  We establish the relation between two objects: an integrable system related
to Painlev\'e II equation, and the symplectic invariants of a certain plane
curve S(TW). This curve describes the average eigenvalue density of a random
hermitian matrix spectrum near a hard edge (a bound for its maximal
eigenvalue). This shows that the s -> -infinity asymptotic expansion of
Tracy-Widow law F_{GUE}(s), governing the distribution of the maximal
eigenvalue in hermitian random matrices, is given by symplectic invariants.
