Mirror Symmetry and the Strominger-Yau-Zaslow conjecture

TL;DR

This survey reviews the development and progress of the Strominger-Yau-Zaslow conjecture in mirror symmetry, connecting differential geometry and algebraic geometry since its inception in 1996.

Contribution

It provides an updated overview of the conjecture's evolution, recent results, and the interplay between geometric and algebraic approaches.

Findings

Progress in understanding the SYZ conjecture since 1996

Development of the algebro-geometric program for mirror symmetry

Recent results fitting into the SYZ framework

Abstract

This survey was written for the Current Developments in Mathematics conference, 2012, and is an updating of my article "The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations," in the Seattle 2005 proceedings. We trace progress and thinking about the SYZ conjecture since its introduction in 1996. We begin with the original differential geometric conjecture and its refinements, and explain how it led to the algebro-geometric program developed by myself and Siebert. After explaining the overall philosophy, I explain how recent results fit into this program.