Andrey Andreyevich Markov (russian: Андре́й Андре́евич Ма́рков;

St. Petersburg Saint Petersburg ( rus, links=no, Санкт-Петербург, a=Ru-Sankt Peterburg Leningrad Petrograd Piter.ogg, r=Sankt-Peterburg, p=ˈsankt pʲɪtʲɪrˈburk), formerly known as Petrograd (1914–1924) and later Leningrad (1924–1991), i ...

, 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 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 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, ...

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

mathematical logic Mathematical logic is the study of 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 forma ...

and the

foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...

. His name is in particular associated with

Markov's principle Markov's 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 not in intuitionistic constructive m ...

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 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 ...

and Markov algorithm in

theoretical computer science 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 circumscribe the ...

. 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 conse ...

was unsolvable;

Emil 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 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 Engel ...

. In 1960, Markov obtained fundamental results showing that the classification of four-dimensional

manifolds 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 ne ...

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 ...

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

Gennady Semenovich Makanin Gennady (or Gennadii or Gennadiy) Semenovich Makanin (1938–2017) was a Russian mathematician, awarded the 2010 I. M. Vinogradov Prize for a series of papers on the problem of algorithmically recognizing the solvability of arbitrary equations in ...

, 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-syste ...

Notes

External links

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