of the Order of the Badge of Honour Recipients of the Order of Lenin Recipients of the Order of the Red Banner of Labour Mathematical logicians Topologists Soviet logicians Soviet mathematicians Burials at Kuntsevo Cemetery Russian scientists {{Mathematician-stub