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Andrey Markov Jr.
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<p>Andrey Andreyevich Markov was a <a href="/facts/Soviet_Union/m7wWsPto">Soviet</a> <a href="/facts/Mathematician/eaFSR8Fx">mathematician</a>, the son of the Russian mathematician <a href="/facts/Andrey_Markov/F409PDqD">Andrey Markov Sr</a>, and one of the key founders of the Russian school of <a href="/facts/Constructivism_(philosophy_of_mathematics)/1VtNuLwT">constructive mathematics</a> and logic. He made outstanding contributions to various areas of mathematics, including <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a>, <a href="/facts/Topology/2EqW47cU">topology</a>, <a href="/facts/Mathematical_logic/SaI7Y3wN">mathematical logic</a> and the <a href="/facts/Foundations_of_mathematics/yjpbbUkt">foundations of mathematics</a>. </p><p>His name is in particular associated with <a href="/facts/Markov%2527s_principle/Dmd1B0J8">Markov's principle</a> and <a href="/facts/Markov%2527s_principle/Dmd1B0J8">Markov's rule</a> in mathematical logic, <a href="/facts/Markov_theorem/v2yPKVzA">Markov's theorem</a> in <a href="/facts/Knot_theory/mb7rtqVP">knot theory</a> and <a href="/facts/Markov_algorithm/3WMjt2As">Markov algorithm</a> in <a href="/facts/Theoretical_computer_science/jSkGGuvy">theoretical computer science</a>. An important result that he proved in 1947 was that the <a href="/facts/Semi-Thue_system/guxDtVd7">word problem for semigroups</a> was unsolvable; <a href="/facts/Emil_Leon_Post/Hl2CKCxw">Emil Leon Post</a> obtained the same result independently at about the same time. In 1953 he became a member of the <a href="/facts/Communist_Party_of_the_Soviet_Union/1PFHfjJE">Communist Party</a>. </p><p>In 1960, Markov obtained fundamental results showing that the classification of four-dimensional <a href="/facts/Manifold/rXPERJtu">manifolds</a> is <a href="/facts/Undecidable_problem/QBxmXBbY">undecidable</a>: 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 <a href="/facts/Halting_problem/Xz1bac40">halting problem</a>. 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. </p><p>His doctoral students include <a href="/facts/Boris_Kushner_(mathematician)/6sE5KUET">Boris Kushner</a>, <a href="/facts/Gennady_Makanin/Sdg6gLgh">Gennady Makanin</a>, and <a href="/facts/Nikolai_Shanin/GP3Ntdb9">Nikolai Shanin</a>. </p>