Complex curves in $CP^2$ from the perspective of bridge trisections Alex Zupan, University of Nebraska Host: Neil Hoffman Contact Neil Hoffman for the meeting link.
Abstract: Peter Lambert-Cole and Jeff Meier revealed that bridge trisections of complex curves in $CP^2$ exhibit elegant structure: Every complex curve admits an inefficient shadow diagram (with respect to the standard genus one trisection) in which shadow arcs form a hexagonal lattice in the torus. Additionally, Lambert-Cole proved a combinatorial classification of symplectic surfaces in $CP^2$: A surface that minimizes genus in its homology class is symplectic if and only if it admits a transverse shadow diagram. We prove a complex version of Lambert-Cole's theorem, that a genus-minimizing surface in $CP^2$ is complex if and only if it admits a transverse hexagonal lattice diagram. In the process, we find infinite families of efficient hexagonal lattice diagrams for complex curves, and we give a combinatorial characterization of the symplectic isotopy problem in $CP^2$.
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