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

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