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.
His doctoral students include Boris Kushner, Gennady Makanin, and Nikolai Shanin.