Tropical cycles and the tropological vertex
Complex manifolds with normal crossing divisors, and normal crossing degenerations, have a natural large scale with a tropical, or piecewise integral-affine structure. I will discuss the case when some useful cohomology theories are described by tropical cycles in this tropical large scale. This case is of interest to me, because relative Gromov-Witten invariants of these spaces correspond to counts of tropical curves in this tropical large scale. If time permits, I will draw some pictures, and explain some beautiful aspects of this tropical correspondence in the Calabi-Yao 3-fold setting, related to the Strominger—Yao—Zaslow approach to mirror symmetry: In this case, the large scale is a 3-dimensional integral affine manifold with singularities along a 1-dimensional graph. The vertices of this singular graph come in two types: positive and negative, and the relative Gromov—Witten invariants around the positive vertices contain the topological vertex of Aganagic, Klemm, Marino and Vafa.
Dr Brett Parker, Monash University