Abstract:
Studying approximations of real numbers by rational numbers, A.A. Markov
introduced a new Diophantine equation in 1879:
$$x^2 + y^2 + z^2 = 3xyz,$$
which later became known as the Markoff equation. Set of its natural
solutions, the "Markoff triples", has a natural tree-graph structure.
In recent years, influenced by the work of Bourgain, Gamburd, and Sarnak, the
Markoff equation has come to be studied over the field
of residues modulo prime $p$. Last year, Chen published the completion of
a very complex proof of the main conjecture, which states that for
sufficiently large primes $p$, all solutions of the Markov equation
over the field of residues modulo $p$ are obtained from its integer solutions by
reduction modulo $p$.
The proof of the conjecture is based on several papers using very different
methods.
I plan to discuss these facts, including Markov's classical results, as well
as completely new generalizations
to the $n$-dimensional case.
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