Yuan Yao

Packing problems have a long history in geometry. The subject 
of ball-packings in symplectic geometry was initiated in Gromov's 
seminal 1985 paper, where it was shown there is much more rigidity than 
simply volume constraints to symplectic packings coming from the 
presence of pseudo-holomorphic curves. The field has greatly developed 
since then, and the problem of symplectically packing k symplectic balls 
into a larger one has been solved in dimension four, i.e. there is now a 
combinatorial criteria of when this is possible. However, not much is 
known about symplectic packing problems in higher dimensions, partly due 
to the absence of powerful gauge theoretic tools in higher dimensions. 
We take a step in this direction in dimension six, by considering a 
“stabilized” packing problem, i.e. we consider symplectically packing a 
disjoint union of  four dimensional balls times a closed Riemann surface 
into a bigger ball times the same Riemann surface. We show this is 
possible if and only if the corresponding four dimensional ball packing 
is possible. The proof is a mixture of geometric constructions, 
pseudo-holomorphic curves, and h-principles. This is based on joint work 
with Kyler Siegel.