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Graham Cox : Unsolvable problems in geometry and topology (Mar 26, 2010 4:25 PM)

The resolution of Hilbert's tenth problem yields the following unsolvability result: there is no algorithm for determining whether or not a given polynomial equation p(x_1,...,x_n) = 0 with integer coefficients will admit an integer solution. After a few definitions and examples, I will discuss another well-known unsolvable problem: the word problem for finitely presented groups. It can be shown that there is no algorithm for determining when an arbitrary word in a finitely presented group is trivial. This has many remarkable topological consequences, including the result that there is no algorithm that will determine when two given manifolds are homeomorphic (provided the dimension is at least four). The unsolvability theorem also has significant geometric applications, allowing one to prove that certain manifolds admit an infinite number of contractible closed geodesics (regardless of the Riemannian structure).

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