Embedded special Legendrian surfaces in the 5-sphere

Cones over special Legendrian surfaces in the 5-sphere provide local models for singularities in degenerating special Lagrangian submanifolds in Calabi-Yau 3-folds, playing a key role in SYZ mirror symmetry. Although many embedded special Legendrian tori have been constructed, until now, no compact embedded examples of higher genus were known. In this talk, I will describe the first examples of compact embedded special Legendrian surfaces of genus $g>1.$ The construction combines integrable systems techniques with gauge-theoretic methods. This talk is based on joint work with Charles Ouyang and Franz Pedit.