**CBMS Regional Conference Series in Mathematics**

Volume: 114;
2011;
317 pp;
Softcover

MSC: Primary 14; 52;

Print ISBN: 978-0-8218-5232-3

Product Code: CBMS/114

List Price: $57.00

Individual Price: $45.60

**Electronic ISBN: 978-1-4704-1572-3
Product Code: CBMS/114.E**

List Price: $57.00

Individual Member Price: $45.60

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#### Supplemental Materials

# Tropical Geometry and Mirror Symmetry

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*Mark Gross*

A co-publication of the AMS and CBMS

Tropical geometry provides an explanation for the remarkable power of
mirror symmetry to connect complex and symplectic geometry. The main
theme of this book is the interplay between tropical geometry and mirror
symmetry, culminating in a description of the recent work of Gross and
Siebert using log geometry to understand how the tropical world relates
the A- and B-models in mirror symmetry.

The text starts with a detailed introduction to the notions of
tropical curves and manifolds, and then gives a thorough description
of both sides of mirror symmetry for projective space, bringing
together material which so far can only be found scattered throughout
the literature. Next follows an introduction to the log geometry of
Fontaine-Illusie and Kato, as needed for Nishinou and Siebert's proof
of Mikhalkin's tropical curve counting formulas. This latter proof is
given in the fourth chapter. The fifth chapter considers the mirror,
B-model side, giving recent results of the author showing how tropical
geometry can be used to evaluate the oscillatory integrals appearing.
The final chapter surveys reconstruction results of the author and
Siebert for “integral tropical manifolds.” A complete
version of the argument is given in two dimensions.

A co-publication of the AMS and CBMS.

#### Readership

Graduate students and research mathematicians interested in mirror symmetry and tropical geometry.

#### Reviews & Endorsements

This book is well-written and provides very useful introductory accounts of many aspects of the highly involved subject of mirror symmetry. This book can be extremely helpful to those who want to understand mirror symmetry and the Gross-Siebert program.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Tropical Geometry and Mirror Symmetry

- Cover Cover11 free
- Title page iii4 free
- Epigraph v6 free
- Contents vii8 free
- Preface ix10 free
- Introduction xi12 free
- Part I. The three worlds 118 free
- The tropics 320
- The A- and B-models 3350
- Log geometry 91108
- Part II. Example: ℙ² 131148
- Mikhalkin’s curve counting formula 133150
- Period integrals 173190
- Part III. The Gross-Siebert program 245262
- The program and two-dimensional results 247264
- Bibliography 307324
- Index of symbols 313330 free
- General index 315332
- Back Cover Back Cover1338