The NSF-CBMS conference on Tropical Geometry and Mirror Symmetry was
held at Kansas State University during the period December 13–17, 2008. It was
organized by Ricardo Casta˜ no-Bernard, Yan Soibelman, and Ilia Zharkov. During
this time, I gave ten hours of lectures. In addition, talks were given by M. Abouzaid,
K.-W. Chan, C. Doran, K. Fukaya, I. Itenberg, L. Katzarkov, A. Mavlyutov, D.
Morrison, Y.-G. Oh, T. Pantev, B. Siebert, and B. Young.
My talks were meant to give a snapshot of a long-term program currently
being carried out with Bernd Siebert aimed at achieving a fundamental conceptual
understanding of mirror symmetry. Tropical geometry emerges naturally in this
program, so in the lectures I took a rather ahistorical point of view. Starting with
the tropical semi-ring, I developed tropical geometry and explained Mikhalkin’s
tropical curve-counting formulas, outlining the proof given by Nishinou and Siebert.
I then explained my recent work in connecting this to the mirror side. Finally,
I sketched the ideas behind recent work by myself and Siebert on constructing
degenerations of Calabi-Yau manifolds from aﬃne manifolds with singularities.
This monograph follows the structure of the lectures closely, filling in many de-
tails which were not given there. Like the lectures, this monograph only represents
a snapshot of an evolving program, but I hope it will be useful to those who may
wish to become involved in this program.
NSF grant DMS-0735319 provided support both for the conference and for this
Mark Gross, La Jolla, 2010