Abstract: 

 Markov numbers are triples \((a,b,c)\) of natural numbers that solve the Markov equation \(a^2+b^2+c^2 = 3abc\).
While these triples parametrize classes irrational numbers that can be approximated only very badly by rational numbers, they recently have played an important role for understanding certain symplectic embeddings. For instance, a Lagrangian \(p\)-pinwheel only embeds into the projective plane if \(p\) belongs to a Markov number, and the Fibonacci staircase describing the problem of symplectically embedding a four-dimensional ellipsoid into a four-ball of minimal size is just the case \(p=1\) of a symplectic embedding problem associated to symplectic neigbourhoods of certain cyclic quotient singularities. 

I will try to explain these things and their relations in a non-technical way.