Geometry and Topology Seminar

The complex projective plane satisfies two special properties that set it apart from its peers: it admits (1) a symplectic structure and (2) a genus one trisection. Together these properties inspire the notion of a ``triple grid diagram,'' the primary object of study in this talk.

Trisections, due to Gay and Kirby, are decompositions of (smooth, closed, connected, oriented) 4-manifolds into three nice pieces. Meier and Zupan showed that surfaces smoothly embedded in trisected 4-manifolds can inherit their own trisection, and can be represented with ``shadow diagrams.'' Triple grid diagrams are specific shadow diagrams of surfaces in the complex projective plane that naturally arise as grid diagrams on the central surface of its standard (genus one) trisection. Diagrams satisfying one extra condition represent Lagrangian surfaces, thus capturing this geometric information combinatorially. This is joint work with David Gay and Peter Lambert-Cole.