Softcover ISBN:  9780821852323 
Product Code:  CBMS/114 
List Price:  $57.00 
Individual Price:  $45.60 
Electronic ISBN:  9781470415723 
Product Code:  CBMS/114.E 
List Price:  $57.00 
MAA Member Price:  $51.30 
AMS Member Price:  $45.60 

Book DetailsCBMS Regional Conference Series in MathematicsVolume: 114; 2011; 317 ppMSC: Primary 14; 52;
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 Bmodels 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 FontaineIllusie 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, Bmodel 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 copublication of the AMS and CBMS.
ReadershipGraduate students and research mathematicians interested in mirror symmetry and tropical geometry.

Table of Contents

Chapters

Part I. The three worlds

Chapter 1. The tropics

Chapter 2. The A and Bmodels

Chapter 3. Log geometry

Part II. Example: $\mathbb {P}^2$

Chapter 4. Mikhalkin’s curve counting formula

Chapter 5. Period integrals

Part III. The GrossSiebert program

Chapter 6. The program and twodimensional results


Additional Material

Reviews

This book is wellwritten 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 GrossSiebert program.
Mathematical Reviews


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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 Bmodels 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 FontaineIllusie 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, Bmodel 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 copublication of the AMS and CBMS.
Graduate students and research mathematicians interested in mirror symmetry and tropical geometry.

Chapters

Part I. The three worlds

Chapter 1. The tropics

Chapter 2. The A and Bmodels

Chapter 3. Log geometry

Part II. Example: $\mathbb {P}^2$

Chapter 4. Mikhalkin’s curve counting formula

Chapter 5. Period integrals

Part III. The GrossSiebert program

Chapter 6. The program and twodimensional results

This book is wellwritten 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 GrossSiebert program.
Mathematical Reviews