Speaker: Tian-Jun Li

Title: Lagrangian spheres, symplectic surfaces and symplectic mapping class groups

Abstract: For any Lagrangian sphere in a symplectic 4-manifold $M$ with $b^+=1$, we find plenty of symplectic surfaces intersecting it minimally. This result turns out to be very useful in both the existence and uniqueness problems of Lag spheres. On the uniqueness side, we show that homologous Lag spheres in a rational manifold are symplectomorphic and smoothly isotopic. On the existence side, when the Kodaira dimension of $M$ is $-\infty$, we show that the obvious necessary condition for the existence of Lag spheres in a given homology class is also sufficient. For the same class of $M$, we give a complete description of the homology action of symplectomorphisms. This is a joint work with Weiwei Wu.