Integral-affine structures and degenerations of K3 surfaces

  I will discuss joint work with V. Alexeev which builds a correspondence between degenerations of polarized K3 surfaces and integral-affine structures on the 2-sphere containing a weighted balanced graph. Under this correspondence, natural questions emerge about the dynamics of the straight line flow on integral-affine manifolds.  In particular, we will explore (1) the relationship between closed trajectories and immersed elliptic curves in the K3 surface, and (2) the possibility of a tropical Yau-Zaslow formula for counting immersed trees.