Preface 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 book. Mark Gross, La Jolla, 2010 ix
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