Lagrangian cell complexes and Markov numbers

Joint work with Ivan Smith. Let p be a positive integer. Take the quotient of a 2-disc by the equivalence relation which identifies two boundary points if the boundary arc connecting them subtends an angle which is an integer multiple of ($2 \pi / p$). We call the resulting cell complex a '$p$-pinwheel'. We will discuss constraints on Lagrangian embeddings of pinwheels. In particular, we will see that a p-pinwheel admits a Lagrangian embedding in $CP^2$ if and only if $p$ is a Markov number. Time permitting, I will discuss nondisplaceability results, which are a purely symplectic analogue of the Hacking-Prokhorov classification of $Q$-Gorenstein degenerations of $CP^2$.