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. Hence, classifying all four-manifolds would imply a solution to Turing's halting problem. 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.
^Kushner, Boris A (2006). "The constructive mathematics of A. A. Markov". Amer. Math. Monthly. 113 (6): 559–566. doi:10.2307/27641983. JSTOR 27641983. MR 2231143.
^Glukhov, M. M.; Nagornyĭ, N. M. (2004). "Andreĭ Andreevich Markov (on the centenary of his birth)". Diskrete Math. Appl. 14 (1): 1–6. doi:10.1515/156939204774148776. MR 2069985. S2CID 120486293.
andrey, markov, other, people, with, same, name, andrei, markov, andrey, andreyevich, markov, russian, Андре, Андре, евич, Ма, рков, september, 1903, saint, petersburg, october, 1979, moscow, soviet, mathematician, russian, mathematician, andrey, markov, found. For other people with the same name see Andrei Markov Andrey Andreyevich Markov Russian Andre j Andre evich Ma rkov 22 September 1903 Saint Petersburg 11 October 1979 Moscow was a Soviet mathematician 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 equations topology mathematical logic and the foundations of mathematics 1 2 His name is in particular associated with Markov s principle and Markov s rule in mathematical logic Markov s theorem in knot theory and Markov algorithm in theoretical computer science An important result that he proved in 1947 was that the word problem for semigroups was unsolvable Emil Leon Post 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 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 Hence classifying all four manifolds would imply a solution to Turing s halting problem 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 EditMedal For Valiant Labour in the Great Patriotic War 1941 1945 1945 Order of the Badge of Honour 1945 Medal For the Defence of Leningrad 1946 Order of Lenin 1954 Order of the Red Banner of Labour 1963 Notes Edit Kushner Boris A 2006 The constructive mathematics of A A Markov Amer Math Monthly 113 6 559 566 doi 10 2307 27641983 JSTOR 27641983 MR 2231143 Glukhov M M Nagornyĭ N M 2004 Andreĭ Andreevich Markov on the centenary of his birth Diskrete Math Appl 14 1 1 6 doi 10 1515 156939204774148776 MR 2069985 S2CID 120486293 External links EditAndrey Markov Jr at the Mathematics Genealogy Project This article about a mathematician is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Andrey Markov Jr amp oldid 1166478278, wikipedia, wiki, book, books, library,
