Title: From polyhedra to Lagrangian fibrations
Abstract: A (singular) Lagrangian fibration of a symplectic manifold induces an integral affine structure on the set of regular values in the base of the fibration. In particular, a generic almost toric fibration of a symplectic K3 surface induces an integral affine structure on the complement of 24 points in a sphere. Within the 44-dimensional space of such integral affine structures are families of structures that enjoy enough symmetry to be defined by a polyhedron. In this talk, I will explain how a rational polyhedron can define a family of Lagrangian fibrations of a symplectic K3 surface and present some initial results and questions about sets of polyhedra that define such families of fibrations. This project is motivated by works in mirror symmetry (by Batyrev, Strominger, Yau, and Zaslow, and Gross and Seibert), relaxing and generalizing constructions in algebraic geometry to the symplectic setting. (Joint work with Liat Kessler.)